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Authors:
L. Martellotta 0873376
S. Moriche Quesada 0885681
STRUCTURALDESIGN
Design Master Project 2014
Tutors:
ir. A.P.H.W. (Arjan) Habraken
prof.Dr.-Ing. P.M. (Patrick) Teuffel
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TABLE OF CONTENTS Terms of reference ...................... ......................... ......................... ...................... ......................... ........... 4
Summary ................................................................................................................................................. 4
Introduction ............................................................................................................................................. 5
Project requirements ........................ ......................... ......................... ....................... ......................... .. 5
Site conditions ..................................................................................................................................... 5
Projects review ......................... ....................... ......................... ........................ ......................... .............. 6
Retractable roofs ...................... ......................... ......................... ...................... ......................... ........... 6
History of retractable roofs ..................... ......................... ...................... ......................... ................... 8
Project overview .................... ...................... .......................... ......................... ......................... ........... 10
Single mast structure .................... ......................... ......................... ....................... ......................... 10
Spoked wheel structure ......................................... ......................... ....................... ......................... 12
Translational systems ...................... ......................... ......................... ......................... .................... 16
Design choice .................................................................................................................................... 19
Reference Project ..................... ....................... ........................ ......................... ......................... ............ 20
“Egg-shaped” spoked wheel-Design proposal ................................ ...................... ......................... ......... 21
equilibrium of the structure ........................ ......................... ...................... ......................... ................. 25
floating columns .................... ...................... .......................... ......................... ......................... ........... 27
shadow path ...................................................................................................................................... 28
Analysis ...................... ......................... ......................... ......................... ....................... ......................... 29
Form finding ..................... ......................... ......................... ...................... ......................... ................. 29
Loading conditions ......................... ...................... ......................... ......................... ...................... ...... 31
Wind Load ...................................................................................................................................... 31
Prestress ........................................................................................................................................ 43
Initial situation .................... ...................... .......................... ......................... ......................... ........... 44
Dead load ..................... ......................... ......................... ...................... ......................... ................. 47
Geometry relations ......................... ...................... ......................... ......................... ...................... ...... 48
Calculation ............................................................................................................................................ 49
Upper roofing cables verification ............................... ......................... ....................... ......................... 53
Pillars verification: .................... ......................... ......................... ...................... ......................... ......... 54
Radial beams verification: ......................................... ......................... ....................... ......................... 57
Compression ring verification: ..................... ......................... .......................... ......................... ........... 59
Connections ..................... ......................... ......................... ...................... ......................... ................. 61
Physical and virtual model ....................................... ......................... ......................... ...................... ...... 73
Conclusion and recommendations ........................ ......................... ...................... ......................... ......... 75
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Literature research ......................... ...................... ......................... ......................... ...................... ...... 75
Design calculations ........................ ...................... ......................... ......................... ...................... ...... 75
Recommendations ......................... ...................... ......................... ......................... ...................... ...... 76
Bibliography .......................................................................................................................................... 77
Acknowledgments ................................................................................................................................. 78
Appendices ........................................................................................................................................... 79
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TERMS OF REFERENCE
This document is mean to be a description of the development of the project design: in Lichtenberg. Whichhas been designed for Luca Martellotta and Sergio Moriche Quesada, and tutored for prof.Dr.-Ing.P.M.
Teuffel and ir. A.P.H.W. Habraken, as a part of the learning portfolio of the Master in Architecture building
and planning in the specialization of Structural design, during the summer semester of 2014 at the
Eindhoven University of Technology.
SUMMARY
The purpose of the project is to design a roof building solution given a determined project. The designshould be consistent with the project in terms of appearance and requirements. The scope of the project
includes a preliminary phase of research of ideas and a design proposal which must be proven to viable.
The design proposal is composed for a description of the structure including plans, the interaction with
loading, and the verification of the structural elements.
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INTRODUCTION The Open air theater in Weert, Lichtenberg (The Netherlands), is a construction of the 1960s, which is part
of an ecclesiastical school complex. The theater is located South-West from the town in a massive greenforest area.
The overall geometry of the theater can be assimilated to a triangular shape, has an audience capacity of
about 2200 people, the seating area runs along a moderate terrain slope with its lowest point at the level
of the stage, where, behind it rises a building of 3 floors.
FIGURE 1-GEOMETRY OF THE TEATHER
PROJECT REQUIREMENTS
The requirements of the project are the followings:
• The structure has to be in harmony with the green surrounding environment • The visual impact of the new structure should be reduced, so that the existing construction it is
not hided.• The covered area should protect the audience and the actors from weather inclemency during
performances.• The roofing solution must permit that the deployment of the roof closes in a relative short time.• The theater needs to be covered in order to plan performances independently of weather
conditions.
SITE CONDITIONS
The limitations of the location are the followings:
• The terrain has a low bearing capacity. (see Annex A1 for further detail)• Due to the architectural value, the building existing elements must be preserved, consequently no
structural elements will be introduced inside the wall perimeter, and modification of any of theoriginal elements is permitted.
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PROJECTS REVIEW This section consist in a small research of information about designed or constructed projects with
equivalent or similar requirements. As a selection criteria, only retractable roofs have been looked at.
RETRACTABLE ROOFS
A retractable roof is a roof system which is able to be pulled or slide backwards, opening and closing thecovering area. The retractable roofs can be divided into two basic types:
• Soft or flexible roofs: which their geometry can be changed by folding, bunching or rollingup.
• Rigid roofs: that consist of several movable segments of fixed shaped parts.
Those two basic types can be further classifies in function of its movement. Frei Otto developed a table in
his book, which is often referred to as the classification of retractable roofs. In this classification theconstruction system is divided by type of movement and it is specified four cases of direction of movement
as well as the above mentioned basic classification.
FIGURE 2-RETRACTABLE ROOF CLASSIFICATION
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When a retractable roof is opened, it must be stored. But in most case the storage space is limited.Therefore the storage size reduction is important. In order to deal with this issue, commonly fourfundamental types of reduction are used: overlapping, folding/bunching, rolling and deforming by air.
If the different directions and the different possibilities of size reduction are combined, plenty of choices arepossible. Besides that independent movable parts or a continuous flexible surface, for instance a membranecan be reduced also using multiple axes.
FIGURE 3-RETRACTABLE ROOF CLASSIFICATION BASED ON THE MOVEMENT DIRECTION AND ON THE AXIS.
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HISTORY OF RETRACTABLE ROOFS
The first well known large-scale movable roof in history is the Collosseum in the Ancient Roman era. A rooffor the sunlight protection could be moved manually. The detailed roof system is unknown, but theirexistence has been confirmed by documents from that period or remains of their columns in the ruins. Itsdimension is assumed to be between 5700 and 23000 m2; anyway this is quite large and such enormousretractable roofs have not been constructed again until the modern times.
FIGURE 4-ROOF SUN PROTECTION OF THE COLLOSSEUM
Since 1930s small movable roofs have been constructed, the roof for a swimming pool in Rotterdam,Netherlands, constructed in 1935, is probably the first modern convertible roof.
Pittsburgh Civic Arena in the United States in 1961 is considered as the first large scale movable roof (127mspan length) that could be operated for opening and closing based on modern technology. Since then a lot
of big stadiums have started to be covered with movable roofs.
At about the 60’s the membrane retractable roofs had also appeared. The most used system since then isthe one which is folded in one single point.
The earliest representative system of that kind of folding movement was a structure based in a single mast.This system found its last application in the Montrèal Stadium, which is the biggest single mast structurefor retractable roof that has ever been built. In this particular case the huge mast caused delay in the buildingprocess besides high cost of realization and maintenance.
In 1989 came up an alternative to the single mast system, Jӧrg Schlaich build in Zaragoza the firstretractable roof with a spoked wheel structure. After him, different projects have been built with the sameor similar basic system.
In the following graph a relation
between the area covered by the
retractable roof and the year of
construction can be seen.
It’s remarkable that the area
covered is increasing from 1990,
where the first spoked wheel
appeared, with exception of
Montreal in the 80’s.FIGURE 5- COVERED AREA DEVELOPMENT IN RETRACTABLE ROOFS
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T RANSITION DIAGRAM FROM SINGLE MAST SYSTEM INTO THE SPOKED - WHEEL.
FIGURE 6-DIAGRAM OF TRANSITION FROM SINGLE MAST TO A SPOKED WHEEL
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PROJECT OVERVIEW A chronological overview of some projects of remarkable importance split first for the sort of structure and
secondly for the movement of the membrane.
SINGLE MAST STRUCTURE
A single mast is a structure composed for one main structural element, a mast is a pole or a column from
which commonly hang the membrane.
FOLDING / BUNCHING - HORIZONTAL TRANSLATION TO THE CENTER
S UMMER F ESTIVAL IN B AD H ERSFELD , G ERMANY .
FIGURE 7- RETRACTABLE ROOF IN THE SUMMER FESTIVAL IN BAD HERSFELD
Year of execution: 1959Engineer/Designer: Frei Otto
Kind of system: Single Mast
Area: Approx. 1300m²Other specifications: Prestressed only by the tractors. Renovate in the 1993 and still in use
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B OULEVARD C ARNOT IN P ARIS , F RANCE
Year of execution: 1967Engineer/Designer: Roger Taillibert
Kind of system: Single Mast
Area: Approx. 1800m²Other specifications: Combination of a winch and a tractor system
M ULTIMEDIA STADIUM IN M UNICH , G ERMANY .
FIGURE 9-SKETCH OF THE MULTIMEDIA STADIUM IN MUNICH
Year of the design: 1970
Engineer/Designer: Frei Otto
Kind of system: Single MastArea: Approx. 60000m²
Other specifications: Not executed due to fund problems.The mast was designed to be 180m. high with a diameter of 5-6m.It is the largest designed covered surface by a fabric membrane roof
FIGURE 8- RETRACTABLE ROOF IN BOULEVARD CARNOT
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M ONTRÉAL STADIUM IN M ONTRÉAL, C ANADA.
Figure 10-Montreal stadium in montreal with a Single mast construction
Year of design-exec: 1972-1978Engineer/Designer: Roger Taillibert-LavalinKind of system: Single Mast
Area: Approx. 20000m²Other specifications: The tower from which the roof was suspended was 168m. high
It signed the end of the single mast construction.
SPOKED WHEEL STRUCTURE
A spoked wheel could be defined as a self-contained structure which consists of a compression ring, acentral hub and radial tension spokes.
The morphology of combination rings comes fromthat a hanging roof with one outer compressionring and one inner tension ring is too flexibleagainst vertical loads. Therefore are two possiblefundamental forms: one consists of onecompression ring and two tension rings. And theother has two compression rings and one tensionring. In both cases a central hub of a spokedwheel could be replaced by a tension ring, which
is required for roofs that demand a centralopening.
FIGURE 11-DEVELOPMENT OF THE SPOKED WHEEL
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FIGURE 12- RETRACTABLE ROOF OF THE BULL-FIGHT
RING
FOLDING / BUNCHING - HORIZONTAL TRANSLATION TO THE CENTER
B ULL- FIGHT RING IN Z ARAGOZA, S PAIN .
Figure 13-Bull-fight ring in zaragoza
Year of execution: 1989Engineer/Designer: Jörg Schlaich and Rudolf Bergermann
Kind of system: Spoked wheelArea: Approx. 1000m²
Other specifications: The shape of the ring is a circle.First spoked wheel system builtThe roof has been removed after about two years cause the wind damages
R OTHENBAUM
S TADIUM IN
H AMBURG
, G
ERMANY
Year of execution: 1999Engineer/Designer: Werner SobekKind of system: Spoked wheel
Area: Approx. 3000m²Other specifications: Asymmetric positioning of the inner roof.
FIGURE 14-RETRACTABLE ROOF OF THE
ROTHENBAUM STADIUM FIGURE 15- ROTHENBAUM STADIUM
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S TADIUM BC P LACE IN V ANCOUVER , C ANADA.
FIGURE 19- RETRACTABLE PNEUMATIC ROOF OF THE STADIUM BC IN VANCOUVER
Year of execution: 2013
Engineer/Designer: Jörg Schlaich, Rudolf BergermannKind of system: Spoked wheelArea: Approx. 8500m²
Other specifications: The membrane is composed of inflatable cushions.
FOLDING / BUNCHING - HORIZONTAL TRANSLATION TO THE PERIPHERY
B ULL- FIGHT R ING IN J AÉN , S PAIN
Year of execution: 1998
Engineer/Designer: F. Escrig and J. SanchezKind of system: Spoked wheelArea: Approx. 3,000 m²
Other specifications: Demolished one year later cause wind damages
FIGURE 20 - RETRACTABLE ROOF IN BULL-FIGHT RING IN JAEN
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TRANSLATIONAL SYSTEMS
Into this group could be encompassed very different and varied sorts of structures, the common parameter
between them is the movement of how they deploy.
FOLDING / BUNCHING - HORIZONTAL TRANSLATIONAL
O PEN AIR THEATER J AEN , S PAIN .
Year of the design: 1998
Engineer/Designer: F. Escrig and J. SanchezKind of system: Sliding foldable arches
Area: -Other specifications: Closing time less than 20 minutes
FIGURE 21- RETRACTABLE ROOF OF OPEN AIR THEATER IN JAEN
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O PEN AIR THEATER IN T ECKLENBURG , G ERMANY .
Year of the design: 1993Engineer/Designer: Carl Nolte
Kind of system: Rigid frame
Area: Approx. 1200 m²Other specifications: Pneumatic system
O PEN S UMMER T HEATRE B URGAS , B ULGARIA.
Year of the design: 1998Engineer/Designer: Proremus Ltd., Tanev and Partners Ltd.
Kind of system: Sliding foldable archesArea: 1,600 m2 Other specifications: Closing time less than 20 minutes
FIGURE 22- RETRACTABLE ROOF OF OPEN AIR THEATER IN TECKLENBURG
FIGURE 23- RETRACTABLE ROOF OF THE OPEN SUMMER THEATER IN BURGAS
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O PEN AIR THEATER IN H ANOI , V IETNAM .
Year of the design: Still designing phaseEngineer/Designer: Nicolai Kugel
Kind of system: Arches and columns
Area: Approx. 2000m²Other specifications: Not executed
O PEN AIR THEATER LAVIS , I TALY
Year of execution: 2012
Engineer/Designer: Alfred ReinKind of system: Arches and columns
Area: Approx. 800 m²Other specifications: The membrane is manually moved
Low cost of realization
FIGURE 24-RENDER OF A RETRACTABLE ROOF IN OPEN THEATER IN HANOI
FIGURE 25-RETRACTABLE ROOF OF OPEN AIR THEATER IN LAVIS
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DESIGN CHOICE
As a consequence of the small research described above and the requirements and limitations of the
project, the shown designs have come up. The left the called “Egg-shaped spoked wheel” and at the right
a composition of 2 spoked wheel.
It has been chosen to develop further the “Egg-shaped spoked wheel”, firstly because it has been
understood that a single structure would provide a cleaner design with lighter and a smaller visual impact,
therefore would reach a higher level of achievement, in terms of project requirements. And secondly,
because been this project a part of a learning portfolio, it did provide the opportunity to learn a such of
new structure system, since no literature has been found about anti symmetric spoked wheel structures.
FIGURE 26-COMPARISON BETWEEN THE TWO DESIGN PROPOSALS
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REFERENCE PROJECT The retractable roof of the Fortress in Kufstein, built in 2006. Has been looked at as a Reference. First it
took our attention for its architectural beauty and after appealed us the static principles of it. Moreover ithas been very convenient to have a comparable example of the design.
The Brochure that has been used as material can be also found in the annex A2
FIGURE 27-DAY LIGHT UPPER VIEW OF THE RETRACTABLE ROOF OF THE
FIGURE 27- NIGHT LIGHT UPPER VIEW OF THE RETRACTABLE ROOF OF THE
FORTRESS IN KUFSTEIN
FIGURE 28-DAY LIGHT UPPER VIEW OF THE RETRACTABLE ROOF OF THE
FORTRESS IN KUFSTEIN
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“EGG-SHAPED” SPOKED WHEEL-DESIGN PROPOSAL The main characteristic of a spoked wheel structure is that it is a close structure. Commonly a spoked wheel
is designed with a circular geometry. In this project, due to the geometry of theater, an “Egg-shaped” spoked
wheel has been chosen.
The structures is radially divided every 18º, consequently is composed of 20 columns, from which 2 are
floating columns. From every column an upper and lower roofing cable hang directed to the center, where
a central hub collects them in a single point. Vertical cables connect the upper and lower cables. Moreover
for static equilibrium every column has an external radial beam at the height of the lower roofing cable. This
radial column links the exterior upper and lower external cables, the former comes from the top outer part
of the column, and the latter from the outer face of the joint between the column and the tension ring. As
mentioned, the 20 columns are linked each other by two horizontal rings, the compression ring which is
located at the same height of the lower roofing cable, and the tension ring which is placed at the same
height that the external lower cable.
The membrane which will be the covering, will be foldable from the perimeter to the center, and will hang
from the lower roofing cable.
Eleven groups of members characterizes the spoked wheel structure. For a succeeding clear understanding
the elements groups will be listed.
P1: U PPER ROOFING CABLES
P2: LOWER ROOFING CABLES
FIGURE 29-UPPER ROOFING CABLES
FIGURE 30- LOWER ROOFING CABLES
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P3: V ERTICAL ROOFING CABLES
P4: C OLUMNS
P5: R ADIAL BEAMS
FIGURE 31-VERTICAL ROOFING CABLES
FIGURE 32- COLUMNS
FIGURE 33- RADIAL BEAMS
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P6: E XTERNAL LOWER CABLES
P7: C OMPRESSION RING
P8: T ENSION RING
FIGURE 34- EXTERNAL LOWER CABLES
FIGURE 35- COMPRESSION RING
FIGURE 36- TENSION RING
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P9: W IND BRACES
P10: C ENTRAL HUB
P11: E XTERNAL UPPER CABLES
FIGURE 37- WIND BRACES
FIGURE 38- CENTRAL HUB
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EQUILIBRIUM OF THE STRUCTURE
In the initial situation, only prestress and dead load act. Inthis case load the upper and lower cables are in tension
(blue arrows on the graphic).
The reaction (green arrow on the graphic) on the node A,
due to the tension created for the prestress in the upper
cable is equilibrated separately, the vertical component
goes into the column creating compression (red arrows on
the graphic), and the horizontal component is equilibrate
with the external upper cable.
1 1 11
1 1 11 1
0 sin sin 0
0 cos cos
0
Fx P P
Fy P P Ry
M
φ ϕ
φ ϕ
∑ = → − =
∑ = → + =
∑ =
where Ry1 is the compressive reaction applied to the
pillar.
The sum of moments is equal to zero because all force meet in a single point.
On the node B, the tension from the external upper and lower cable should be equilibrated, the vertical
components of each other are in equilibrium while the horizontal component of both are equilibrated for the
radial beam which is in compression.
FIGURE 40- STRESS FLOW
SYSTEM
FIGURE 39 - EQUILIBRIUM OF THE
NODE A
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11 6 1
11 6
0 sin cos
0 cos sin 0
0
Fx P P Rx
Fy P P
M
ϕ α
ϕ α
∑ = → + =
∑ = → + =
∑ =
where Rx1 is the compressive reaction applied to the
radial beam.
The vertical reaction on the node C comes from a
compression of the upper part of the column and from
the vertical component of the lower roofing cable, both
give a compression in to the lower part of the column.
Regarding the horizontal reaction components, a
compression coming from the radial beam can beadded to a tension force from the lower roofing cable,
those forces are directed through the compression ring,
as if it would be an arch. Been the compression ring a
close structure, the reaction of those virtual arches
equilibrate each other.
2 2 11 6 2
1 1 11 2 2 2
0 cos sin cos
0 cos cos sin
0
Fx P P P Rx
Fy P P P Ry
M
φ ϕ α
φ ϕ φ
∑ = → + + =
∑ = → + − =
∑ =
Where Rx2 is the compressive reaction applied to the compression ring, and Ry2 is the compressive
reaction applied to the pillar.
FIGURE 41 - EQUILIBRIUM IN NODE B
FIGURE 42 - EQUILIBRIUM IN NODE C
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On the node D, the equilibrium of vertical forces is given by the sum of the compression given in the upper
part of the column, a vertical component of the tension given by the external lower cable and a compressive
reaction that acts in the lower part of the column, this reaction is equal to the axial force given by the
foundation. Regarding equilibrium of horizontal forces in the node D
Where Rx3 is the tensile reaction applied to the
tension ring, and Ry3 is the compressive reaction
applied to the pillar.
FLOATING COLUMNS The floating columns have been introduced in the design for geometric reasons. Due to the triangular shape
of the theater, the columns where located inside the perimeter of the theater, as already mentioned in
section PROJECT REQUIREMENTS no structural element should be placed in the inner perimeter in order
to not affect the appearance of this monumental space.
As an alternative to increase the covered area, the design of a floating is introduced.
The equilibrium of those floating columns is different than the general, in node the wind braces take the
vertical reaction before taken by the column, moreover the stiffness of the connection between the
compression ring and the floating column contributes.
6 3
1 1 11 2 2 6 3
0 cos
0 cos cos sin sin
0
Fx P Rx
Fy P P P P Ry
M
α
φ ϕ φ α
∑ = → =
∑ = → + − − =
∑ =
9 9
1 1 11 2 2 6 9
0 sin sin 0
0 cos cos sin sin 2 sin
0
Fx P P
Fy P P P P P
M
β β
φ ϕ φ α β
∑ = → − =
∑ = → + − − =
∑ =
FIGURE 43 EQUILIBRIUM OF THE FLOATING PILLAR
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SHADOW PATH It has been taken into consideration that been a foldable covering, the membrane may be either open or
close. To make sure that this fact will not disturb the performances, the shadow path has been checked.
A utilization of the space from March to September has been supposed. It can be seen from the graph
below that, the fact that the membrane is stored in the center will hardly disturb any performance.
Shadow path during a day in July
FIGURE 44 - DAILY SHADOW PATH IN JULY
Shadow path overview March-September
FIGURE 45 - SHADOW PATH DURING THE WORKING SEASON
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ANALYSIS
FORM FINDING
The form-finding process start defining the area to be covered, the kind and place of the supports and the
boundary conditions under which the membrane should be in equilibrium. The area to be covered is the
internal ellipse that the structure forms, the supports are determined as hinges in the location where the
columns are placed. And the boundary conditions at which the membrane should be in equilibrium are
found from load paths.
In order to simplify the process the form finding has been done with the cables and not with the membrane
which in practice makes no difference.
The load paths that determine the form-finding, are shaped from the mean loads that the structure has to
deal with, overall wind pressure and dead load. Due to the geometry of the design, the wind load path onthe cables has a triangular shape with its highest point on the external part of the spoked wheel, which
represents that the area of membrane which supports is bigger.
Note that the path load for the lower cable has been taken in the upward direction.
FIGURE 47 - STRAIGHT CABLES WITH PATH LOADFIGURE 46- FORM FINDING CABLES
FIGURE 48- ASYMMETRIC SHAPE
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Should be noticed that since the hub it is not placed on the highest point, the height of the hub plays a role
to avoid interference between upper and lower cables.
The upper cables are symmetric to the lower ones respect to the XY plane because equivalent and opposite
wind loading conditions are applied.
FIGURE 49- FORM FINDING UPPER AND BOTTOM CABLES
The vertical cables disposed between the upper and lower cable let them collaborate defining the final
shape. The amount of hangers depends on the length of the cables, a distance of about 3.50m between
the hangers has been determined, that distance has been taken from the referenced project.
FIGURE 50- UPPER, BOTTOM AND VERTICAL CABLES AFTER FORMFINDING
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LOADING CONDITIONS
Due to the nature of the structure, external loads have a higher impact than conventional building structures,
does should lead to an accurate study of external loading patterns.
In this case the governing external load are wind forces and the prestress, that is because the design itself
allows the covering to be deployable in winter time, therefore snow will not be taken into account.
WIND LOAD
Wind actions act directly as pressures on the external surfaces, in the case of an open structure will also
act directly on the internal surface. Pressures will result into forces normal to the surface in which they are
acting.
In order to get a pressure acting on the fabric, input may be collected from the location where the project is
located. In accordance with EN-1991-1-4:2005+A1, Eurocode 1: Actions on Structures – Part 1-4: General
Actions, Wind actions. Section 4.2 and with EUROCODE 1991.Stuwdrukken berekenen volgens nieuwe
norm windbelasting.Bowen met Staal 201
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In accordance with the EN 1991-1 and with the Dutch national Annex, the wind velocity is found, note that
an interpolation for the season coefficient has been made, expecting to have a complete use of the structure
from middle April until middle September.
0 24.5 /
1 0.9 24.5 22, 05 /
b
b
v m s
v m s
=
= ⋅ ⋅ =
From the Dutch national Annex is found that the peak velocity pressure q p is described as follows.
where:
ρ is the air density defined as:
ce is the exposure factor defined as:
where:
kr is the terrain factor described as:
g is the peak factor described as:
Iv(z) is the wind turbulence intensity at
height z,
described as:
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z0 is the roughness length described as:
In accordance with EN-1991-1-4:2005+A1, Eurocode 1: Actions on Structures – Part 1-4: General Actions
– Wind actions. Section 4.3 table 4.1 – Terrain categories.
In relation with the mentioned parameters:
2
0.07 17 10.19(0.3 / 0.05) ln 1 2 3.5 2.07
170.3ln
0.3
ec
= ⋅ + ⋅ =
2
2
12.06 1.25 22.05 628.18
2 p
N qm
= ⋅ =
The wind pressure is described as follows from EN-1991-1-4:2005+A1, Eurocode 1: Actions on Structures
– Part 1-4: General Actions – Wind actions. Section 5.2 Wind pressure on surfaces
, p p net W q C = ⋅
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Where:
qp is the peak velocity pressure
Cp,net is the net pressure coefficent
The roof has been subdivided into load (or wind exposure) zones by means radial or concentric partition
lines. The subdivision follows a study of the wind tunnel based results, from European Design Guide for
Tensile Surface Structures: Appendix 2: Cp Values for open stadium roofs. Section A2.4: Standardisation
of roof zones.
The radial subdivision has been done into radial zones of 18° spacing each. For the concentric zoning, the
following scheme has been used:
Concentric ring Number Percentage
Outer ring Ring A 12% of the roof depth
First inner zone Ring B 26% of the roof depth
Second inner zone Ring C 62% of the roof depth
The following numbering scheme for the roof zones has been used:
1. Concentric rings a,b,c.
2. Ring surface elements ( radial ) 1 to 20 ( each ring )
Code derived from the scheme:
e.g. “ a20 stands for ring a ( outer ring ), element 20
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The location of the zones is shown below:
FIGURE 51- DIVISION OF THE ZONE PRESSURES
For the determination of the coefficient Cp,net the canopy roofs settlement has been used, in accordance
with EN-1991-1-4:2005+A1, Eurocode 1: Actions on Structures – Part 1-4: General Actions – Wind actions.
Section 7.3 – Canopy roofs
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For
complex roof shapes the most accurate method of wind pressure coefficient determination is demonstrated
to be the wind tunnel test. Anyway, being a double curved roof, it can be considered as a double pitch
canopy roof, analyzing every direction of the wind, assuming different values of ϕ, which leads to different
values of Cp,net.
Since the structure presents elliptic anti-symmetric shape, five different wind direction cases has been
studied.
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For each wind direction case, the degree of obstruction has been derived, and a value for the roof slope
has been computed. The roof slope has been found approximating the curved roof to a double pitch roof
for every wind direction.
Therefore for every wind case a perpendicular to the wind direction cross section has been made and a
respective value of α and ϕ have been determined. In case of an antysimmetric roof slope approximation ,
the mean value of the two angles has been assumed.
Can be seen from the graphs below, that the area of the cross section taken into account starts from theground level, the reason of this assuption, is to do not underestimate the blockage coefficient, which leads
the factor towards safety.
FIGURE 52- WIND DIRECTIONS
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Wind direction 1:
α=1
Not obstructed area= 689.1 m2 Obstructed area = 327.5 m2
Total area = 1016.6 m2
ϕ=0.32
Wind direction 2:
α=6
Not obstructed area= 818.8 m2
Obstructed area = 233.6 m2
Total area = 1052.4 m2
ϕ=0.22
Wind direction 3:
α=5.5
Not obstructed area= 779.3 m2
Obstructed area = 531.4 m2
Total area = 1310.7 m2
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ϕ=0.40
Wind direction 4:
α=6
Not obstructed area= 578.8 m2
Obstructed area = 634 m2
Total area = 1212.8 m2
ϕ=0.52
Wind direction 5:
α=7
Not obstructed area= 672.4 m2
Obstructed area = 233.6 m2
Total area = 1017.3 m2
ϕ=0.66
Since the angle α varies between 5° and 7°, the values of Cp,net in the following table have been taken for
every wind case in the line related to α=5°
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In order to find out the Cp,net values related to the different ϕ cases listed above, a linear interpolation has
been done.
Zone A Zone B Zone C Zone D
Cp,net Cp,net Cp,net Cp,net
Maximum all ϕ 0,6 1,8 1,3 0,4
Minimum ϕ=0 -0,6 -1,4 -1,4 -1,1
Wind case Minimum ϕ=1 -1,3 -2,0 -1,8 -1,5
1 ϕ 0,3 -0,8 -1,6 -1,5 -1,2
2 ϕ 0,2 -0,8 -1,5 -1,5 -1,2
3 ϕ 0,4 -0,9 -1,6 -1,6 -1,3
4 ϕ 0,5 -1,0 -1,7 -1,6 -1,3
5 ϕ 0,7 -1,1 -1,8 -1,7 -1,4
As mentioned above, for every wind case, the roof has been subdivided into load (or wind exposure) zones
by concentric partition lines. Following the schematization of the double pitch canopies, the net pressure
zone coefficients correspond to the concentric area with this order:
Concentric zone A = Zone C
Concentric zone B = Zone AConcentric zone C = Zone D
Therefore the wind pressure related for every wind direction has been
listed. Furthermore the maximum uplift Cp,net value, ( -1.8) found by the linear interpolation has been used
to find the overall uplift wind pressure and respectively with the overall downward wind pressure (1.3). The
following values are listed in N/m2.
Wind CaseUplift pressure Downward pressure
Zone A Zone B Zone C Zone A Zone B Zone C
1 -959,9 -517,6 -771,4 816,6 376,9 251,3
2 -934,7 -473,6 -746,3 816,6 376,9 251,3
3 -980,0 -552,8 -791,5 816,6 376,9 251,3
4 -1010,1 -605,6 -821,7 816,6 376,9 251,3
5 -1045,3 -667,1 -856,8 816,6 376,9 251,3
6 -1045,3 816,6
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FIGURE 53- WIND PRESSURES
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PRESTRESS
The level of prestress is fundamental for the behavior of the structure, because of that, an intense research
for the ideal prestress has been carried out.
On a spoked wheel system, of the characteristics of our design, the prestress in the upper cable, lower
cable, external cables, tension ring and wind braces have a close relationship creating a loop. Moreover,
due to the asymmetric “Egg-shape” the relation between the prestress on the different cables is not equal
but varies in function of their length.
FIGURE 55- ELEMENTS AFFECTED IN ONE COLUMN FOR THE
PRESSTRESS
In order to find a prestress equilibrium between the different elements that are interrelated, an interaction
thermal method was employed.
This method consist in to apply a thermal loading to the elements, the elements independently of its length
will have an equal strain. Then the equilibrium of forces is found through a second order analysis, and in
accordance with the cross section and stiffness of the elements, a relation of stresses on the cables is
found.
th
th
T
F E
A
ε α
σ ε
= ⋅ ∆
= = ⋅
Taking that relation of stresses as a unit prestress and interactively increasing it so that no cables are loose
under the ULS load combinations brings the ideal prestress.
FIGURE 54- ASYMMETRY BETWEEN CABLES
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FIGURE 56- FORCE ON THE UPPER CABLE IN THE INITIAL SITUATION
FIGURE 57-FORCE ON THE LOWER CABLE IN THE INITIAL SITUATION
INITIAL SITUATION U PPER CABLE
LOWER CABLE
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FIGURE 58-FORCE ON THE EXTERNAL UPPER CABLE IN THE INITIAL SITUATION
FIGURE 59- FORCE ON THE EXTERNAL LOWER CABLE IN THE INITIAL SITUATION
E XTERNAL UPPER HANGER
E XTERNAL LOWER HANGER
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FIGURE 60- FORCE ON THE TENSION RING CABLE IN THE INITIAL SITUATION
FIGURE 61- FORCE ON THE WIND BRACES IN THE INITIAL SITUATION
T ENSION RING
W IND BRACES
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The cable members have been thermally loaded with a magnitude of -100 degrees Cº, and the factor
interactively found of the ideal prestress corresponds to 4.1 times the unit prestress.
This factor has been determined with the most critical case. That corresponds to the load combination48(for a further detail of this combination refer to the annex A3). Analyzing this combination the 10% of the
introduced prestress is still present on the most critical cables, which are the shortest. Because of the strict
relation between prestress on the different cables, that effectiveness of 10% of the prestress left is hardly
achievable over all elements.
FIGURE 62 - FORCE ON THE LOWER CABLE IN THE MOST CRITICAL SITUATION
DEAD LOAD
The dead load has been taken into consideration in two differentiated ways, every element modeled in GSA
has a determinate material properties which include density, also the geometry of the elements is set, it
can be considered that the compute of the dead load of the mentioned elements is quasi automatic. For
the elements not modeled in GSA, more specifically for the membrane, a conservative value of the density
has been taken 1.5 Kg/m2. As the membrane is not modeled, the dead load corresponding to every surface
has been logically divided to the correspondent supporting cables.
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GEOMETRY RELATIONS
It should be mentioned that the geometry of any members with exception to the “elliptic egg shape” has
been design to minimize internal forces.
The first relation of the geometry is given with the stress. Can be seen that the height of the upper part of
the column (from the compression ring upwards) is mainly given to increase the curvature in the upper and
lower cable, with this action is possible to decrease the horizontal reaction on the supports and the stress
in the cables.
The length of the radial beams increases the angle between the column and the external cables, that factdecrease substantially the force in the cable needed to equilibrate the nodes.
The second relation of the geometry is given with the stiffness, it is evident that the stiffness of the elements
is related with its geometry. Due to the nature of an iterative process, the cross sections change during the
design in order to achieve an optimum and effective cross section, should be noted, that any change in a
cross sectional element will affect the stiffness of the elements and consequently the strict relationship of
the prestress.
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CALCULATION
Tensile structural members have been realized by cable elements. For the homogeneity of the structure
profile, the choice of the cable cross section has been done following the most stressed cable for eachgroup of tensile members.
Cable elements are standardized product, which properties are given by the producer.
The Pfeifer Company Catalogue furnishes its produced elements features including the characteristic
breaking load ( NB,K), and the max allowable load (NR,D).
The solicitation values for each group of tensile members have been found, by the GSA software
elaboration, which listed the maximum axial force. The maximum axial force used for the cross section
design has been taken by the result of the load combinations analyzed by the software.
You can find the axial force solicitation resume in the Annex A4.
E l e m e n t c o d e
Element
Designvalue
N u m b e r o f e l e m e n t s
Size
Charact.Breaking load
Limittension Metallic
crosssection
Weight
C o n s t r u c t i o n
Nominaldiameter
Ne,d Nb,k NR,d ds
KN KN KN mm2 Kg/m mm
P1Upperroofing
cables
883 1 PV-150 1520 921 1060 8,9VVS-
2
40
P2Lowerroofingcables
1372 2 PV-115 1170 709 808 6,8VVS-
235
P3Verticalroofingcables
70 1 PV-40 405 245 281 2,4VVS-
121
P6External
lowercables
2741 1 PV-490 4890 2964 3390 27,9VVS-
370
P8Tension
ring1856 1 PV-360 3590 2176 2490 20,5
VVS-3
60
P9
Wind
braches 1120 1 PV-195 1930 1170 1340 11,2
VVS-
2 45
P11External
uppercables
2098 1 PV-360 3590 2176 2490 20,5VVS-
360
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The tensile members are composed by cables and sockets. Pfeifer supplies geometry of each socket
corresponding to the relative cable.
FIGURE 63- CROSS SECTION OF THE CABLES
FIGURE 64- DIMENSIONS OF THE SOCKET
The connection between tensile members and the others elements of the structure is obtained by welded
steel plates. The steel plates and the corresponding weldings have been designed following the Eurocode
1993.1.8.2005. Section 3.13.1 type A, which has been used, gives geometrical requirements respect the
distance between the hole border and the steel plate edge respect to a fixed thickness.
The welding has been designed for every connection as filled welds respecting the Eurocode 1993.1.8.2005
guidelines in the section 4.5.3.2.
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Since the used material for the design is the steel S355 the βw factor has been chosen as equal to 0.9 for
every case.
UPPER ROOFING CABLES VERIFICATION Since the upper cables have different length, the angle between the cable and the column in the connection
is different as well. Therefore, the steel plate geometry is different for every cable; the following calculation
has been done for a generic plate related to the cable 1 and 20.
Socket geometry:
Type A B2 C L1 L B1 E ds db
mm mm mm mm mm mm mm mm mm
PV-150 160 77 105 98 295 70 82 40 64
Steel plate
Fe,d 883000 N Euro code
γM0 1 a> = 76,0 mm
fy 345 N/mm2
ø = db+ 2 66 mm c >= 54,0 mm
t 40 mm
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Welding
F ┴ 407692,7 N σ ┴ 145,605 N/mm2
Fǁ 169437 N τǁ 60,5132 N/mm2
a = 0,5 t min 14 mm
t plate 40 mm[ σ2 + 3τ2 ]0,5 < fu/(βw γM2)
t support 40 mm
γM2 1,25 179,41 < 435,56
βw 0,9σ ┴ < 0,9 fu/ γM2
fu 490 N/mm2
Lenght > 200 mm 145,6045372 < 352,8
s 20 mm
For the steel plate detail reference: see plan 04-A1 ; 05-A1 ; 06-A1
The following table lists the resume of the calculation done for every tensile member. The specification of
the calculations can be found in the Annex 5.
Code Element description
AxialForce
Hole ø
Thicknessplate
Thicknesssupport
amin
cmin
Weldingthroat
thickness
Reference welding
length
KN mm mm mm mm mm mm mm
P1 Upper roofing cables 883 66 40 40 76 54 14 200
P2 Lower roofing cables 1372 66 40 55 69 47 14 200
P6
External lower cables( lower )
2741 120 40 40 179 139 14400
External lower cables(upper )
450
P8 Tension ring 1856 100 40 40 134 100 14 250
P9Wind branches ( lower )
1120 75 40 40 90 65 14200
Wind branches (upper ) 200
P11
External upper cables( lower )
2098 100 40 40 143 109 14350
External upper cables(upper ) 400
PILLARS VERIFICATION:The 20 pillars have been realized by a circular hollow cross section element s355 with an external diameter
of 711 mm and a thickness of 40 mm and 55 mm. Indeed the thickness increases from 40 to 55 mm in the
range that is subjected by the compression ring pressure. The profiles are 33 meters length c.t.c. joined to
the steel plate foundation by a pin connection. Two of them are floating pillars.
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Stressed value are given by the GSA elaboration which can be found in the Annex A4.
The pillars verifications have been done following the guidelines of the Eurocode 1993.1.1.2005.
PROFIL PROPERTIES
d t fy fu A Aholes Anet ε class section
partial factor
mm mm N/mm² N/mm² mm² mm² mm² 0 1 2
711 40 335 470 84320 0 84320 0,83755151 1 1,00 1,00 1,25
E Ixx Iyy Wel,xx Wel,yy Wpl,xx Wpl,yy
N/mm² mm4 mm4 mm3 mm3 mm3 mm3
210000 4762423729 4762423729 13396409,9 13396409,9 18030973,33 18030973,3
Tension verification according to Eurocode 1993.1.1.2005. Section 6.2.3.
TENSION
verified NEd Nt,Rd Npl,Rd Nu,Rd YES N N N N
0,00286399 80900 28247316,2 28247316,2 28534005,4
Compression verification according to Eurocode 1993.1.1.2005. Section 6.2.4.
COMPRESSION
verified ? NEd Nc,Rd
YES N N
0,11509058 3251000 28247316,2
Bending moment verification according to Eurocode 1993.1.1.2005. Section 6.2.5.
BENDING MOMENT
verified ? MEd Mc,Rd
YES Nmm Nmm
0,45344859 2739000000 6040376067
Shear verification according to Eurocode 1993.1.1.2005. Section 6.2.6.
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SHEAR
verified ? VEd Vc,Rd Vpl,Rd Av Vpl,T,Rd
YES N N N mm² N
0,08243393 536000 6502176,91 10382374,4 53680 6502176,91
Torsion verification according to Eurocode 1993.1.1.2005. Section 6.2.7.
TORSION
verified ? TEd TRd Ip WT
YES Nmm Nmm mm4 mm3
0,00067906 6095000 8975594650 9524847457 26792819,9
Bending and Shear verification according to Eurocode 1993.1.1.2005. Section 6.2.8.
BENDING AND SHEAR
verified ? MEd My,V,Rd ρ VEd Vpl,Rd
YES Nmm Nmm N N
0,45344859 2739000000 6040376067 0 536000 10382374,4
Bending and Axial force verification according to Eurocode 1993.1.1.2005. Section 6.2.9.
BENDING AND AXIAL F.
verified ? MEd My,N,Rd Mx,N,Rd n
YES Nmm Nmm Nmm
0,43289024 2739000000 5887326677 5887326677 0,11509058
Bending, Shear and Axial force verification according to Eurocode 1993.1.1.2005. Section 6.2.10.
BENDING,SHEAR AND
AXIAL F.
verified ? MEd My,V,N,Rd Mx,V,N,Rd n ρ VEd Vpl,Rd
YES Nmm Nmm Nmm N N
1,6578E-0853600
0588732667
7588732667
70,1150905
80
536000
10382374,4
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Uniform members in compression, buckling resistance verification according to Eurocode 1993.1.1.2005.
Section 6.3.1.
BUCKLING RESISTANCE OF MEMBERS
UNIFORM INCOMPRESSIO
N
verified ? NEd Nb,Rd χ φ λ̄ Ncr K L α i λ
YES N N N mm
0 , 7
3 6 7 3 8
3 2 5 1 0 0 0
4 4 1 2 6 9 3
0 , 1
5 6
3 , 6
5 8
2 , 4
1 9
4 8 2 9 2 3 2
1 , 3
7
3 3 0 0 0
0 , 2
1
2 3 7 , 6
5 5
7 8 , 6
5 7
RADIAL BEAMS VERIFICATION:
The 20 radial beams have been realized by a circular hollow cross section element s355 with an externaldiameter of 273 mm and a constant thickness 25 mm. The profiles are 6 meters length c.t.c. joined to the
steel pillars by pin connection. Radial elements result to be fully compressed by the external cables.
Stressed value are given by the GSA elaboration which can be found in the Annex A4.
The beams verifications have been done in accord to the guidelines of the Eurocode 1993.1.1.2005.
PROFIL PROPERTIES
d t fy fu A Aholes Anet ε class section partial factor
mm mm N/mm² N/mm² mm² mm² mm² 0 1 2
273 25 335 470 19478 0 19478 0,838 1 1,00 1,00 1,25
E Ixx Iyy Wel,xx Wel,yy Wpl,xx Wpl,yy
N/mm² mm4 mm4 mm3 mm3 mm3 mm3
210000 151267608 151267608 1108187,6 1108187,6 1542808,333 1542808,33
Compression verification according to Eurocode 1993.1.1.2005. Section 6.2.4.
COMPRESSION
verified ? NEd Nc,Rd
YES N N
0,41700587 2721000 6525087,94
Bending moment verification according to Eurocode 1993.1.1.2005. Section 6.2.5.
BENDING MOMENT
verified ? MEd Mc,Rd
YES Nmm Nmm
0,02966097 15330000 516840792
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Shear verification according to Eurocode 1993.1.1.2005. Section 6.2.6.
SHEAR
verified ? VEd Vc,Rd Vpl,Rd Av Vpl,T,Rd
YES N N N mm² N
0,00259278 6078 2344200,34 2398313,02 12400 2344200,34
Torsion verification according to Eurocode 1993.1.1.2005. Section 6.2.7.
TORSION
verified ? TEd TRd Ip WT
YES Nmm Nmm mm4 mm3
0,00011448 85000 742485694 302535215 2216375,2
Bending and Shear verification according to Eurocode 1993.1.1.2005. Section 6.2.8.
BENDING AND SHEAR
verified ? MEd My,V,Rd ρ VEd Vpl,Rd
YES Nmm Nmm N N
0,02966097 15330000 516840792 0 6078 2398313,02
Bending and Axial force verification according to Eurocode 1993.1.1.2005. Section 6.2.9.
BENDING AND AXIAL F.
verified ? MEd My,N,Rd Mx,N,Rd n
YES Nmm Nmm Nmm
0,00293763 15330000 399998928 399998928 0,41700587
Bending, Shear and Axial force verification according to Eurocode 1993.1.1.2005. Section 6.2.10.
BENDING,SHEAR AND
AXIAL F.
verified ? MEd My,V,N,Rd Mx,V,N,Rd n ρ VEd Vpl,Rd
YES Nmm Nmm Nmm N N
4,6178E-10 6078 399998928 399998928 0,41700587 0 6078 2398313,02
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Uniform members in compression, buckling resistance verification according to Eurocode 1993.1.1.2005.
Section 6.3.1.
BUCKLING RESISTANCE OF MEMBERS
UNIFORM INCOMPRESSIO
N
verified ? NEd Nb,Rd χ φ λ̄ Ncr K L α i λ
YES N N N mm
0 , 5
5 1 4 7 7
2 7 2 1 0 0 0
4 9 3 4 0 2 2
0 , 7
5 6
0 , 9
4 5
0 , 8
6 6
8 7 0 8 8 8 3
1 6 0 0 0
0 , 2
1
8 8 , 1
2 6
7 8 , 6
5 7
COMPRESSION RING VERIFICATION:
The 20 beams have been realized by a circular hollow cross section element s355 with an external diameterof 508 mm and a constant thickness 40 mm. The profiles length varies respect to the distance between the
relative pillars. For verification purpose, the longest length has been assumed, equal to 12.2 meters. The
connection between the compression ring beam and pillar has been realized by fillet weld.
Stressed value are given by the GSA elaboration which can be found in the Annex A4.
The beams verifications have been done in accord to the guidelines of the Eurocode 1993.1.1.2005.
PROFIL PROPERTIES
d t fy fu A Aholes Anet ε class section partial factor
mm mm N/mm² N/mm² mm² mm² mm² 0 1 2508 40 335 470 58811 0 58811 0,838 1 1,00 1,00 1,25
E Ixx Iyy Wel,xx Wel,yy Wpl,xx Wpl,yy
N/mm² mm4 mm4 mm3 mm3 mm3 mm3
210000 1621879126 1621879126 6385350,89 6385350,89 8782293,333 8782293,33
Compression verification according to Eurocode 1993.1.1.2005. Section 6.2.4.
COMPRESSION
verified ? NEd Nc,Rd
YES N N
0,41605851 8197000 19701555,8
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Bending moment verification according to Eurocode 1993.1.1.2005. Section 6.2.5.
BENDING MOMENT
verified ? MEd Mc,Rd
YES Nmm Nmm
0,20995434 617700000 2942068267
Shear verification according to Eurocode 1993.1.1.2005. Section 6.2.6.
SHEAR
verified ? VEd Vc,Rd Vpl,Rd Av Vpl,T,Rd
YES N N N mm² N
0,0005391 141100 261730496 7241358,02 37440 261730496
Torsion verification according to Eurocode 1993.1.1.2005. Section 6.2.7.
TORSION
verified ? TEd TRd Ip WT
YES Nmm Nmm mm4 mm3
0,09875683 422500000 4278185096 3243758252 12770701,8
Bending and Shear verification according to Eurocode 1993.1.1.2005. Section 6.2.8.
BENDING AND SHEAR
verified ? MEd My,V,Rd ρ VEd Vpl,Rd
YES Nmm Nmm N N
0,20995434 617700000 2942068267 0 141100 7241358,02
Bending and Axial force verification according to Eurocode 1993.1.1.2005. Section 6.2.9.
BENDING AND AXIAL F.
verified ? MEd My,N,Rd Mx,N,Rd n
YES Nmm Nmm Nmm
0,14685781 617700000 2279523475 2279523475 0,41605851
Bending, Shear and Axial force verification according to Eurocode 1993.1.1.2005. Section 6.2.10.
BENDING,SHEAR AND
AXIAL F.
verified ? MEd My,V,N,Rd Mx,V,N,Rd n ρ VEd Vpl,Rd
YES Nmm Nmm Nmm N N
7,6629E-09 141100 2279523475 2279523475 0,41605851 0 141100 7241358,02
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Uniform members in compression, buckling resistance verification according to Eurocode 1993.1.1.2005.
Section 6.3.1.
BUCKLING RESISTANCE OF MEMBERS
UNIFORM INCOMPRESSIO
N
verified ? NEd Nb,Rd χ φ λ̄ Ncr K L α i λ
YES N N N mm
0 , 4
4 5 6 7 6
8 1 9 7 0 0 0
1 8 3 9 2 2 6 7
0 , 9
3 4
0 , 6
3 8
0 , 4
6 9
8 9 4 5 7 4 6 3
0 , 5
1 2 2 6 0
0 , 2
1
1 6 6 , 0
6 6
7 8 , 6
5 7
CONNECTIONS
The welded connections between steel plates and hollow cross section elements have been verified inaccordance to Eurocode 1993.1.8.2005 section 7.4.
Indeed every connection has been verified for normal stress, bending moment and punching shear failure:
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Connection between steel plate of P1 ( Upper roofing cable ) and P4 ( Pillar ).
The detail can be found in plan 04-A1 Detail 1
Resistance of connection gusset plates to CHS members
h1 790 mm N1,Rd 3526667 N yes
d0 711 mm M1,Rd 2786066667 Nmm yes
η = h1/d0 1,111 Shear stress
σp,ED 14,100 N/mm2 564 N < 15935 N yes
fy,0 345 N/mm2
γM5 1
np 0,0409
kp 1
Moment lever 708,5 mm
Mp,1,Ed 315674429,3 Nmm
Wel 6,769E+12 mm3
N1,Ed 4,456E+05 N
Connection between steel plate of P2 ( Lower roofing cable ) and P4 ( Pillar ).
The detail can be found in plan 05-A1 Detail 3
Resistance of connection gusset plates to CHS members
li 600 mm N1,Rd 3245401 N yes
d0 711 mm M1,Rd 1,95E+09 Nmm yes
η = li/d0 0,844 Shear stress
σp,ED 13,197 N/mm2 528 N < 15473 N yes
fy,0 335 N/mm2
γM5 1
np 0,0394
kp 1
Moment lever 708,5 mmMp,1,Ed 224406899,8 Nmm
Wel 6,769E+12 mm3
N1,Ed 3,167E+05 N
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Connection between steel plate of P6 ( External lower cable ) and P4 ( Pillar ).
The detail can be found in plan 06-A1 Detail 4
Resistance of connection gusset plates to CHS members
li 835 mm N1,Rd 3570338 N yes
d0 711 mm M1,Rd 2,98E+09 Nmm yes
η = li/d0 1,174 Shear stress
σp,ED 28,991 N/mm2 1160 N < 15935 N yes
fy,0 345 N/mm2
γM5 1
np 0,0840
kp 1Moment lever 1052 mm
Mp,1,Ed 1018648099 Nmm
Wel 6,769E+12 mm3
N1,Ed 9,683E+05 N
Connection between steel plate of P6 ( External lower cable ) and P5 ( Radial beam ).
The detail can be found in plan 04-A1 Detail 2
Resistance of connection gusset plates to CHS members
li 840 mm N1,Rd 4883077 N yes
d0 273 mm M1,Rd 4,1E+09 Nmm yes
η = li/d0 3,077 Shear stress
σp,ED 28,866 N/mm2 1155 N < 15935 N yes
fy,0 345 N/mm2
γM5 1
np 0,0837
kp 1
Moment lever 708,5 mm
Mp,1,Ed 687161657 Nmm
Wel 1,116E+11 mm3
N1,Ed 9,699E+05 N
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Connection between steel plate of P11 ( External upper cable ) and P8 ( Pillar ).
The detail can be found in plan 04-A1 Detail 3
Resistance of connection gusset plates to CHS members
li 1440 mm N1,Rd 4157468 N yes
d0 711 mm M1,Rd 5,99E+09 Nmm yes
η = li/d0 2,025 Shear stress
σp,ED 6,775 N/mm2 271 N < 15935 N yes
fy,0 345 N/mm2
γM5 1
np 0,0196
kp 1
Moment lever 708,5 mm
Mp,1,Ed 276504183 Nmm
Wel 6,769E+12 mm3
N1,Ed 3,903E+05 N
Connection between steel plate of P11 ( External upper cable ) and P5 ( Radial beam ).
The detail can be found in plan 04-A1 Detail 2
Resistance of connection gusset plates to CHS members
li 580 mm N1,Rd 4225934 N yes
d0 273 mm M1,Rd 2,45E+09 Nmm yes
η = li/d0 2,125 Shear stress
σp,ED 41,970 N/mm2 1679 N < 15935 N yes
fy,0 345 N/mm2
γM5 1
np 0,1217
kp 1
Moment lever 708,5 mm
Mp,1,Ed 689866801 Nmm
Wel 1,116E+11 mm3
N1,Ed 9,737E+05 N
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Connection between steel plate of P8,P9 (Tension ring, Wind brances) and P8 ( Pillar ).
The detail can be found in plan 06-A1 Detail 4
Resistance of connection gusset plates to CHS members
li 562 mm N1,Rd 3305401 N yes
d0 711 mm M1,Rd 1,86E+09 Nmm yes
η = li/d0 0,790 Shear stress
σp,ED 62,785 N/mm2 2511 N < 15935 N yes
fy,0 345 N/mm2
γM5 1
np 0,1820
kp 1
Moment lever 708,5 mm
Mp,1,Ed 999988563,1 Nmm
Wel 6,769E+12 mm3
N1,Ed 1,411E+06 N
Connection between steel plate of P9 (Wind braces) and P8,P7 ( Pillar, Compression ring ).
The detail can be found in plan 06-A1 Detail 3
Resistance of connection gusset plates to CHS members
li 200 mm N1,Rd 2954093 N yes
d0 711 mm M1,Rd 5,91E+08 Nmm yes
η = li/d0 0,281 Shear stress
σp,ED 35,335 N/mm2 1413 N < 15935 N yes
fy,0 345 N/mm2
γM5 1
np 0,1024
kp 1
Moment lever 708,5 mm
Mp,1,Ed 200279460,2 Nmm
Wel 6,769E+12 mm3
N1,Ed 2,827E+05 N
The connection between the radial beam and the column has been designed as an hinged connection.
Indeed the joint is realized by an union of two welded steel plates 20 mm thick on the beam and a 40 mm
thick steel plate welded to the pillar. The plates are joined together by a pin. The tolerance between the
plates has been designed to be 2 mm. The bending moment assume an important value in the analysis of
antisymmetric wind load.
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See detail 3 plan 05-A1
The design node solicitant stress are:
Ned = 2767 KN
Ved= 6 KN
Med = 15.33 KNm
The following tables show the design geometry of the two steel plates welded to the radial beam and the
steel plate welded to the pillar according to the Eurocode 1993.1.8.2005. Section 3.13.1 type A.
Beam steel plate
Fe,d 1383500 N Eurocode
γM0 1 a> = 168,3 mmfy 345 N/mm2
ø = dp+ 2 102 mm c >= 134,3 mm
t 20 mm
Pillar steel plate
Fe,d 2767000 N Eurocode
γM0 1 a> = 168,3 mm
fy 345 N/mm2
ø = db+ 2 102 mm c >= 134,3 mm
t 40 mm
For the design of the pin, the guideline of the Eurocode 1993.1.8.2005 Section 3.13.2 has been followed.
The pin has been realized using a not treated bolt 8.8.
Characteristics of the pin:
Material:
R = 50 mm
fu = 800 N /mm2
fy = 800 N/mm2
Bending moment in the pin:
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( )2767000
, 40 8 40 30.48
Ed M KNm= + + =
Shear resistance of the pin:
,
0.6 (7850 800)3014
1.25V Rd
F KN ⋅ ⋅
= =
Bearing resistance of the plate and of the pin:
B,
1.5 80 100 3454140
1 Rd F KN
⋅ ⋅ ⋅= =
Bending resistance of the pin:
,
1.5 98174.77 64094
1 Rd M KNm
⋅ ⋅= =
Combined shear and bending resistance of the pin22
0.8715.33
94.25
2767
3015+ =
The welding connection has been designed as filled welds respecting the Eurocode guidelines in the section
4.5.3.2.
The perpendicular force is given by the bending moment Med on the beam.
Beam steel plates welding
F perpendicular 25,55 N σ perpendicular 0,04 N/mm2
F paralell 3000 N τ paralell 4,3E+00 N/mm2
a = 0,5 t min 7 mm
t plate 20 mm[ σ2 + 3τ2 ]0,5 < fu/(βw γM2)
t support 40 mm
γM2 1,25 7,42 < 435,56
βw 0,9σ perpendicular < 0,9 fu/ γM2
fu 490 N/mm2
Reference lenght 100 mm 0,04 < 352,8
s 10,0 mm
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Pillar steel plate welding
F perpendicular 25,55 N σ perpendicular 0,01 N/mm2
F paralell 6000 N τ paralell 1,4E+00 N/mm2a = 0,5 t min 14 mm
t plate 40 mm[ σ2 + 3τ2 ]0,5 < fu/(βw γM2)
t support 55 mm
γM2 1,25 2,47 < 435,56
βw 0,9σ perpendicular < 0,9 fu/ γM2
fu 490 N/mm2
Reference lenght 300 mm 0,01 < 352,8
s 20,0 mm
The welded connections between steel plates and hollow cross section elements ( beam and pillar) have
been verified in accordance to Eurocode 1993.1.8.2005 section 7.4.
Indeed every connection has been verified for normal stress, bending moment and punching shear failure:
Resistance of connection gusset plates to CHS members
li 300 mm N1,Rd39163
70N yes
d0 711 mm M1,Rd1,17E+09
Nmm yes
η = li/d0 0,422 Shear stress
σp,ED 223,917 N/mm2 0 N < 21910 N yes
fy,0 345 N/mm2
γM5 1
np 0,6490
kp 0,67891639
Moment lever 708,5 mm
Mp,1,Ed 15330 Nmm
Wel 8,727E+12 mm3
N1,Ed 2,555E+01 N
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The connection between the compression ring and the pillars have been designed had filled welding
connection, respecting the Eurocode guidelines in the section 4.5.3.2.
Welding
r 254 mm σ ┴ 55,3926 N/mm2
t 40 mm τǁ,v 4,5459 N/mm2
a 14 mm τǁ,torque 33,0376 N/mm2
γM2 1,25
βw 0,9[ σ2 + 3τ2 ]0,5 < fu/(βw γM2)
fu 490 N/mm2
F v 141,1 kN 85,47 < 435,56
Av 31038,93542 mm2
σ ┴ < 0,9 fu/ γM2Myy 617,6 kNm
Fmoment 1215,748031 kN 55,39260066 < 352,8
AM 31038,93542 mm2
Mxx 421,7 kNm
Zw 7,83439E-08 mm3
Since the compression ring by the welding transmits forces as axial, shear, bending moment ( in plane and
out of plain ), torsional moment, the verification for hollow section joints has been done.
The design forces magnitude con be found in the Annex A4.
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The verification is in accordance to Eurocode 1993.1.8.2005 section 7.4.2.
Resistance of connection between CHS members
F Ed 8197 kN N1,Rd*1 10690 kN
Myy, Ed 617,0 kNm N1,Rd*2 17484 kN
Mxx, Ed 465,6 kNm Mip,1,Rd*3 3875,92 KNm
r1 254,0 mm Mop,1,Rd*4 2819,73 KNm
t1 40,0 mm
r0 355,5 mm
t0 55,0 mm
A 58810,6 mm2 0,96 < 1,00
σp,ED 139,380 N/mm2 *1:Chord face failure resistance
fy,0 345 N/mm2 *2:Punching shear failure r.
γM5 1 *3:Chord face failure in plane moment r.
np 0,4040 *4:Chord face failure out of plane moment r.
kp 0,83
θ 90 º
β 0,7145
γ 6,4636
The connection between the column and the foundation plate has been designed as an hinged connection.
Indeed the joint is realized by an union of two welded steel plates 40 mm thick on the pillar and a 80 mm
thick steel plate welded to the pillar. The plates are joined together by a pin. The tolerance between the
plates has been designed to be 2 mm. The design bending moment is given by the load acting on the pin,
indeed during the analysis a 3d pin has been set as connection between column and foundation plate.
See Datail 5 plan 06-A1
The design node solicitant stress are:
Ned = 1930 KN
Ved= 254 KN
Med = 26.6 KNm
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The following tables show the design geometry of the two steel plates welded to the column and the steel
plate welded to the foundation plate according to the Eurocode 1993.1.8.2005. Section 3.13.1 type A.
Column steel plates
Fe,d 965000 N Eurocode
γM0 1 a> = 89,6 mm
fy 345 N/mm2
ø = db+ 2 82 mm c >= 62,3 mm
t 40 mm
Foundation steel plate
Fe,d 1930000 N EurocodeγM0 1 a> = 102,7 mm
fy 335 N/mm2
ø = db+ 2 82 mm c >= 75,3 mm
t 60 mm
For the design of the pin, the guideline of the Eurocode 1993.1.8.2005 Section 3.13.2 has been followed.
The pin has been realized using a not treated bolt 8.8.
Characteristics of the pin:
R = 40 mm
fu = 800 N/mm2
fy = 800 N/mm2
Shear resistance of the pin
verified Fv,Ed Fv,Rd
YES N N
0,52999839 1023000 1930194,53
Bearing resistance of the plateand the pin
verified Fv,Ed Fv,Rd
YES N N
0,61775362 1023000 1656000
Bending resistance of the pinverified Med,max MRd
YES Nmm N
0,55119833 26598000 48254863,2
Combined shear and bendingresistance of the pin
verified Fv,Ed Med Fv,Rd MRd
YES N Nmm² N N
0,5847179 1023000 26598000 1930194,53 48254863,2
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The anchors bolt have been verified according to Eurocode 1993.1.8.2005 section 3.6 table 3.4 regarding
the shear and tension resistance. The basic anchorage length has been designed according to
1992.1.1.2004 section 8.4.3.
Characteristics of the anchor bolts:
Material B 500 S
d = 20 mm
fu = 550 N/mm2
fy = 500 N/mm2
Tension resistance
verified Fv,Ed Ft,Rd n k2 Ft,Rd
YES N N N
0,00014453 89,9 622035,345 5 0,9 124407,069
Shear resistance
verified Fv,Ed Ft,Rd n αv Ft,Rd
YES N N N
0,73500646 254000 345575,192 5 0,5 69115,0384
design bond resistance
verified Fv,Ed Fb,Rd nfbd lb
YES N N mm
0,00030282 89,9 296880,506 5 1,89 500
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PHYSICAL AND VIRTUAL MODEL
FIGURE 66- CONTROL PANEL VIEW OF THE THEATER
FIGURE 65- PHISICAL MODEL
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FIGURE 68- SPECTATOR TOP VIEW OF THE THEATER
FIGURE 67- ENTRANCE VIEW OF THE THEATER
FIGURE 70- ACTORS VIEW OF THE THEATERFIGURE 69- ENTRANCE VIEW OF THE THEATER
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CONCLUSION AND RECOMMENDATIONS
The goal of this project was aimed at designing and visualizing the possibilities of a covering of The Open
air theater in Weert, Lichtenberg. This has been realized by carrying out both literature research and design
calculations.
Overall it can be stated that the goal has been achieved; a first step has been made in research, existing
project review of lightweight structures and in particular for retractable canopy roofing. Base on the
completed work, some conclusion and recommendations for future development to the project can be
made.
LITERATURE RESEARCH The literature reviewing has been done starting from what is a lightweight structure and which are the
possibilities of this constructive principle.
The first survey has been carried out by the history of the lightweight structure and how they have been
developed from the first one built on the Colosseum during the Roman Empier to nowadays.
The literature research has been of fundamental importance in order to understand the different solutions
that can be adopted for a retractable canopy. Indeed the choice of the spoked wheel as been thought to be
a valid answer to the project requests. The spoked wheel idea has come up after the viewing of the
reference of the Fortress in Kufstein, which, as the project demanding, shows a perfect harmony between
a light weight structure as the spoked wheel structure and the historical and artistic back ground of the open
theatre in Kufstein. Furthermore the reference project shown some important solutions which has been of
strongly important value for some fundamental alternatives adopted for the spoked wheel in Weert. First of
them the idea of floating pillars, in order to do not affect the theatre structure, being and historicalmonument.
The second step of the literature research has been to find out which are the limits for a not circular spoked
wheel. Indeed, wanting the realization of an only one spoked wheel to cover the entire theatre area, which
presents a trapezoidal shape, further survey have been implemented. The analyzed reference shown how
in stadiums and other canopies the circular shape can be converted in an elliptical shape, or as the central
hub can be moved from the central part to an anti-symmetric point respect to the structure. No one studied
reference project presents an anti-symmetric shape as an egg shape, which has been adopted for the
spoked wheel in Weert.
DESIGN CALCULATIONS The anti-symmetric egg shape spoked wheel presents many issues in terms of equilibrium and design
calculation.
The first of them has been met for the form finding process. In fact the necessity to find a balanced shape
for the membrane roofing and therefore the cable prestress have presented a needing of deep knowledge
to form finding principle and of the GSA software.
Indeed being every roofing cable different in length an intense research of the prestress using the thermal
load has been done, since the thermal load presents an equilibrium situation of stress.
Secondly since the inexperience of these kind of structures many solutions have been analyzed in order to
define the best geometry which aims to reduce the stress in the elements. In fact the clear relation between
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the angles of the cables and the axial force acting on the cables has played an important role in the first
step of design calculations.
Every calculation and verification has been effectuated in according to the Eurocode guidelines and for thedetermination of loads the Dutch National Annex.
RECOMMENDATIONS Successive developments for the project can be thought in terms of refining some element cross sections
in order to further enphasize the meaning of lightweight structure and a deeper design regarding the moving
connection details between membrane and roofing cables.
Indeed the verified steel plate elements for the connection of the cables although
being from a structural safety right they can be improved by reduzing the size and
therefore their weights.
Further more an initial survey has been started in order to realize a slender pillarwhich answer with the own shape to the stress requirements. The pillar which need
a farer investigation for the stability aspect has been design as hollow circular cross
section elements, but a natural and more beauty shape can be defined.
Successive design solutions need to be found for the connections between
membrane and roofing cable, which allow the retract of the roof and a prestress of it
when it is open. Also the hub connection needs a particular attention, since a specific
research need to be done in order to define the best shape solution between circular
or following the egg shape of the structure. Eventually some farther clarifications are
needed for the water collecting system and for the engines system act to the
membrane movement.
FIGURE 72 - PROPOSALFOR COLUMN DESIGN
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BIBLIOGRAPHY
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ACKNOWLEDGMENTS Special appreciation to our tutors Patrick and Arjan, our group mates, and the educational and professional
staff, who gave us support during our design.
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Master design 2Luca Martellotta and Sergio Moriche Quesada Summer semester 2014
APPENDICES
Annex A1: Soil profil
Annex A2: Reference project
Annex A3: Load cases
Annex A4:Forces on the structure
Annex A5: Plans
Annex A6: GSA file