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Technical University of Madrid

School of Civil Engineering

Department of Continuum Mechanics and Theory of Structures

An integrated approach

to conceptual design of arch bridges

with curved deck

Author

Leonardo Todisco

Civil Engineer

Technical University of Madrid

Supervisor

Hugo Corres Peiretti

PhD MEng Prof. In Civil Engineering

Technical University of Madrid

E.T.S.I.C.C.P.

U.P.M.

-I-

ABSTRACT

An integrated approach to conceptual design of arch bridges

with curved deck

by

Leonardo Todisco

MS THESIS IN ENGINEERING OF STRUCTURES, FOUNDATIONS AND MATERIALS

Spatial arch bridges represent an innovative answer to demands on functionality, structural

optimization and aesthetics for curved decks, popular in urban contexts.

This thesis presents SOFIA (Shaping Optimal Form with an Interactive Approach), a

methodology for conceptual designing of antifunicular spatial arch bridges with curved deck in a

parametric, interactive and integrated environment.

The approach and its implementation are in-depth described and detailed examples of

parametric analyses are illustrated. The optimal deck-arch relative transversal position has been

investigated for obtaining the most cost-effective bridge.

Curved footbridges have become a more common engineering problem in the context of urban

developments when the client is looking for a strong aesthetics component: an appropriate

conceptual design allows to obtain an efficient and elegant structure.

Antifunicular, hanging models, form-finding, spatial arch bridges, structural design, geometry, Spatial

analysis, structural response, structural forms, curvature, curved deck, compression-only structures,

structural optimization.

Thesis supervisor: Hugo Corres Peiretti, Professor of Structural Concrete and Conceptual Design

at the School of Civil Engineering at the Technical University of Madrid

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

-II-

RESUMEN

An integrated approach to conceptual design of arch bridges

with curved deck

de

Leonardo Todisco

TRABAJO FIN DE MÁSTER EN INGENIERÍA DE LAS ESTRUCTURAS, CIMENTACIONES Y

MATERIALES

El conceptos de funicularidad se puede extender a estructuras lineales espaciales como, por

ejemplo, los puentes arco con tablero curvo. Estas estructuras, especialmente pasarelas

peatonales, son consecuencia de la necesidad de encajar trazados exigentes y de dar respuesta a

nuevas demandas arquitectónicas. En las estructuras curvas el diseño conceptual juega un papel

absolutamente esencial. Siempre ha sido así, pero en el caso presente, cabe resaltar que una

errónea elección de la geometría conlleva una serie de problemas que se irán acumulando a lo

largo del proceso de proyecto, de la construcción y de la vida de la estructura.

En este trabajo se presenta SOFIA (Shaping Optimal Form with an Interactive Approach), una

herramienta capaz de, conocida la geometría del tablero, de buscar automáticamente la forma

del arco antifunicular correspondiente. El planteamiento seguido es conceptualmente el mismo

que el utilizado en la búsqueda de formas óptimas en estructuras en dos dimensiones: el arco

antifunicular es el que representa, para unas cargas dadas, el lugar geométrico de los puntos con

momento flector nulo.

La herramienta ha sido desarrollada en un entorno integrado, interactivo y paramético. Su

implementación está ilustrada y unos ejemplos de análisis paramétricos están desarrollados. La

posición transversal relativa entre tablero y arco ha sido investigada para obtener la configuración

del puente estructuralmente más eficiente.

Las pasarelas curvas se han convertido en un problema de ingeniería más común de lo habitual

en el contexto de los desarrollos urbanos cuando el cliente está buscando un fuerte componente

estético: un diseño conceptual adecuado permite obtener una estructura eficiente y elegante.

-III-

TABLE OF CONTENTS

ABSTRACT ...................................................................................................................................................................................................... I

RESUMEN ..................................................................................................................................................................................................... II

FIGURES INDEX............................................................................................................................................................................................ V

TABLES INDEX ........................................................................................................................................................................................... VI

NOTATION ................................................................................................................................................................................................ VII

1 INTRODUCTION AND MOTIVATION ............................................................................................................................ 9

1.1 Overview of the thesis .......................................................................................................................................................... 9

1.2 Objectives................................................................................................................................................................................. 9

1.3 Conceptual design: need for new design tools ............................................................................................................. 11

1.4 Organization and structure of the work .......................................................................................................................... 12

2 STATE OF THE ART: THE CONCEPT OF FUNICULARITY ........................................................................................ 14

2.1 Introduction ........................................................................................................................................................................... 14

2.2 Historical overview............................................................................................................................................................... 16

2.3 The funicular curve ............................................................................................................................................................... 20

2.4 The search of the funicular curve ..................................................................................................................................... 26

2.4.1 Method of moments ........................................................................................................................................................... 26

2.4.2 Graphic statics ....................................................................................................................................................................... 27

2.4.3 Physical models ..................................................................................................................................................................... 27

2.4.4 Numerical methods ............................................................................................................................................................. 29

2.4.5 Particle-spring systems ....................................................................................................................................................... 29

2.4.6 Final comments ..................................................................................................................................................................... 29

2.5 Historical and modern examples ..................................................................................................................................... 29

2.5.1 The Pantheon ......................................................................................................................................................................... 30

2.5.2 Salginatobel Bridge .............................................................................................................................................................. 32

2.5.3 The Tiemblo bridge .............................................................................................................................................................. 34

2.6 Final remarks .......................................................................................................................................................................... 35

3 ANTIFUNICULAR SPATIAL ARCH BRIDGES FORM FINDING USING AN INTEGRATED AND

INTERACTIVE APPROACH ...................................................................................................................................................................... 37

3.1 Introduction ........................................................................................................................................................................... 37

3.2 State-of-the-art of spatial arch bridges ........................................................................................................................... 39

3.3 Designing spatial arch bridges .......................................................................................................................................... 42

3.3.1 An interactive and integrated environmental workspace: Rhinoceros, Grasshopper, Karamba .................... 42

3.3.2 Form finding methodology for spatial antifunicular arches ...................................................................................... 43

4 PARAMETRIC ANALYSIS ON THE INFLUENCE OF DECK-ARCH RELATIVE TRANSVERSAL POSITION 48

4.1 Cases of study ....................................................................................................................................................................... 48

4.2 Discussion of results ........................................................................................................................................................... 60

4.3 Conclusions ............................................................................................................................................................................ 67

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

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5 CONCLUSIONS AND FUTURE STUDIES ..................................................................................................................... 69

5.1 Summary of results ............................................................................................................................................................. 69

5.1.1 The funicular principle ....................................................................................................................................................... 69

5.1.2 Designing antifunicular spatial arch bridges.................................................................................................................. 70

5.2 Further studies ....................................................................................................................................................................... 71

5.2.1 Introduction ............................................................................................................................................................................ 71

5.2.2 Torsional behaviour of curved decks .............................................................................................................................. 72

5.2.3 Arrangement of layout of cables ...................................................................................................................................... 73

BIBLIOGRAPHY .......................................................................................................................................................................................... 75

PUBLICATIONS BY AUTHOR ................................................................................................................................................................ 80

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FIGURES INDEX

Figure 1-1: Antifunicular spatial arch for C- and S-shaped deck ....................................................................................................... 11

Figure 2-1: Funicular and anti-funicular geometries for known distribution of loads ............................................................... 15

Figure 2-2: Inca suspended bridge. Taken from Squier (Squier 1877) ........................................................................................... 16

Figure 2-3:Examples of compression-only structures built B.C. .................................................................................................... 18

Figure 2-4: Hanging chain and correspondent inverted arch used by Poleni for the structural safety assessment of San

Peter dome in Rome. Taken from Poleni (Poleni 1982). .................................................................................................................. 20

Figure 2-5:. infinitesimal element of rope loaded with a set of vertical loads ............................................................................ 20

Figure 2-6: Horizontally distribution of load for a constant load per unit length for arches with different rises .............. 21

Figure 2-7: Maximum value of the horizontal projected load for different values of λ .......................................................... 22

Figure 2-8: Difference between catenary (dot line) and parabola (continue line) for different values of ...................... 23

Figure 2-9: Distribution of loads corresponding to different funicular geometries ................................................................. 24

Figure 2-10: Influence of the backfill disposition on the minimal thickness of a stable circular shell. Taken from Ramos

Casquero (Ramos Casquero 2011) ......................................................................................................................................................... 25

Figure 2-11: Funicular geometry for a general distribution of loads .............................................................................................. 26

Figure 2-12: Physical models used by Gaudí and Isler ....................................................................................................................... 28

Figure 2-13: Concrete density of Pantheon dome. ............................................................................................................................. 30

Figure 2-14:. Independent arches used for analysis ............................................................................................................................ 31

Figure 2-15: Line of thrust for different load cases ............................................................................................................................. 32

Figure 2-16: Salginatobel Bridge by Maillart ........................................................................................................................................ 33

Figure 2-17: Analysis of the bridge using graphic statics. Taken from Fivet (Fivet and Zastavni 2012). ............................... 34

Figure 2-18: Front view and cross section of the Tiemblo bridge. Courtesy of Fhecor Ingenieros Consultores ............. 34

Figure 2-19: Graphic statics applied for finding the thrust line of the Tiemblo bridge due to own weight. ..................... 35

Figure 2-20: Main Train Station in Berlin. Photo taken from SBP (SBP 2014). ........................................................................... 36

Figure 3-1: Examples of curved bridges ................................................................................................................................................ 38

Figure 3-2:sketch of the Golden Horn Bridge designed by Leonardo da Vinci in 1502 (Biblioteque Institute Paris ). .... 40

Figure 3-3:Ripshorst Footbridge by SBP (SBP 2014) .......................................................................................................................... 41

Figure 3-4:: spatial arch footbridge proposals .................................................................................................................................... 41

Figure 3-5:Description of the form-finding process SOFIA ............................................................................................................ 45

Figure 3-6: Resulting antifunicular arch for bridges with C- and S-shape curvature of the deck ......................................... 46

Figure 4-1: Cases of studies for the parametric analysis .................................................................................................................. 49

Figure 4-2: Antifunicular configuration for C-shape deck ............................................................................................................... 54

Figure 4-3: Antifunicular configuration for S-shape deck ................................................................................................................ 59

Figure 4-4: Axial forces along the arch ................................................................................................................................................. 61

Figure 4-5: Summation of axial force of cables for different values of y. .................................................................................... 62

Figure 4-6: Axial forces in the deck ....................................................................................................................................................... 63

Figure 4-7: Axial forces along the arch ................................................................................................................................................. 64

Figure 4-8: Summation of axial force of cables for different values of y. .................................................................................... 65

Figure 4-9: Axial forces in the deck ...................................................................................................................................................... 66

Figure 5-1: different arrangements of cables layout .......................................................................................................................... 74

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NOTATION

la span of the arch

ld span of the deck

fa rise of the arch

fd sag of the arch

N axial force

H horizontal component of axial force

q load along the curved geometry

p horizontally distribution of load

E elastic modulus

I area moment of inertia

s curvilinear coordinate of the arch

C torsion centre

G centre of gravity

Kt tangent stiffness

ΔF load step

Miy bending moment for the i-section around the y-local axis

Mix bending moment for the i-section around the x-local axis

ei value of the eccentricity for the i-section

e* maximum allowable value of the eccentricity

y deck-arch transversal distance

1. INTRODUCTION AND MOTIVATION

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1 INTRODUCTION AND MOTIVATION

Architecture is too important to be left to architects.

Giancarlo de Carlo

1.1 Overview of the thesis

Spatial arch footbridges represent an innovative answer to the demands on functionality,

structural optimization and aesthetics for curved decks, popular in the urban context.

This work focuses on a sound conceptual design of arch footbridges with curved deck, using an

integrated and interactive approach developed in a parametric environment. A concatenation of

parametric design with structural analysis is proposed.

Curved structures are characterized for having the geometry as an essential factor involved in

their structural behaviour: a proper conceptual design of the structural shape is a key aspect for

obtaining a cost-effective bridge.

Like in a curved inverted chain, forces should flow in axial compression toward the supports

minimizing moments to obtaining structural efficiency: in this thesis a new integrate approach for

finding funicular geometries is presented in detail, some results are discussed and future works

are proposed.

Curvature can be symbol of grace: a designer who is able to understand the fundamental

relationships between shape and structural behaviour will take advantage of curvature in order to

design more efficient and sustainable structures.

1.2 Objectives

The aim of this work is to illustrate an integrated approach in order to perform form finding

analysis of arches characterized for supporting a curved deck.

It is not unusual that the footbridge deck has to adapt to a complex geometry in plan as well as in

elevation, due to regulations on disabled peoes, which often entails the use of long

ramps. Usually designers incorporate external ramps to decrease the slope as much as possible

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

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to fulfil the admissible slope limit. An alternative to that solution is to increase the length of the

deck, by making it horizontally curved.

Structurally, the curved deck could be supported by an arch, or can be cable-stayed or suspended

by a cable system. In this work only spatial arch bridges which support a curved deck using

secondary ties are considered. The proposed approach is valid for any geometry of the deck.

Differently to comprehensive previous works on the topic (Jorquera Lucerga 2007), a interactive

and integrated approach has been investigated in a parametric environment.

The conceptual design of an arch footbridge involves choosing an appropriate geometry among

many possibilities. A shape finding process, in which the geometry of the arch is generated, has

been developed in order to simulate numerically a chain which works as an compression-only

configuration in static equilibrium with the design loading.

Examples of resulting antifunicular arches for a C- and S-shaped decks are shown in Figure 1-1.

a) C-shaped deck

1. INTRODUCTION AND MOTIVATION

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b) S-shaped deck

Figure 1-1: Antifunicular spatial arch for C- and S-shaped deck

The thesis includes an in-depth look at how funicularity of form and new computational tools can

unlock hidden architectural and aesthetic potential in order to explore new efficient geometries.

1.3 Conceptual design: need for new design tools

As it has been remarked in the previous section, in designing curved structures, i.e. shells, arches,

domes, vaults, the most important phase is the choice of the structural shape: conceptual design

is main issue for designers involved in spatial structures.

Conceptual design is the approach that creates an idea in order to find a solution to a new

proposal for a structure or solve a detail in a specific structure (Corres 2013). The aim of this

section is not to illustrate how conceptual design should be carried out because has it been

described by several authors during recent years (Schlaich 1996; Corres 2013; Muttoni 2011; Allen

and Zalewski 2009), and it is not the main topic of this thesis. The goal of this section is to point

out the existence and the need of new tools that can help designers to perform sound

conceptual designs of curved structures. A more general discussion on the topic is proposed in

Clune (Clune et al. 2012).

Structural engineers usually have strong engineering knowledge but, despite architects, little

creative capacity. Nevertheless, taking in mind 'classical proportions' and having previous design

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

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experience, new structural solutions can be found in order to obtain more structural efficiency

(Romo 2013).

In order to compare different solutions and to explore new ones, designers do not need

advanced structural analysis programs, but they need tools for helping to design and to image

new solutions having more freedom changing geometry, supports, cross-sections, etc.. in a three-

dimensional environment.

During the last few years new tools have been developed in order to correlate geometry and

structural behaviour in a dynamic and interactive way. An introduction to concatenating

parametric design with structural analysis was proposed by Georgiou (Georgiou 2011).

A combined use of these tools has been performed for finding the most efficient spatial arch

bridge, as shown in section 3.

There is a strong need of parametric tools that allow to better manage data approximations

typical in the first stage of a project and to perform quick structural evaluations obtaining the

most efficient structural solution to the problem. An integration of empirical, traditional and

innovative structural design techniques could be an advantage for designing cost-effective

structures in a truly rational way.

Experience, knowledge of structural engineering (spatial structures cannot exist without

mechanical basis (Sasaki 2005)), ambition and the use of new tools are without doubt the

elements valuables for increase creativity and innovation in structural design.

1.4 Organization and structure of the work

This dissertation presents a novel computational methodology to design antifunicular arch

bridges with a curved deck and its implementation with a parametric study. The thesis has been

divided in four main parts:

Part 1: State of the art.

In this section the concept of funicularity is presented not only for the design of new structures,

but for safety assessment of historic arches and vaults too. Literature on the topic has been in-

depth reviewed and methodologies, more or less contemporary, have been illustrated and

critically analyzed. Some case studies have been described and discussed.

Part 2: Designing antifunicular spatial arch bridges.

This part focuses on the methodology and its implementation. Traditional techniques used by

engineers of the 20th century have been extended and implemented in a parametric

environment. The used tool has been described in detail in order to evidence the advantages

related to that novel integrated approach.

1. INTRODUCTION AND MOTIVATION

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Part 3: Examples of parametric analysis.

In this section some examples of parametric analysis using the new tool have been performed.

The influence of relative position between arch and deck on structural behaviour has been

investigated. The optimal position of a curved deck minimizing the axial forces of arch and cables

has been found.

Part 4: Conclusion and further studies.

In this section general conclusions have been remarked and further studies that are currently

under development are illustrated.

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

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2 STATE OF THE ART: THE CONCEPT OF FUNICULARITY

The resistant virtues of the structure that we make depend on their form; it is through their form that

they are stable and not because of an awkward accumulation of materials. There is nothing more

noble and elegant from an intellectual viewpoint than this; resistance through form.

Eladio Dieste

2.1 Introduction

Curved structures, if they are the outcome of an appropriate design, have the capacity to carry

large loads and cover important areas saving costs and can come to represent examples of the

highest structural efficiency. In designing shells or arches, geometry plays a primary role, so

enough energy should be spent for a proper design of the structural shape.

Arches and cables seem to be structures apparently different, but they are very similar under the

structural behaviour point of view. As it is well known, the funicular is the geometry of a chain,

pure in tension and free of bending, loaded with a determined distribution.

The word funicular comes from the Latin word funiculus, diminutive of funis, meaning 'slender

rope' (Treccani 2014). The basic idea to design an arch is to use the principle of inversion,

adopting its dual form: the main difference is that the dual geometry is a compression-only

geometry.

In figure 2.1 different symmetric funicular and anti-funicular geometries are shown for some load

distributions.

2. STATE OF THE ART: THE CONCEPT OF FUNICULARITY

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Figure 2-1: Funicular and anti-funicular geometries for known distribution of loads

A pedagogic way to find intuitively a compression-only geometry is to use a chain without

bending stiffness and to hang the chain with a known set of loads.

As previously shown in figure 2-1, to one distribution of loads corresponds an infinite series of

funicular and antifunicular curves. Changing the horizontal reaction, o similarly, the rise of the

arch, o its length it is possible to obtain the family of compression-only or tension-only

geometries corresponding to one set of loads.

The physical concept that arises from the principle of the funicularity is the lack of bending

moments: ropes, free bending, find automatically their unique equilibrium geometry.

The principle of funicularity is important for several reasons:

- to allow designing efficient structures in which the use of material is optimized. The choice of

the funicular is conditioned by functional, economic, technological, architectural aspects.

- to allow assessing the safety of an existing construction that work principally with compression

forces, in which funicular curves coincide with thrust lines. In this case a thrust line included in

the thickness of the structure (arch, columns, wall, foundation, etc..) has to be found (Heyman

1966).

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

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The funicularity principle and its implications should be indispensable part of the expertise of a

structural designer. These ideas are well known but an everyday more frequently use of model

computer analysis has contributing to eclipse this vital concept.

In the following section, the concept of funicularity is in-depth investigated and its historical

evolution is illustrated. Several tools, more or less recent, for finding that geometry are

presented. Finally some compression-only structures are described and critically discussed.

2.2 Historical overview

Curved structures (arches, shells, domes, etc..) are constructions in which, as previously stated,

the thrust line has to be included in the structural thickness in order for such structures to work

primarily in compression.

First funicular structures for crossing a gorges built over the centuries were tension-only

geometry because was easier to find long fiber ropes for crossing a river; furthermore there was

no need of geometrical-mechanical knowledge because, due to their no bending stiffness, ropes

adopt automatically their optimal and equilibrated configuration.

Figure 2-2 show an example of an Inca suspended footbridge, where technology was well suited

to the problem and their resources.

Figure 2-2: Inca suspended bridge. Taken from Squier (Squier 1877)

2. STATE OF THE ART: THE CONCEPT OF FUNICULARITY

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Figure 2-3 shows some examples of ancient structures working only with compression paths of

forces. Compression-only three-dimensional structures were spread in Cyprus in 5500 B.C, in

1700 B.C. in Sardinia (Italy) and during the VII century B.C. in Lazio (Italy).

(a) Cyprus. Image taken form Fernandez Casado ndez Casado 2006).

(b) Sardinia. Image taken from Wikipedia

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

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(c) Etruscan civilization in Lazio (Italy). Image taken from Canino (Canino 2014)

Figure 2-3:Examples of compression-only structures built B.C.

The majority of curved structures build over two thousand years ago were corbelled structures,

where starting from the base, the stones that form the internal structure were arranged

projecting towards the inner part of the dome. On this first ring, other rings are added in the

subsequent projection, with smaller diameter, until reaching a minimum opening, which is closed

by a last stone. An extensive archive of corbelled domes is provided by Lobbecke (Lobbecke

2012). The corbelling construction technique needs a self supporting structure, so that the hoop

forces are mobilized in order to ensure equilibrium (Allen 1984). An interesting study on this kind

of structures has been developed by Como (Como 2007) for Mycenaean Tholoi and by Sanitate

(Sanitate et al. 2014) for trulli in Apulia (Italy).

Before the XV century the use of suitable only-compression or only-tension structure was based

on the empirical acceptation of its good structural behaviour without an explicit formulation of

its mechanical behaviour.

Starting from the XV century, the first documents on investigations about the structural

mechanics behind the behaviour of chains and arches started to arise: a theoretical

approximation began to combine the experimental evidence. The first attempts defining the

structural behaviour of curved structures were due to Leon Battista Alberti (1404-1472), Palladio

(1508-1580) and Leonardo da Vinci (1452-1519), who proposed the basis of the scientific analysis

attempting to develop a theory with the concept of static analysis. In the following are reported

some writing of Leonardo da Vinci (Da Vinci and Richter 1970). He realized the basis of the

structural mechanics.

"The arch is nothing else than a force originated by two weaknesses, for the arch in buildings is

composed of two segments of a circle, each of which being very weak in itself tends to fall; but as each

opposes this tendency in the other, the two weaknesses combine to form one strength."

2. STATE OF THE ART: THE CONCEPT OF FUNICULARITY

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"As the arch is a composite force it remains in equilibrium because the thrust is equal from both sides;

and if one of the segments weighs more than the other the stability is lost, because the greater pressure

will outweigh the lesser."

"Next to giving the segments of the circle equal weight it is necessary to load them equally, or you will

fall into the same defect as before."

"An arch breaks at the part which lies below half way from the centre."

"If the excess of weight be placed in the middle of the arch at the point a, that weight tends to fall

towards b, and the arch breaks at 2/3 of its height at c e; and g e is as many times stronger than e a,

as m o goes into m n."

"The arch will likewise give way under a transversal thrust, for when the charge is not thrown directly

on the foot of the arch, the arch lasts but a short time."

The difference between funicular due to self weight (constant weight per unit length) and due to

a uniformly distribution horizontally load, respectively a catenary and a parabola, could be very

small: Galileo (1564-1642) got it wrong in his writing 'Dialogues Concerning Two New Sciences"

(Galilei 1638), where he stated that the funicular geometry formed by a chain was a parabola. The

difference between the two curves was demonstrated and published by Jungius (1587-1657) after

his death. The first scientific essay on the theory of arches was formed diffused by La Hire (1640-

1718) in "Traitè de Mecanique" in 1695.

The idea to use the inversion principle was published by Hooke (1635-1703) in his famous

anagram to find the ideal compression-only geometry for a rigid arch (Hooke 1676):

"Ut pendet continuum flexile sic stabit contiguum rigidum inversum"

("As hangs the flexible line, so but inverted will stand the rigid arch.")

After Hook, several scientific of the past studied the problem (Bernoulli, Leibniz, Huygens) and

Gregory published an important essay on the topic (Gregory 1697) where he described

mathematically the catenary geometry.

The idea of obtaining the geometry of the arch as the inversion of a hanged cable encouraged the

use of reduced models. It is interesting to evidence that the shape of an hanging chain is

independent of scale, as the statical equilibrium of compression structures (Adriaenssens et al.

2014).

The idea of using physical reduced models was adopted by Poleni (Poleni 1982) in his assessment

of the structural stability of San Peter dome in Rome as shown in figure 2-4.

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

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Figure 2-4: Hanging chain and correspondent inverted arch used by Poleni for the structural safety assessment of San

Peter dome in Rome. Taken from Poleni (Poleni 1982).

2.3 The funicular curve

Funicular curves represent, as previously described, the shape of a hanging chain for a known set

of loads. These curves are compression- or tension-only. Infinite number of funiculars exist for

one distribution of loads. Each curve corresponds to one horizontal reaction, or one rise, or one

total length. An important property of that geometry is its independence from the typology of

supports. Supports become relevant when the funicular does not match the axis of the structure.

Figure 2-5 shows an infinitesimal element of rope loaded with a set of vertical loads: the

horizontal component (H) of the axial force (N) is constant along the curve because only vertical

loads are imposed.

q(x)

AB

x

y

q dx

a

b

dx

Na

H

Nb

H

Figure 2-5:. infinitesimal element of rope loaded with a set of vertical loads

2. STATE OF THE ART: THE CONCEPT OF FUNICULARITY

-21-

The differential equation of the funicular curve can be expressed as:

2

2()d y q x

dx H

(2.1)

Equation 2.1 is similar to the differential equation of the classical beam theory:

2

2()d y M x

dx EI

(2.2)

The previously analogy shows that the funicular shape has the same geometry of the bending law

for a beam with same span and loaded with the same loads (Timoshenko 1953).

An important aspect that should always taken into account is the real distribution of loads; the

difference between a constant weight per unit length and an uniformly distribution horizontally

load can be relevant in some cases. Figure 2-6 illustrates three arches, with the same thickness,

span and different inclination, on which there is a load q uniformly distributed along the axis. The

consequent horizontally distribution of load is far from a constant value, as higher is the rise of

the arch.

Figure 2-6: Horizontally distribution of load for a constant load per unit length for arches with different rises

Figure 2-7 illustrates a plot that allows to calculate the maximum variation of the horizontally

distribution of loads respect to a constant value obtained for a flat beam. A parabolic arch has

been used for plotting the graph and the value of the maximum load, in proximity of hanging

lines, has been plotted versus the parameter = l / f. For high values of , i.e. arch with small rise,

the maximum values of load coincides.

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

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Figure 2-7: Maximum value of the horizontal projected load for different values of λ

Considering

2

1 ( )

dy

ds dx

(2.3)

and the equation of the arch as

2

4

( ) ( )

4

x

y x x

(2.4)

it is possible to evaluate that curve as

() ds

p x q dx

(2.5)

For low values of , as for example =2 (possible value in designing shells), the horizontally

distribution of loads can be 120% higher close to support respect to the centre of the span.

Parabola

2. STATE OF THE ART: THE CONCEPT OF FUNICULARITY

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Parabola is the funicular curve corresponding to a horizontally constant distribution of load. If the

load is constant per unit length but the geometry has low inclination (high values of ), the

differences between parabola and real funicular can be neglected. Many times the parabola

equation has been used for designing suspended bridges: rigorously speaking parabola does not

represent the funicular curve for dead load and self weight for that kind of bridge but if loads due

to deck are really high compared to cables ones, the difference between the real funicular curve

and a parabola can be disregarded.

Catenary

The word catenary comes from the Latin word catenarus and it represents the funicular

geometry obtained with a load constant per unit length, as for example is the self weight if the

cross section is the same along the curve. Catenary represent the real geometry of an hanging

chain. The use of catenary for architectural purposes was diffuse in Middle East (Dome of the

Rock in Jerusalem). The catenary equation is well known:

//

cosh( ) ( )

2x h x h

xh

y h e e

h

(6)

Developing the equation using Taylor series, it can be observed that catenary and parabola share

the first three terms, with the differences only starting from the fourth. In figure 2-8 the

difference between catenary and parabola is shown for arches with different values of = span /

rise. For high values of , i.e. arch with low rise, the difference is minimal.

=10

=5

=3.3

=2.5

=2

=10

=5

=3.3

=2.5

=2

Figure 2-8: Difference between catenary (dot line) and parabola (continue line) for different values of

Circumference

Roman master builders designed curved structures, as arches and shells, starting from

geometries derived from the circumference a priori less efficient, but easier to built. There are

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

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symbolic reasons too: circumference represent the perfection, the unity, the harmony. The

distribution of loads for a circular arch in order to have a perfect compression-only behaviour

needs to be either constant in the radial direction or with a infinite value close to the supports as

is shown in figure 2-9.

qqq

parabola catenaria arco de circulo arco de circulo

q

a) parabola

b) catenary

qqq

parabola catenaria arco de circulo arco de circulo

q

c) arch of circumference

Figure 2-9: Distribution of loads corresponding to different funicular geometries

It seems that Roman master builders were aware that for these kind of geometries additional

loads had to be added close to the support. This necessity does not entail a strong obligation

because abutments and backfill were necessary to guarantee an horizontal slope of roads,

aqueducts and buildings: in this way, probably unconsciously, circumference was similar to

funicular of dead loads, absolute predominant in this kind of bridges.

Backfills have a key role: to increase loads, approach thrust line to barycenter axis and prestress

the structure in order to improve their structural behaviour in front of bending moments due to

live load (important for railway bridges, and starting from the XX century, for roadway bridges).

2. STATE OF THE ART: THE CONCEPT OF FUNICULARITY

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The importance of filling has been studied by many authors. In Ramos Casquero (Ramos

Casquero and León González 2011) the different 'missions' of the fill are described, mainly to

guarantee the stability of the dome. As an example showing the importance of filling, the figure

2-10 (Ramos Casquero 2011) shows how an adequate disposition of fill allows that the minimal

thickness of a circular shell is stable for a fill high of 0.63 of the rise.

Figure 2-10: Influence of the backfill disposition on the minimal thickness of a stable circular shell. Taken from Ramos

Casquero (Ramos Casquero 2011)

The funicular curve for real structures

The funicular (or anti-funicular) curve is tension-only (or compression-only) if the distribution of

loads does not change, and if the geometry remains exactly the same, during the life of the

structure.

This condition is true only if the axial stiffness is infinite and there are not rheological effects. If

the real geometry is different from the funicular curve, bending moments arise.

In real structures the axial stiffness could have high values but never be infinite so if there are no

coactions (flexural stiffness) and loads do not change their position or magnitude, the deformed

curve will be similar to the starting one (there is no variation of curvature between sections) and

it is equally funicular (or anti-funicular), but with different rise (higher in funicular and lower for

anti-funicular).

General speaking, especially for concrete and masonry structures, flexural stiffness is not zero,

producing bending moments, in order to ensure the compatibility condition for fully restrained

structures, equal to the axial forces multiplied the distance between the barycentre axis and the

line of thrust. If the arch is fully restrained, the support rotation is not allowed, so compatibility

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

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moments arise close to supports. If the arch is simply supported, bending moments arise in order

to satisfy the condition of compatibility due to the fix length of the arch.

2.4 The search of the funicular curve

2.4.1 Method of moments

During last centuries different methodologies more or less efficient have allowed designers to

find the shape of the funicular for a known distribution of load.

Remembering the definition of funicular curve, bending moments in each point of the curve have

to be zero. If the rise is known, it is possible to find the points that generate the funicular

configuration. Bending moments, M(x) for a generic arch could be expressed as:

( ) ( ) ( )

o

M x M x Hy x

(2.7)

where Mo(x) is the moment produced by the same distribution of load for a simply supported

beam, y(x) is the vertical coordinate of points and H is the horizontal reaction. If the funicular

geometry is found, M(x) has to be equal to zero, so the vertical coordinate of point y(x) is:

()

() o

Mx

yx H

(2.8)

That result is of paramount importance because illustrates how the funicular configuration

corresponds perfectly to the moment diagram of a beam with the same span and distribution of

loads. In a more general case, if the distribution of load is variable, as shown in figure 2-11, the

vertical position of node y(x) corresponding to the funicular geometry is expressed by equation

2.9:

q(x)

M(x)

Figure 2-11: Funicular geometry for a general distribution of loads

2. STATE OF THE ART: THE CONCEPT OF FUNICULARITY

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00

00

00

( ) ( )

1

( ) ( ( ) ) (1 ) ( ) ( )

( ) ( )

lx

lx

lx

q x xdx q x xdx

y x q x dx q x dx x

Hq x dx q x dx

(2.9)

An improvement of this methodology could be, for example, to take into account that the self

weight of the arch changes for different values of the rise. In that case, it should be necessary to

iterate in order to find the right funicular configuration.

2.4.2 Graphic statics

Varignon introduced the concept of funicular polygon and force polygon in his work "Nouvelle

Mécanique ou Statique" (Varignon 1725). Based on this principle, starting form the XIX century, a

new technique using the reciprocal relationship between funicular polygon and the force

polygon in order to find compression- and tensile-only geometries. Probably the most

comprehensive work on that topic is 'Graphical Calculus'(1890) by Cremona (1830-1903). Graphic

statics is an approach in which only equilibrium equation are taken into account and it is a very

powerful tool for designing new structures and to analyze the existing ones.

Important designers, i.e. Cullmann (1821-1881), Ritter (1847-1906), Maillart (1871-1940), Isler (1926-

2009) y Menn (1927), used that technique for designing elegant and efficient structures.

Graphic statics has been a little used during the last years because a variation of the geometry or

loads changes the whole model. However, during the last years new computational tools have

been developed in order to use the graphic statics in a very easy and intuitive way. Important

improvements have been reached thanks to works by Ochsendorf and Block, respectively at MIT

(Masonry Group 2009) and ETHZ (Block Research Group 2014).

2.4.3 Physical models

As it has been previously described, the idea to find the geometry of an arch using the inversion

approach of a hanging chain boosted the use of reduced physical models. Poleni used this

technique for a safety assessment of the Saint Peter's dome in Rome.

During the first decades of the XX century, some of the most important engineers and architects

faced the problem of finding a funicular geometry using physical models. The following figure 2-

12 show the models used by Gaudí for the Sagrada Familia in Barcelona, and by Isler for one of his

well known shells in Swizerland. Gaudí physical models were models in which a set of loads were

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suspended to a network of wires. Inverting and 'freezing' the system made it possible to obtain

compression-only structures (Popovic Larsen and Tyas 2003).

a) Physical models by Gaudí

b)Physical models by Isler

Figure 2-12: Physical models used by Gaudí and Isler

Main disadvantages of this approach are the impossibility to take into account variable thickness

and loads different from the vertical ones. Furthermore, it is not easy to obtain values of stress

and strain in different point of the hanging geometry and materials used for physical models are

different from real building materials. Today thanks to the incredible developing of 3D printer

and CNC machines, it is possible to promote a return to that technique.

2. STATE OF THE ART: THE CONCEPT OF FUNICULARITY

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2.4.4 Numerical methods

Numerical methods have been developed in order to find the solution to a problem

characterized by a strong geometric non-linearity. They are numerical algorithms where an

iterative process stops when a static equilibrium is reached (Lewis 2003). An interesting overview

of these methodologies is illustrated in Veenendaal (Veenendaal and Block 2012). An in-depth

state-of-art review on these methods can be found in Ramm (Ramm 2004).

These methodologies could be divided in three different groups: methods using dynamic

relaxation, stiffness matrix and force density. An example of the utilization of computational

tools for finding the funicular geometry is included in the work of Galafel lez 2011)

where equilibrium equations and the theory of plasticity have been used for finding funicular

geometries. Differently to the previous described approaches, these tools cannot allow an easy

exploration of new geometries.

2.4.5 Particle-spring systems

Particle-spring systems are an interesting alternative for form finding tension-only geometries.

That approach was developed by Kilian (Kilian and Ochsendorf 2005). The use of spring system is

the basis of an algorithm able to find funicular geometries. Conceptually this methodology is

similar to the physical models with hanging elements. The stiffness of the springs corresponds to

the axial stiffness of the material used with reduced physical models. Changing the stiffness of

the spring it is possible to explore the infinite funicular configurations corresponding to one set

on loads. In the context of graphic statics that is equivalent to change the position of pole in

order to obtain different funicular polygons.

2.4.6 Final comments

Five different approaches for finding funicular geometries have been briefly described. All of

these methods allow to obtain different funicular configurations for one distribution of load.

Changing the horizontal reaction in the method of moments, moving the pole in graphic statics,

using materials with different properties in physical and numerical models, changing the constant

of elasticity in particle-spring systems, are all equivalent ways to explore different configuration

of compression or tension-only geometries.

2.5 Historical and modern examples

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Throughout history a large number of compression- and tension-only structures have been

designed, after a form finding process or simply using intuition or experience, in order to get an

efficient use the materials. In the following section few structures are described and an

implementation of some of the previously approaches is illustrated.

2.5.1 The Pantheon

The building devoted to all the gods, the Pantheon in Rome, represents the top of the

architectonic and structural revolution or Romans. It was rebuild, after a fire, between 118 AD and

128 AD, by Apollodorus of Damascus, and it has been, for different reasons, a revolutionary

construction. The most important are: the use of concrete, opus caementicium, the construction

of the 43-meters dome (record of span dome for several centuries), and the use of light

aggregates wisely distributed.

From a geometrical point of view, the Pantheon, has been built with a spherical geometry. That

geometry is far from being the funicular configuration due to self weight and it has been chosen

because it represents the heaven with evident reference to gods, to which the building is

devoted.

In order to improve the structural behaviour of the dome a series of strategies have been

adopted for changing the distribution of load: the variation of the dome thickness, the use of

concrete with different weight, the addition of external rings. As shown in figure 2-13, the density

of the concrete is between 1350 kN/m3 close to the crown dome and 1600 kN/m3 of bottom part

of the dome and walls. Romans had a large experience in the design of circular arches so they

were aware of the importance of adding more loads close to walls in order to 'verticalize' the

load.

Figure 2-13: Concrete density of Pantheon dome.

2. STATE OF THE ART: THE CONCEPT OF FUNICULARITY

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The Roman Pantheon, as described by Robert (Robert and Hutchinson 1986), actually is cracked

along vertical meridians in the bottom part, creating independent meridian elements in that area;

actually its structural behaviour is a mix between independent arches in the bottom part and a

dome in the top part. In the top area the structural behaviour has to been considered in three

dimensions, otherwise it is not possible to have an oculus at the crown.

An approximated analysis has been performed, as if the structure is working as a series of

independent arches with a section corresponding meridian planes of 22,55° as shown in figure 2-

14.

Figure 2-14:. Independent arches used for analysis

In the following figure 2-15 three antifunicular geometries corresponding to different conditions

are illustrated. In the first case (1) a constant thickness has been considered and the different

density of concrete has not being taken into account. In the second case (2) the real thickness of

the dome has been considered, but ignoring the different density. In the third case (3) both

effects, related to different density and thickness, have been considered. Figure 2-15 shows the

three funicular geometries corresponding to the previous described cases. The eccentricity

between line of centroids and funicular is 1,07 m for the first case, 0,57 m for the second one and

0,54 m for the third one.

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Blue line: Line of centroids

Red curve: case 1

Green curve: case 2

Violet curve: case 3

Figure 2-15: Line of thrust for different load cases

The previous comparative analysis about the distribution of loads on the structural behaviour

illustrates the great knowledge of Romans, who employed a new material in an efficient way with

an extraordinary intuition and talent for changing the thrust line in order to adapt it to the chosen

geometry.

Positioning loads at the top of elements addressed to receive horizontal thrust (walls, flying

buttress), with the aim of 'verticalize' the resulting axial force is an example of prestressing used

in roman and gothic constructions. The main difference, with the modern prestressed system, is

that in the latter case the system is auto-equilibrated, while the in first case the reaction are

increased.

2.5.2 Salginatobel Bridge

Arch bridges with hinges at the crown and at the springing lines have been developed firstly by

French and German engineers at the start of the second half of the XIX century in order to solve

the difficulties met designing and building fully restrained arch bridges.

In order to minimize bending in the arch due to dead load, the axis of the arch should be located

along the dead load pressure line (Menn 1990). If the bridge is hinged at the springing lines and at

the crown, the calculation of forces is easier and the funicular line is unique because three point

of its geometry are identified. Arch bridges, fully or partially restrained at the springing lines,

show different structural behaviour if the thrust line does not match with the axis of the arch.

Figure 2-16 illustrates a recent photo of the Salginatobel Bridge, a 132 meter span concrete bridge

hinged at the springing lines and at the crown. It was designed by Maillart [1871 1940] in 1929.

2. STATE OF THE ART: THE CONCEPT OF FUNICULARITY

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Figure 2-16: Salginatobel Bridge by Maillart

The structural design of the bridge was faced by its author using graphic statics: a first geometry

of the arch was chosen using that approach and after several more in-depth analysis the

geometry did not change. A recent work has been developed by Fivet (Fivet and Zastavni 2012),

who, starting from original sketches of the designer, illustrated the importance of graphic statics

and its development with recent tools.

Figure 2-17a shows the distribution of permanent loads of the Salginatobel bridge, while figure 2-

17b illustrates the thrust line and its comparison with the axis of the bridge. There is a perfect

match close to springing lines, while when there is no perfect match, such as at the middle half of

the span, bending moments arise but the cross-section is bigger.

a) Distribution of permanent loads

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b) Axis line of the arch compared with thrust line.

Figure 2-17: Analysis of the bridge using graphic statics. Taken from Fivet (Fivet and Zastavni 2012).

The above briefly described bridge represents a fantastic example of beauty, efficiency and

economy. The previous analysis evidences that the funicular curve could not match perfectly the

axis of the arch: it is important to find a suitable solution according to theorems of plasticity.

2.5.3 The Tiemblo bridge

The Tiemblo bridge above the Burguillo river is a 165 meter concrete arch bridge supporting a

roadway. The total length is 268 meters. It has been designed by FHECOR Ingenieros

Consultores.

As it is shown in figure 2-18, the geometry of the arch evidences some singularities: variable cross

section with depth between 3,10 meters (l/53) at the springing lines and 1,75 meters (l/94) at the

crown, a constant width of 4.00 meters. The rise of the bridge is 22 meters and the ratio λ=l/f is

7.5.

Figure 2-18: Front view and cross section of the Tiemblo bridge. Courtesy of Fhecor Ingenieros Consultores

2. STATE OF THE ART: THE CONCEPT OF FUNICULARITY

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Using graphic statics, as shown in figure 2-19, it is possible to find a line of thrusts obtained for

loads due to arch, girder and columns that approximately matches the axis of the bridge.

Figure 2-19: Graphic statics applied for finding the thrust line of the Tiemblo bridge due to own weight.

2.6 Final remarks

The definition of the ideal form for a given set of loads, known as the funicular or anti-funicular

curve, depending on the sign of the axial force to be mobilised, has been a very used principle

over last centuries in a more or less intuitive way.

In this section an historical modus operandi has been described: its utilization is revitalized thanks

to modern tools, that help designer to think about the structural behaviour, something that not

always happens due to the massive presence of advanced finite element software.

The Pantheon in Rome, as the roof of the Main Train Station in Berlin shown in figure 2-20,

designed by SBP (SBP 2014), represents the answer to the same problem: established a geometry

(a sphere and a three-centred arch, respectively, for cultural and functional reasons) it is possible

to change the distribution of the load in order to have the line of thrusts corresponding with the

line of centroids. The problem is the same, but it has been solved with different strategies. In the

first case adding external loads, changing the density of the material; in the second case adding

prestressed cables.

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Figure 2-20: Main Train Station in Berlin. Photo taken from SBP (SBP 2014).

During last decades several interesting tools have been developed in order to perform form

finding analysis, saving computational cost, within friendly and innovative environments.

Analysis illustrated in this section have been performed using some of these modern tools,

Geogebra (GeoGebra 2013) and Grasshopper (McNeel 2014) that, working into an interactive

environment, allow the user to change parameters involved into the geometry and distribution

of loads, in order to evaluate different structural solutions.

Virtual tools, together with physical models, could have an enormous benefit to understand the

behaviour of different kind of structures. Physical models, totally disused during the last decades,

could be another time interesting tool thanks to the development of 3d printers and CNC

machines. The idea of exploring, designing, printing and testing a structure is a ambitious goal,

but probably achievable in few years.

The integration of conceptual design and CAD/CAM techniques can allow to build formwork in a

easy way and with a multitude of materials (wood, plastic, corrugated paper, that could be

recycled or not), allowing the development of novel construction systems that make these

fascinating structures more cost-effective.

Finally, the concept of funicularity could be extended easily to a three-dimensional problems: the

physical concept is exactly the same.

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3 ANTIFUNICULAR SPATIAL ARCH BRIDGES FORM FINDING USING AN

INTEGRATED AND INTERACTIVE APPROACH

El nacimiento de un conjunto estructural, resultado de un proceso creador, fusión de técnica con arte,

de ingenio con estudio, de imaginación con sensibilidad, escapa del puro dominio de la lógica para

entrar en las secretas fronteras de la inspiración. Antes y por encima de todo cálculo está la idea,

moldeadora del material en forma resistente, para cumplir su misión.

Eduardo Torroja

3.1 Introduction

Curved structures, if they are the outcome of an appropriate design, have the capacity to carry

large loads and cover important areas saving costs and can represent examples of the highest

structural efficiency. In designing shells or arches, geometry plays a primary role: enough energy

should be spent for a proper design of the structural shape. Acting loads and geometry are the

essential factors involved in the structural behaviour.

Pedestrian bridges are peculiar structures because they allow designer to create a contextualized,

unique, beautiful structure considering the human scale of the user, covering both functional and

aesthetical aspects (Stein 2010). It is not unusual that the footbridge deck has to adapt to a

gulations,

which entails the use of long ramps. Usually designers incorporate external ramps located to

decrease the slope as much as possible to fulfil the admissible slope limit. An alternative to that

solution is to increase the length of the deck, by making it curved in plant. Structurally the curved

deck could be supported by an arch, or can be cable-stayed or suspended by a cable system.

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Curvature can be a symbol for grace and beauty: a designer who is able to take advantage of

curvature will design structures with efficiency of performance and elegance of form. Some

example are shown in figure 3-1.

a) Sassnitz Bridge, Germany, designed by SBP (SBP 2014).

b) Ponte del Mare, Italy, designed by Miranda. Taken from (Studio de Miranda Associati 2013)

Figure 3-1: Examples of curved bridges

The purpose of this section is to present a integrated and interactive approach to shape

antifunicular spatial arches that support a curved decks using secondary cables. Sarmiento-

Comesias (Sarmiento-Comesias et al. 2013) defines spatial arch bridges, as bridges in which

vertical deck loads centred on the deck induce internal forces not contained in the arch plane.

3. ANTIFUNICULAR SPATIAL ARCH BRIDGES

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To design efficient spatial arch bridges, the same concepts used for planar arch bridges,

described in the previous section, should be maintained: the most important is to choice the

geometry of the arch coinciding with one of the antifunicular curves.

As described above, the funicular curve is subjected to tensile loads only if the geometry and

loads do not change during the life of a structure; this situation is incompatible with real

structures showing the first limit of this design method: a geometry can be funicular only for a

unique distribution of loads. For each distribution of loads there are infinite funicular curves: the

variation of the horizontal reaction or of the rise or of the entire length allows to obtain infinite

configurations of compression-only geometries.

As already mentioned, this section illustrates an integrated approach to design antifunicular

spatial arch bridges developed in a parametric, three-dimensional and innovative environment.

In this work only spatial arches which support a curved deck using secondary ties will be

considered. In this section first the state-of-the-art of spatial arch bridges is presented; then, a 3-

dimensional design method and its implementation is described.

3.2 State-of-the-art of spatial arch bridges

The history of curved bridges is rather recent. Even more is so is the history of arch bridges with

curved decks. An in-depth overview on spatial arch bridges has been published by Sarmiento-

Comesias (Sarmiento-Comesias et al. 2013) covering over 80 spatial arch bridges designed and

built around the world.

The first idea of conceiving a spatial curved bridge came from the genius of Leonardo da Vinci

who in 1502 sent to Turkish Sultan Bayezid II his designing proposal to erect a bridge between

Galata and Istanbul with a span of 240 m, as shown in figure 3-2.

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Figure 3-2:sketch of the Golden Horn Bridge designed by Leonardo da Vinci in 1502 (Biblioteque Institute Paris ).

The first constructed bridges of this nature were the Ziggenbach Bridge (1924), Landquart Bridge

(1930), Bohlbach Bridge (1932) and Schwandbach Bridge (1933), all designed by Maillart

(Laffranchi and Marti 1997).

Some of the best known contemporary designers of inclined arches are Santiago Calatrava, Jiri

Strasky, Wilkinson Eyre with Flint & Neil and Javier Manterola (Baus and Schlaich 2008). Several

curved pedestrian bridges have been constructed in UK before the end of the Millennium and

valid examples are the Millennium Bridges in York and in Gateshead (Davey and Forster 2007;

Curran 2003). Other examples in UK are the Merchants' bridge in Manchester by Whitbybird

and the Butterfly Bridge in Bedford by Wilkinson Eyre Architects and Jan Bobrowski & Partners.

Due to the strong need for urban bridges in Spain, several arch spatial bridges have been

designed by Calatrava (La Devesa footbridge in Ripoll, the Campo de Volatin bridge in Bilbao,

Port of Ondarroa bridge, Alameda Bridge in Valencia) and by Manterola, from Carlos Fernandez

Casado (the bridge over the Galindo River in Bilbao). Another example of spatial arch bridge is

the Footbridge Rari-Nantes in Padua, with span 75m, designed by Enzo Siviero. An outstanding

example of an existing funicular curved arch bridge with superior deck is the Ripshorst

Footbridge, shown in figure 3-3, designed by SBP (SBP 2014). As described in Schober (Schober

2003) its funicular geometry was found using a hanging model.

3. ANTIFUNICULAR SPATIAL ARCH BRIDGES

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Figure 3-3:Ripshorst Footbridge by SBP (SBP 2014)

The first in-depth theoretical studies on this topic have been developed by Jorquera (Jorquera

Lucerga and Manterola Armisen 2012; Jorquera Lucerga 2007; Jorquera Lucerga 2009), Lachauer

(Lachauer and Kotnik 2012) and by Sarmento-Comesias (Sarmiento-Comesias et al. 2013). Figure

3-4 shows two funicular spatial arch footbridges proposed, respectively, by Romo (Romo 2014)

for a competition in Salford and by SBP for a competition in Mettingen (Stein 2010).

a) Salford footbridge proposal by Romo. Taken from Romo (Romo 2014)

b) Mettingen footbridge proposal by SBP. Taken from Stain (Stein 2010)

Figure 3-4:: spatial arch footbridge proposals

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3.3 Designing spatial arch bridges

3.3.1 An interactive and integrated environmental workspace: Rhinoceros,

Grasshopper, Karamba

The procedure to design antifunicular spatial arch bridges has been developed thanks to a

combination of different software (Rhinoceros, Grasshopper and Karamba) in a parametric,

three-dimensional and innovative environment.

The development of generative design methods enables the designer to describe mathematically

complex form and explore new ones (Georgiou 2011; Georgiou et al. 2011).

Rhinoceros is a commercial NURBS-based 3-D modelling software; NURBS (non-uniform rational

B-splines) are mathematical representations that can accurately model any shape from a simple

2-D line, circle, arc, or box to the most complex 3-D free-form organic surface or solid. Because of

their flexibility and accuracy, NURBS models can be used in any process from illustration and

animation to manufacturing. NURBS geometry is an industry standard for designers who work in

3-D where forms are free and flowing (Robert McNeel & Associates 2013). During last decade

Rhinoceros has gained popularity in architectural design thanks to its plug-in Grasshopper

(Robert McNeel & Associates 2014).

Grasshopper provides for a new way for creating parametric geometries. It is a graphical

generative design editor that allows users, without knowledge of scripting, to generate

parametric complex geometries in a very intuitive way. This tool allows to create relations

connecting parameters and component with cables. Components are designed with input and

output, representing the input data and the output after performing script stored into the

component. The interface is enjoyable to use and encourages the development of new ideas. The

geometry generated using Grasshopper is transferred to Rhino in real time and this allows to

perform a fast visual check of the model. As briefly described in section 1, there is a strong need

for this kind of tools, devoted to design, to explore and not only to analyze structures.

Grasshopper, different to typical other software, is not a 'black box' because the user has total

control of what the software is doing because the user has programmed it.

Grasshopper allows to perform integrated finite element analysis (FEA) thanks to its plug-in

Karamba (Preisinger and Heimrath 2014). Karamba allows to interactively calculate the response

of three-dimensional structures under the action of external loads. Starting from generated

geometries in Grasshopper, Karamba is able to convert geometry into structural elements and to

assemble a model defining geometrical and mechanical properties of beams, loads and supports.

The FE Analysis performed by Karamba is linear elastic, but geometrical non-linear behaviour

could be taken into account using a "large deformation" component. This component performs

geometric non-linear analysis by an incremental approach: all external loads are applied in steps

starting from an initial geometry. Calculations and loads are not considered as a whole, but are

gradually increased until successive states of equilibrium are solved: after each step the model

geometry updates to the deformed state. Reducing the step size it is possible to have a better

3. ANTIFUNICULAR SPATIAL ARCH BRIDGES

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approximation of geometric non-linearity. Using the incremental method, inevitably the solution

will drift from the true equilibrium curve (Lewis 2003): considering this tool as a fast way to

produce shapes during conceptual design, these kind of approximations are acceptable because

they are conceptually correct.

The incremental approach makes use of the tangent stiffness Kt (instead of the stiffness matrix

used in static analysis) which relates small changes in load to small changes in displacement. At

each load step ΔF, Karamba calculates the incremented displacement Δu, which is:

t

F

uK

(3.1)

After each step the tangent stiffness matrix is taken as the one obtained from the previous step /

iteration. The procedure continues until all the load is applied. Convergence checks are made and

iterations are stopped as soon as the state of equilibrium is reached. The final solution is reached

when all (internal and external) forces reach equilibrium.

In real time software for structural applications, like Karamba + Grasshopper, user interaction,

visualization of the model and structural response occur simultaneously, without the need for

the user to explicitly run the analysis (Clune 2010). This kind of tools are valuable for exploring

new shapes and pushing a deeper understanding of structural behaviour. A geometric modeller

and an analysis software converge in a unique tool allowing the user to explore a wide range of

possible geometric configurations to design sustainable and efficient structures. The combined

use of these tools allows the user to explore new forms and shapes. Substantially there is a

revolution with respect to the classical procedure: create geometry - analyze - review results

because there is a real-time interaction between geometry and results.

3.3.2 Form finding methodology for spatial antifunicular arches

In this section, the methodology SOFIA (Shaping Optimal Forms with an Interactive Approach)

for the generation of antifunicular arch spatial bridges in a integrated approach is in-depth

described.

A well known method for finding the antifunicular shapes of an arch consists in loading a cable

(very low flexural stiffness compared with axial stiffness) with a known set of loads; the cables

have to be subdivided in elements connected with hinges, like a chain, in order to find a

compression-only geometry. This method is equal to the hanging models used by Gaudí and Isler

(Adriaenssens et al. 2014).

This kind of analysis is characterized by a strong geometric non-linearity. Starting from a generic

geometry, the tool, performing an incremental analysis, allows to obtain different funicular

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

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geometries depending on the applied load. Some tests have been performed for simple

configurations of loads. Starting from a simple circular arch with the following sets of loads:

projected distributed loads directed along the global z-axis, distributed loads directed along the

global z-axis, projected distributed loads directed along the local z-axis. Expected geometries,

respectively, a parabola, a catenary and a circular arch have been obtained.

The above illustrated methodology has been implemented for shaping three-dimensional

geometries with loads acting with any value and direction in order to find a funicular geometry

for a spatial arch bridge. Since the loads acting on the arch depend on shape of the deck, this

method is iterative. Usually after few iterations the solution is reached. The method is described

step-by-step in figure 3-5.

In the following paragraph the steps are described:

- the geometry of deck, the number and the position of hangers, and the initial geometry of arch

are necessary data for the first model of the bridge. The geometry of the deck is arbitrarily

curved. The starting geometry of the arch is arbitrarily chosen;

- loads are evaluated based on the dominant load case. All dead loads, related to arch and deck,

and one half of live loads are taken into account for designing the geometry of the arch. That

combination of loads is frequently used in order to decrease effects of live load during the life of

the bridge;

- forces acting on cables are evaluated taking into account an infinite stiffness of the hangars;

- the form finding analysis of the arch is performed taking into account the axial forces of the

hangars and the dead load of the arch. The final geometry has the total length of the initial one,

i.e. the an infinite axial stiffness has been considered. The analysis is stopped when a funicular

curve is reached, i.e. the maximum value of eccentricity ei is less than an arbitrary allowable value

of the eccentricity, e*. The eccentricity is calculated for each i section as:

22

yz

ii

ii

MM

eN

(3.2)

where Ni is the axial force in each section of the arch, Miy and Miz are the bending moments for

each section of the arch, respectively, around the y- and z-local axis. The methodology is

developed, not with real areas in order to avoid eccentricities due to the real deformability of the

structure; as already it has pointed out by Jorquera (Jorquera Lucerga and Manterola Armisen

2012) considering that there are no external torsional moments affecting the arch, the torsional

moments automatically vanish if the bending moments are set to zero. In other words: torsional

effects can be neglected;

- once the analysis has been completed, the bridge is regenerated with the deck and the updated

geometry of the arch. The funicular behaviour of the arch is checked. If the bending moments are

too big, the updated geometry of the arch is used for a new iteration. If the funicular

configuration has been reached the form finding analysis is completed.

3. ANTIFUNICULAR SPATIAL ARCH BRIDGES

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- once the final geometry of the arch is found, preliminary sizes of the members can be

determined using the computed forces; prestressing loads are imposed on cables in order to

obtain the small displacement of the deck determined from analysis with axially rigid members

adopting real properties of sections and materials.

Figure 3-5:Description of the form-finding process SOFIA

The resulting funicular arch is not only curved in elevation, but also double-curved in plan.

Examples of antifunicular geometries for bridges with 'C' and 'S' shape curvature of the deck are

shown in figure 3-6.

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a) antifunicular arch for bridges with C-shaped deck

b) antifunicular arch for bridges with S-shaped deck

Figure 3-6: Resulting antifunicular arch for bridges with C- and S-shape curvature of the deck

Since hand calculations cannot be performed for these complex structures with sufficient

accuracy because curvature results in complication in computations, a validated finite element

3. ANTIFUNICULAR SPATIAL ARCH BRIDGES

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software, Sofistik, has been used for checking the funicular geometries obtained with the new

design tool. There are no relevant differences between the results.

After the geometry has been created, since the funicular configuration of the arch has been

found for one set of loads, the designer should test results with all possible scenarios to consider

other variables such as real properties of materials, different distribution of loads, non-linear

behaviour, vibrations, construction stages and time-dependent effects.

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4 PARAMETRIC ANALYSIS ON THE INFLUENCE OF DECK-ARCH

RELATIVE TRANSVERSAL POSITION

Più che parlare di rapporti tra Architetto, Ingegnere e Costruttore mi pare che si dovrebbe esaminare

come queste tre mentalità e competenze possano e debbano fondersi per raggiungere quella unità da

cui nasce la vera Architettura

Pier Luigi Nervi

4.1 Cases of study

As an example illustrating a case of SOFIA, parametric analyses have been performed for spatial

arch footbridges with C- and S-shapes of the deck. The starting geometries of the bridge before

form finding analysis are shown, in figure 4-1a and 4-1b, respectively, for deck with C and S

geometries.

a) C-shaped configuration

4. PARAMETRIC ANALYSIS

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b) S-shaped configuration

Figure 4-1: Cases of studies for the parametric analysis

In its C-shaped configuration, the bridge has one span of 100m, while for the S-shaped

configuration the bridge has two spans, both of 50m. The rise (fa) for both bridges is 45m.

The geometry of the deck is usually fixed by functional constrains (Romo 2014) while the position

of the arch could be slight variable: the analysis has been carried on C- and S-shaped bridges

using as parameter the distance y between supports of deck and abutments of the arch. The

effect of the variation of y on deck, arch and cables has been evaluated.

Referring to figure 4-1, the main dimensions and parameters considered in the study are shown in

table 4-1.

Table 4-1: Main dimensions and parameters considered in the parametric analysis

[m]

C-Shape

S-Shape

da

100

100

fa

45

45

dd

100

50

fd

10

10

y

-10<y<3

-10<y<3

The bridge is only subjected to vertical loads, which are assumed to be uniformly distributed on

both deck and arch. The considered loads are 8.55 kN/m and 30 kN/m, respectively for arch and

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deck. Dead load of the arch corresponds to an arch with a steel hollow circular transversal section

with diameter of 2m and thickness of 17mm. The load of the deck corresponds to a steel deck

with a width of 5m, 3,50 kN/m2 of dead load and one half of live load, 2,50 kN/m2. The adopted

loads and geometries are similar to the real one of the Millennium Bridge in Gateshead (Davey

and Forster 2007).

The following figure 4-2 shows the starting geometry of the deck with C-shape and the

corresponding funicular geometry of the arch, obtained varying the value of y. The figures on the

left shows a perspective view while the images on the right show a lateral view.

y=-10

y=-9

4. PARAMETRIC ANALYSIS

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y=-8

y=-7

y=-6

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y=-5

y=-4

y=-3

4. PARAMETRIC ANALYSIS

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y=-2

y=-1

y=0

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y=+1

y=+2

y=+3

Figure 4-2: Antifunicular configuration for C-shape deck

4. PARAMETRIC ANALYSIS

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Fgure 4-3 shows the starting geometry of the deck with S-shape and the corresponding funicular

geometry of the arch, obtained varying the value of y. The figures on the left shows a perspective

view while the images on the right show a lateral view.

y=-10

y=-9

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y=-8

y=-7

y=-6

4. PARAMETRIC ANALYSIS

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y=-5

y=-4

y=-3

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

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y=-2

y=-1

y=0

4. PARAMETRIC ANALYSIS

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y=+1

y=+2

y=+3

Figure 4-3: Antifunicular configuration for S-shape deck

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

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4.2 Discussion of results

The proposed form-finding method for shaping antifunicular spatial arch bridges, called SOFIA,

has been used to study the influence of the relative position between deck and arch on the

structural behaviour of the bridge. The aim of this analysis is to find the optimal position of the

arch's abutments with respect to the deck in order to minimize the cost of the bridge.

In figure 4-4 the axial forces along the arch have been plotted versus the curvilinear coordinate of

the arch for different values of y, respectively for C-and S-geometries of the decks.

a) C-shaped deck

4. PARAMETRIC ANALYSIS

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b)S-shaped deck

Figure 4-4: Axial forces along the arch

The minimal axial force in the arch correspond to values of y between 7 and 5 for the C-shaped

deck. Figure 4-4b shows the asymmetrical distribution of the axial forces in the arch due to the

asymmetric position of the S-shaped deck. Only for the case y=0 the position of the deck is

symmetrical respect to the arch and therefore the distribution of the axial forces is symmetrical.

In figure 4-5 the summation of axial forces of all j cables has been plotted for different values of y,

respectively for C- and S-shaped deck.

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

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a) C-shaped deck

b)S-shaped deck

Figure 4-5: Summation of axial force of cables for different values of y.

Figure 4-5 illustrates that the optimal position of the deck, corresponds to y=-6 and y=o for,

respectively, C- and S-shaped decks.

The values of axial forces in the C- and S-shaped deck are plotted versus the curvilinear

coordinate of the deck for different values of y in figure 4-6.

a) C-shaped deck

4. PARAMETRIC ANALYSIS

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b)S-shaped deck

Figure 4-6: Axial forces in the deck

Figure 4-6a shows a symmetrical distribution of axial forces in the deck. The minimum absolute

values correspond to the situation in which y is included between 5 and 7. The difference

between the maximum and minimum axial force for each value of y, is almost constant for all

cases.

It is interesting to observe that the deck could be compressed or tensioned depending on the

direction of the cable respect to the curvature of the deck. If the cable direction is toward the

inner side of the curvature, the horizontal thrust compresses the deck, while for the opposite

case the horizontal thrust results in tension in the deck.

The previous comments of results illustrate that the optimal position of the arch's abutments to

design the most efficient spatial arch with C-shaped deck is around y=6. In this case the distance

between the centre of gravity of the deck and the vertical plane passing from the abutments of

the arch is minimum.

The same conclusion could be drawn for a spatial arch with S-geometry of the deck. The most

efficient solution coincides with y=0, the unique value of y for which the deck is symmetrical with

respect to the arch. Furthermore, the centre of gravity of the deck is contained in the vertical

plane passing through the abutments of the arch.

In order to validate the conclusions obtained for fd=10m, more parametric analysis have been

performed with fd=5m and fd=15m. Relevant plots only for fd=5m are shown in the following

figures.

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Figures 4-7, 4-8 and 4-9 show, respectively, the distribution of axial forces in the arch, the

summation of axial force of cables and the distribution of axial forces in the deck for different

values of y in the case of fd=5m.

a)C-shaped deck

b)S-shaped deck

Figure 4-7: Axial forces along the arch

4. PARAMETRIC ANALYSIS

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b)C-shape deck

b)S-shape deck

Figure 4-8: Summation of axial force of cables for different values of y.

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b)C-shape deck

b)S-shape deck

Figure 4-9: Axial forces in the deck

The previous figures 4-7, 4-8 and 4-9 illustrate that for the C-shaped deck, the most cost-effective

bridge is obtained for y=3, while for the S-shaped deck, the best solution corresponds to y=0.

Both results point out that the deck optimal position minimizing the axial forces of arch and

4. PARAMETRIC ANALYSIS

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cables corresponds to the situation for which the distance between deck's centre of gravity and

vertical plane passing from arch's abutments is minimum. The same conclusion has been

obtained with fd=15m.

4.3 Conclusions

In this section, SOFIA, a methodology to approach the conceptual design of spatial arch bridges

in order to obtain a funicular configuration of the main arch, has been presented. The main

parameters affecting the antifunicular shape of the arch are the geometry of the deck and the

arch-deck transversal relative position. Using SOFIA, parametric studies have been performed in

this section.

The presented parametric analyses allow designers to find the optimal position of the deck with

respect to the arch in order to minimize the cost of the bridge. Results show that the optimal

position of the deck minimizing the axial forces of arch, deck and cables corresponds to the

situation for which the distance between deck's centre of gravity and vertical plane passing from

arch's abutments is minimum.

The parametric analyses have been performed for different values of curvature for both, pointing

out the independence of conclusions from the curvature of the deck.

This design tool, in combination with profound knowledge, could be an efficient method for the

conceptual design of arch spatial bridges and a source of inspiration and creativity for the

designer.

5. CONCLUSIONS AND FUTURE STUDIES

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5 CONCLUSIONS AND FUTURE STUDIES

The overall quality of many structures today leaves much to be desired. The rapid technological

progress does not reflect adequately in their variety, beauty and sensitivity. Too often structural

engineers neglect the creative conceptual design phase by repeating standard designs and not

sufficiently contributing with own ideas to the fruitful collaboration with architects. Engineers thus

often waste the chance to create building culture.

Jörg Schlaich

5.1 Summary of results

5.1.1 The funicular principle

The world is faced with challenges in all three dimensions of sustainable development: economic,

social and environmental. In cities of middle- and high-income countries, investment in

infrastructure, renewable energy, buildings, and improved electricity and water efficiencies is

important (DESA United Nations 2013). There is a great awareness on saving energy and

resources designing efficient structures.

Curved structures, if outcome of an appropriate design, have the capacity to carry elevate loads

and cover important areas saving costs and representing the highest examples of structural

efficiency: the prima donna among all structural typologies. Furthermore, curved structures

combine high structural performance and beauty, so are strongly used as architectural and civil

structures.

In designing shells or curved bridges, geometry plays a primary role, so a high attention should be

spent for a proper design of the structural shape. An inappropriate choice of the geometry

causes a series of problems that will be grow during the project, the construction and the life of

the structure.

The definition of the ideal shape for a given set of loads, known as the funicular or anti-funicular

curve, depending on the sign of the axial force to be mobilised, has been a very used principle

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over last centuries in a more or less intuitive way. That concept has been in-depth investigated

and illustrated.

Before the digital era, a lot of designers were interested in finding optimal geometries of curved

structures: using different form finding techniques, intuition and great knowledge of structural

behaviour they were able to construct great structures. Some examples of those constructions

have been shown in this work. An old modus operandi has been recovered, which actualization is

revitalized thanks to modern interactive tools, that invite users to think about structural

behaviour: something that does not always happen with more conventional tools, that use finite

element analysis.

During the last decades several interesting tools have been developed in order to perform shape

finding analysis, saving computational cost, within friendly and interactive environments. These

virtual tools together with physical models can have a positive effect on the degree of

understanding the structures behaviour. This way of investigating structures would also be a task

for education and research. It would be desirable that the Spanish Technical Universities, as well

as other Technical Schools in the world, such as MIT (Masonry Group 2009), Cambridge, USA,

ETH (Block Research Group 2014), Zurich and EPFL Lausanne (IBETON EPFL 2013), Switzerland,

strengthen teaching and management of these tools.

Engineers and architects design approaches should be carried out in parallel, in order to allow a

holistic solution which would be optimized in terms of both environmental and structural

performance (Shepherd and Richens 2012).

5.1.2 Designing antifunicular spatial arch bridges

In this work a new integrated and interactive approach for designing spatial antifunicular arch

bridges has been presented. Spatial arch footbridges represent an innovative answer to demands

on functionality, structural optimization and aesthetics for curved decks, quite popular in urban

contexts.

A new computational strategy, called SOFIA, has been developed in order to find an anti-

funicular arch configuration for any geometry of the deck: an extension of physical hanging

models in space. The resulting geometry has a double curved spatial configuration integrating

structural efficiency and an esthetical appeal.

The method is suited for the conceptual design phase of a project. After a funicular geometry has

been found for the predominant load case, more refined analysis must be performed in order to

check the structure for deflections, vibration, live loads, fatigue, time-dependent effects, etc...

Parametric analyses have been performed in order to evaluate the influence of deck-arch relative

transversal position on the structural behaviour of the arch. It has been show that the eccentricity

5. CONCLUSIONS AND FUTURE STUDIES

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between the arch springings and the deck abutments plays a key role in finding the antifunicular

configuration.

The optimal position of the arch, in order to build a cost-effective bridge, has been found. Results

show that the best position of the deck minimizing the axial forces of arch and cables

corresponds to the situation for which the distance between deck's center of gravity and vertical

plane passing from arch's abutments is minimum.

The presented design tool, in combination with deep knowledge, could be an efficient method

for the conceptual design of antifunicular arch spatial bridges, and a source of inspiration and

creativity for the designer.

5.2 Further studies

5.2.1 Introduction

The developed interactive approach, SOFIA, has been used to study the influence of deck-arch

relative transversal position. It can be employed to analyze the effect on the arch due to the

variation of deck curvature. Superior deck antifunicular spatial arch bridges can be studied, with

an easy improvement of SOFIA.

In-plane and out-of-plane buckling, dynamic, temperature and fatigue effects have not been

taken into account in this study: they can be object of further researches.

In this work only uniformly distributed loads have been considered, however it can be interesting

to take into account point loads or distributed loads applied only in a limited part of the bridge.

Furthermore it can be interesting to study non-funicular spatial arch bridges, as for example

spatial arch bridges with imposed curvature (where the arch has the same curvature of the deck

(Sarmiento-Comesías et al. 2012)) or curved bridges characterized by a vertical arch.

In order to build a double curved arch the feasibility of manufacturing of its shape must be taken

into account. Fabrication aspects could be introduced as parameters in the form finding process.

A concrete spatial curved arch would require expensive formworks, complicated scaffolding and

manual casting; using steel it would be possible to use different kind of profiles that can be easier

changed in size and thickness. For these reasons concrete arches are rarely used in pedestrian

bridges (Keil 2013).

Nowadays there is an incredible development of CNC machines and 3D printers that use CAD-

CAE-CAM technology to create any geometry and shape; formworks could be directly printed

with any geometry, or real scale steel elements of the bridge could be produced. This is a very

promising field of research that merits strong consideration.

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

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Two lines of research have been started to investigate the influence of deck cross-section, cables'

position and arrangement on the bridge behaviour.

5.2.2 Torsional behaviour of curved decks

The curved deck of an antifunicular spatial arch, as for classical cable-stayed bridges, can be

suspended on its outer or inner edge, or on both of its edges or on its central axis (Strasky 2005).

The cross section behaviour depends fundamentally on the supporting system, whether it is

placed inside or outside the curve, taking into account that a fluid route trajectory of pedestrians

and cyclist requires to have the supporting system placed on one side only (Keil 2013).

It is well-known that the statical behaviour of a curved decks is completely different compared to

straight ones. The transfer of loads cannot be separated clearly into longitudinal and transverse

actions because of three dimensional interactions: in curved bridges there is a reciprocal

relationship between torsion and bending. In the case that the radius of curvature is very large,

the two situations could become similar. In bridges with curved deck, torsional moments in

addition to longitudinal bending moments should be taken into account; for bridges with an

angle of curvature <30° coupling effect can be neglected (Kolbrunner and Basler 1969).

A clear understanding of the effects of torsion on structural members is essential to design

efficient curved structures. A structural element, in general, exhibits two ways to resist torsion.

The first, commonly called Saint-Venant torsion, generates a shear flow in the transversal cross

section, while the second, commonly named warping torsion, generates normal and shear

stresses.

In the majority of practical applications, only one structural mechanism is relevant, the other

could be neglected. In closed cross sections (solid or hollow) the warping mechanism could be

neglected, while for open sections that mechanism is the most relevant. It is important to point

out that the preponderance of one of the two structural mechanisms depends on cross section

properties, length of the structural element and support conditions.

There are two main strategies to design structures in which relevant torques are applied: the first

is the use of common cross sections adequately strengthened in order to resist torques, the

second consists in varying the geometry of the cross section (centre of torsion cannot match the

barycentre) and/or modifying the position of supports in order to generate torques for

decreasing torsional effects.

5. CONCLUSIONS AND FUTURE STUDIES

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Transversal position of cables plays a key role in deck behaviour and cross section design. Varying

the cables transversal arrangement it is possible to change totally the structural behaviour of the

bridge. That topic is currently under development.

5.2.3 Arrangement of layout of cables

A research is currently ongoing to analyze analyzing the effects of hangers arrangement on the

antifunicular geometries including buckling effects and the effect of live loads.

The first investigated hanger configurations are shown in figure 5-1.

a) vertical hangers

b) radial hangers

c) hangers distributed according a growing linear law

d) hangers distributed according a decreasing linear law

An integrated approach to conceptual design of arch bridges with curved deck - Leonardo Todisco

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e) Nielson hangers arrangement

e) Nielson-network hangers arrangement

Figure 5-1: different arrangements of cables layout

PUBLICATIONS BY AUTHOR

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PUBLICATIONS BY AUTHOR

- Todisco, L., (2011) "Test evidence for applying strut-and-tie models to deep beams and D-

regions of beams,"

October 2011.

- Sanitate, G., Todisco, L., and Monti, G. (2014). "Effective assessment methodology for trulli in

Apulia, Italy." Proceedings of 9th International Masonry Conference, Guimares.

- Reineck, K.-H.; Todisco, L.; (2014) -slender Reinforced

ural Journal. V. 111, No. 1-6, January-December 2014.

- Todisco, L.; Reineck, K.-H.; Bayrak, O.; -slender beams with

- Todisco, L.; Reineck, K.-H.; Bayrak, O.; Database of Shear Tests for Non-slender Reinforced

In preparation.

- la comprobación de

- . In

preparation.

- An integrated approach to conceptual design of arch bridges with curved deck

I.S.B.N.

E.T.S.I.C.C.P.

U.P.M.