egg shaped spoked wheel

8/10/2019 EGG SHAPED SPOKED WHEEL

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Authors:

L. Martellotta 0873376

E: [email protected]

S. Moriche Quesada 0885681


STRUCTURALDESIGN

Design Master Project 2014

Tutors:

ir. A.P.H.W. (Arjan) Habraken


prof.Dr.-Ing. P.M. (Patrick) Teuffel




Master design 2Luca Martellotta and Sergio Moriche Quesada Summer semester 2014

2

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




3

Literature research ......................... ...................... ......................... ......................... ...................... ...... 75

Design calculations ........................ ...................... ......................... ......................... ...................... ...... 75

Recommendations ......................... ...................... ......................... ......................... ...................... ...... 76

Bibliography .......................................................................................................................................... 77

Acknowledgments ................................................................................................................................. 78

Appendices ........................................................................................................................................... 79




4

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.




5

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.




6

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




7

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.




8

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




9

T RANSITION DIAGRAM FROM SINGLE MAST SYSTEM INTO THE SPOKED - WHEEL.

FIGURE 6-DIAGRAM OF TRANSITION FROM SINGLE MAST TO A SPOKED WHEEL




10

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




11

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




12

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




13

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




15

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


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




18

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


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




19

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




20

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




21

“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




22

P3: V ERTICAL ROOFING CABLES

P4: C OLUMNS

P5: R ADIAL BEAMS

FIGURE 31-VERTICAL ROOFING CABLES

FIGURE 32- COLUMNS

FIGURE 33- RADIAL BEAMS




23

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




24

P9: W IND BRACES

P10: C ENTRAL HUB

P11: E XTERNAL UPPER CABLES

FIGURE 37- WIND BRACES

FIGURE 38- CENTRAL HUB




25

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




26

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




27

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




28

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




29

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




30

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




31

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




32

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:




33

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 = ⋅




34

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




35

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




36

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.




37

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




38

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


ϕ=0.22

Wind direction 3:

α=5.5







39

ϕ=0.40

Wind direction 4:

α=6


Obstructed area = 634 m2


ϕ=0.52

Wind direction 5:

α=7




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°




40




41

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




42

FIGURE 53- WIND PRESSURES




43

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




44

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




45

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




46

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




47

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.




48

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.




49

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




50

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.




51




52




53

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




54

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.




55

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.




56

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




57

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.


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



210000 151267608 151267608 1108187,6 1108187,6 1542808,333 1542808,33


COMPRESSION


YES N N

0,41700587 2721000 6525087,94


BENDING MOMENT


YES Nmm Nmm

0,02966097 15330000 516840792




58


SHEAR


YES N N N mm² N

0,00259278 6078 2344200,34 2398313,02 12400 2344200,34


TORSION


YES Nmm Nmm mm4 mm3

0,00011448 85000 742485694 302535215 2216375,2


BENDING AND SHEAR


YES Nmm Nmm N N

0,02966097 15330000 516840792 0 6078 2398313,02




YES Nmm Nmm Nmm

0,00293763 15330000 399998928 399998928 0,41700587


BENDING,SHEAR AND

AXIAL F.


YES Nmm Nmm Nmm N N

4,6178E-10 6078 399998928 399998928 0,41700587 0 6078 2398313,02




59


Section 6.3.1.



N


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.


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



210000 1621879126 1621879126 6385350,89 6385350,89 8782293,333 8782293,33


COMPRESSION


YES N N

0,41605851 8197000 19701555,8




60


BENDING MOMENT


YES Nmm Nmm

0,20995434 617700000 2942068267


SHEAR


YES N N N mm² N

0,0005391 141100 261730496 7241358,02 37440 261730496


TORSION


YES Nmm Nmm mm4 mm3

0,09875683 422500000 4278185096 3243758252 12770701,8


BENDING AND SHEAR


YES Nmm Nmm N N

0,20995434 617700000 2942068267 0 141100 7241358,02




YES Nmm Nmm Nmm

0,14685781 617700000 2279523475 2279523475 0,41605851


BENDING,SHEAR AND

AXIAL F.


YES Nmm Nmm Nmm N N

7,6629E-09 141100 2279523475 2279523475 0,41605851 0 141100 7241358,02




61


Section 6.3.1.



N


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:




62

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 ).



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




63

Connection between steel plate of P6 ( External lower cable ) and P4 ( Pillar ).



li 835 mm N1,Rd 3570338 N yes



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 ).



li 840 mm N1,Rd 4883077 N yes



σp,ED 28,866 N/mm2 1155 N < 15935 N yes

fy,0 345 N/mm2

γM5 1

np 0,0837

kp 1


Mp,1,Ed 687161657 Nmm

Wel 1,116E+11 mm3

N1,Ed 9,699E+05 N




64

Connection between steel plate of P11 ( External upper cable ) and P8 ( Pillar ).



li 1440 mm N1,Rd 4157468 N yes



σp,ED 6,775 N/mm2 271 N < 15935 N yes

fy,0 345 N/mm2

γM5 1

np 0,0196

kp 1


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 ).



li 580 mm N1,Rd 4225934 N yes



σp,ED 41,970 N/mm2 1679 N < 15935 N yes

fy,0 345 N/mm2

γM5 1

np 0,1217

kp 1


Mp,1,Ed 689866801 Nmm

Wel 1,116E+11 mm3

N1,Ed 9,737E+05 N




65

Connection between steel plate of P8,P9 (Tension ring, Wind brances) and P8 ( Pillar ).



li 562 mm N1,Rd 3305401 N yes



σp,ED 62,785 N/mm2 2511 N < 15935 N yes

fy,0 345 N/mm2

γM5 1

np 0,1820

kp 1


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 ).



li 200 mm N1,Rd 2954093 N yes



σp,ED 35,335 N/mm2 1413 N < 15935 N yes

fy,0 345 N/mm2

γM5 1

np 0,1024

kp 1


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.




66

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


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:




67

( )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 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




68

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 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:


li 300 mm N1,Rd39163

70N yes

d0 711 mm M1,Rd1,17E+09

Nmm yes


σp,ED 223,917 N/mm2 0 N < 21910 N yes

fy,0 345 N/mm2

γM5 1

np 0,6490

kp 0,67891639


Mp,1,Ed 15330 Nmm

Wel 8,727E+12 mm3

N1,Ed 2,555E+01 N




69

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


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




76

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.




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

egg shaped spoked wheel

Documents