Nacalculatie van een StaalBeton Brug

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ERASMUS PROJECT ••••••••••••••••••••••••••••••••••••• PROJECTO ERASMUS

An DEJONGH June 2014

Modeling and structural analysis of the lower deck of Ponte D. Luis I

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ERASMUS PROJECT ••••••••••••••••••••••••••••••••••••• PROJECTO ERASMUS

An DEJONGH June 2014

T he s i s s ubm i t t e d t o ob ta in t h e Deg r ee o f M a s t e r o f S c i en c e i n E ng i n ee r i n g T e c hno l o g y C ons t r uc t i on

Modeling and structural analysis of the lower deck of Ponte D. Luis I

S uperv i so r ••• Ir. Ricardo Santos Co-‐supervisor ••• Ir. Stef Pillaert

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Preface

When the opportunity of studying abroad was offered by the Catholic University of Leuven, I didn’t hesitate for a moment, I needed to do this. The opportunity of discover a new country, city, culture and language and this all combined with an educational enrichment was too good to refuse.

Arriving all alone in Porto, I had no idea what to expect from the school, neither of the city itself and his inhabitants. Luckily for me I had a warm welcoming from the school. I quickly started to discover the city with all its beauty and hidden places. I was impressed by the friendliness of the Portuguese people and the pleasant atmosphere of the city.

Being taking out of my comfort zone, had a big impact on my live and it was not always easy. Although the experience I got in replace is indescribable. What I learned here on this short period of time I would never been able to learn if I had stayed at home. This experience taught me thinks that will have an influence on the rest of my live.

I was very honored to do a project about the Ponte D. Luis I, the symbol of Porto and at the same time a connection with my Belgian roots, since it was engineered by the Belgian engineer Théophile Seyrig. The structure is impressive and a real master piece. The fact that it is situated in the middle of the city center gave me the opportunity to make multiple visits which only increased my admiration and interested for this structure.

To realize this project I got the great help of some people which I would sincerely like to thank for that. First of all a special thank you to professor Ricardo Santos, to be my supervisor and helping me realize this project. It was not always easy and a lot of struggling came together with creating the 3D-‐model. I was always welcome in his office and a lot of hours were spend on realizing the correct model and solving my problems and questions. Thank you a lot professor Santos for all these hours of great help, sharing your interested and knowledge about bridges with me and teaching me about the Portuguese culture and traditions. Secondly a big thank you to professor Stef Pillaert to be my co-‐supervisor in Belgium and for helping me with my questions regarding the Scia Engineering software. I could always count on a quick, structured and comprehensive answer. I would also like to thank both schools, the catholic university of Leuven campus Ghent and Instituito Superior de Engenharia do Porto, for making this exchange possible. A special thank you to Guido Kips, Barbara Wauman, Ilse Roelands and Gudrun Van den Abeele for taking care of the paper work and maintained the international relations. Also Goreti Araújo , head of international office at ISEP, for the warm welcoming and guiding me through the school. Last but not least I would like to thank Pedro Ribeiro da Silva, for offering me a work spot in his architect office, Anarchlab. Roisin Slattery and Jan Reynders for correcting the grammatical English. A special thank you to engineer Thomas Alaert for the help that he offered me and Héctor Termenon Lopez for borrowing me his computer, when mine was broken and of course my parents and boyfriend for the emotional support.

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Copyright

The authors give permission to make this Master's thesis available, on paper and digital, for

consultation and to copy parts of the Master's thesis for personal use. Any other use falls under the

limitations of the copyright, especially with regard to the obligation of mentioning the source

explicitly on quoting the results of this Master's thesis.

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Os autores dão autorização para consultar esta tesenas versões impressa e digital, e para copiar

partes para uso próprio. Qualquer outro uso está sujeito aos limites estritos dos direitos de autor; em

partucular, é feita referência à obrigação de indicar a fonte explicitamente ao citar os resultados

desta tese.

Toelating tot bruikleen

De auteurs geven de toelating deze masterproef op papier en digitaal voor consultatie beschikbaar te

stellen en delen ervan te kopiëren voor eigen gebruik. Elk ander gebruik valt onder de strikte

beperkingen van het auteursrecht; in het bijzonder wordt er gewezen op de verplichting de bron

uitdrukkelijk te vermelden bij het aanhalen van de resultaten van deze masterproef.

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Abstract

This work is in sequel to the thesis “Modelling and structural analysis of Ponte D. Luis I” written by my colleagues Jan Daniëls and Sara Pieters in the academic year 2012-‐2013. In this dissertation the lower deck of the Ponte D. Luiz I bridge is modelled and structurally analyzed.

A full 3D model is created, based on the original survey conducted in 1954 and local observations made on visits to the site. The calculations were performed using a finite element model and the software Scia Engineering, with the loads defined on Eurocode 1, Parts 1 and 2.

Stresses and deformations —considering second order effects—were calculated in order to check the safety of this sub-‐structure to the loads defined on the Eurocodes. These values are closer to the loads the bridge has to endure nowadays than the loads it had to bear at the time of its construction.

keywords: Porto; Ponte D. Luis I; Theophile Seyrig; Iron arch bridge; finite element method

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Abstrato

Este trabalho, que surge na sequência da tese “Modelling and Structural analysis of Ponte D. Luís I” realizado pelos meus colegas Jan Daniels and Sara Pieters durante o ano lectivo 2012-‐2013. Nesta dissertação, o tabuleiro inferior da Ponte D. Luis I é modelado e analisado estruturalmente.

Foi criado um modelo 3D baseado no projecto original levado a cabo em 1954 e em observações efectuadas ao local. Os cálculos foram realizados utilizando o Método de Elementos Finitos e o software de Engenharia SCIA, com as cargas definidas no Eurocódigo 1, Partes 1 e 2.

O stress e as deformações – considerando os efeitos de segunda ordem – foram calculados de forma a verificar a segurança desta sub-‐estrutura em relação ás cargas definidas no Eurocódigo. Estes valores estão mais próximos das cargas que a ponte tem de suportar hoje em dia, do que as cargas que teve de suportar na altura da sua construcção.

Palavras-‐chave: Porto; Ponte D. Luís I; Theophile Seyrig; Ponte em arco de ferro; Método dos Elementos Finitos

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Abstract

Dit werk is een vervolg op de thesis “Modeling and structural analysis of Ponte D. Luis I” geschreven door mijn collegas Jan Daniëls en Sara Pieters in het academiejaar 2012-‐2013. In dit werk wordt het onderdek van de, Ponte D. Luiz I-‐brug gemodeleerd en structureel geanalyseerd.

Een volledig 3D model werd gemaakt, gebaseerd op originele plannen uit 1954 en locale observatie gedaan ter plaatsen. De berekeningen worden uitgevoerd gebruikmakend van de eindige elementen methode and de software Scia Engineering, met de ladingen gedefineerd volgens Eurocode 1, deel 1 en 2.

Spanningen en vervorming -‐ 2de orde effecten in rekening gebracht -‐ zijn berekend in functie van het controleren van de veiligheid van deze onderbouw met de belastingen erop, gedefinieerd volgens de Eurocodes. De waarden hiervan leunen meer aan tegen de ladingen die de brug hedendaags moet aankunnen dan de ladingen waarvoor ze oorspronkelijk ontworpen was.

Sleutelwoorden: Porto; Ponte D. Luis I; Theophile Seyrig; Iron arche bridge; EEM

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Table of Contents Preface .......................................................................................................................................... 3

Abstract ......................................................................................................................................... 5

Abstrato ........................................................................................................................................ 6

Abstract ......................................................................................................................................... 7

List of illustrations ....................................................................................................................... 10

List of tables ................................................................................................................................ 11

1. Introduction ........................................................................................................................ 12

2. Modeling ............................................................................................................................. 13

2.1. Description .................................................................................................................. 13

2.2. Measurements ............................................................................................................ 14

2.3. Sections ....................................................................................................................... 15

2.4. Simplifications and contraditions ................................................................................ 15

2.5. Properties .................................................................................................................... 17

3. Loads ................................................................................................................................... 20

3.1. Self weight ................................................................................................................... 20

3.2. Vertical loads ............................................................................................................... 20

3.2.1. Load model 1 ....................................................................................................... 21

3.2.2. Load model 4 ....................................................................................................... 24

3.3. Horizontal loads .......................................................................................................... 24

3.4. Wind ............................................................................................................................ 25

3.4.1. Wind Load on the deck ....................................................................................... 25

3.4.2. Wind effect on the pillars .................................................................................... 30

3.5. Thermal actions ........................................................................................................... 30

4. Combination of action ........................................................................................................ 34

5. Controlling the model ......................................................................................................... 36

5.1. Classification ............................................................................................................... 36

5.2. Buckling ....................................................................................................................... 39

5.3. Lateral torsional buckling ............................................................................................ 45

5.4. Relative deformation .................................................................................................. 45

5.5. Model 0 ....................................................................................................................... 46

6. Calculations ......................................................................................................................... 49

6.1. General simplifications ................................................................................................ 49

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6.2. Calculation Load model 1 ............................................................................................ 50

6.2.1. Position of the tandem system ........................................................................... 52

6.2.2. Percentage dead load ......................................................................................... 58

6.2.3. General ................................................................................................................ 59

6.3. Calculation Load model 4 ............................................................................................ 60

6.3.1. Percentage dead load ......................................................................................... 64

6.3.2. General ................................................................................................................ 66

7. Conclusion ........................................................................................................................... 67

Bibliography ................................................................................................................................ 68

Annexes ....................................................................................................................................... 69

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List of i l lustrations

Figuur 1: measurement of tietruss ............................................................................................. 14

Figuur 2: arch formed deck, picture (left) – flat formed deck, 3D model (right) ........................ 15

Figuur 3: section of the deck by the original plans (left) – photo of the deck from bottom view (right) .......................................................................................................................................... 16

Figuur 4: bottom view deck, 3D model ....................................................................................... 16

Figuur 5: bridge – picture (up) – model (down) .......................................................................... 18

Figuur 6: detail pillar bottom view – picture (left) – model (right) ............................................. 18

Figuur 7: structural reinforcement-‐ picture (left) – model (right) .............................................. 19

Figuur 8: pillar, bottom view –picture (left) – model (right) ....................................................... 19

Figuur 9 Application of TS and UDL along the longitudinal axis -‐Application of LM1 on the notional lanes-‐ Location of tandem system for the verification of short structural members ... 21

Figuur 10 directions of the axes of the wind on bridges ............................................................. 25

Figuur 11: solidity ratio ............................................................................................................... 29

Figuur 12 terrain category and terrain parameters ................................................................... 29

Figuur 13: diagrammatic representation of constituent components of a temperature profile 30

Figuur 14: Thermal map of Portugal for Tmin (left) and Tmax (right) ............................................. 31

Figuur 15: effective mimimum and maximum temperature ...................................................... 31

Figuur 16: possibility too expand ................................................................................................ 32

Figuur 17: classification of the web ............................................................................................ 38

Figuur 18: classification of the flanges ........................................................................................ 38

Figuur 19: bridge, buckling control ............................................................................................. 39

Figuur 20: cross section T1 .......................................................................................................... 39

Figuur 21: buckling curves .......................................................................................................... 41

Figuur 22: cross section, T2 ......................................................................................................... 42

Figuur 23: Von Mises stress, self weight ..................................................................................... 48

Figuur 24: differences between the bridge and the 2D/3D model ............................................. 49

Figuur 25: maximum internal forces on beams, LM1 ................................................................. 52

Figuur 26: combination key ........................................................................................................ 53

Figuur 27: representation of beam s14367 ................................................................................. 54

Figuur 28: Stress in beam s14367, model1 ................................................................................. 54

Figuur 29: Stress in beam s16149 ............................................................................................... 55


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Figuur 31: representation of beam s 10783 ................................................................................ 56

Figuur 32: Stres in beam s10783 ................................................................................................. 56


Figuur 34: stress in beam s15017 ................................................................................................ 57

Figuur 35: Reactionforces ........................................................................................................... 58

Figuur 36 Von Mises stress in LM1 .............................................................................................. 59

Figuur 37: maximal internal froces, LM4 .................................................................................... 60

Figuur 38: Combinationkey, LM4 ................................................................................................ 61




Figuur 42: stress in beam s10783, LM4 ....................................................................................... 63

Figuur 44: von mises stress, LM4 ................................................................................................ 66

List of tables

Tabel 1: projcet data (left) – wrought iron properties (right) ..................................................... 17

Tabel 2: number and width of notional lanes ............................................................................. 22

Tabel 3: load model1, characteristic values ................................................................................ 22

Tabel 4: load classes for road bridges ......................................................................................... 23

Tabel 5: uniform linear gradient over a vertical direction .......................................................... 33

Tabel 6: desing valeus of actions (STR/GEO) (Set B) ................................................................... 34

Tabel 7: recommended values of ψ factor for road bridges ....................................................... 35

Tabel 8: selection of buckling curve for a cross-‐section ............................................................. 41

Tabel 9: Reaction forces of the deck subjected to dead loads .................................................... 47

Tabel 10: dead load of the suspension trusses ........................................................................... 48

Tabel 11: description of the different loadcases, LM1 ................................................................ 51

Tabel 12: description of the different load cases, LM4 ............................................................... 60

Tabel 13: Reaction forces LM4 .................................................................................................... 65

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1. Introduction

This assignment is a sequence to “Modeling and structural analysis of Ponte D. Luis I”, written by my colleagues Jan Daniëls and Sara Pieters. In this sequel on their work the lower deck of Ponte D. Luis I is modeled and analyzed. For the first part of this work, a model was created in scia engineer. A lot of time was needed to obtain the right skills to tune the model into a very detailed model.

Secondly , the determination of the different loads and description of the different load models was done according to Eurocode 1.

For the third part classifications and other general controls are done by hand.

The loads described in the second part are now put together on different models were their effect is generated and analyzed.

In the last part, the model is fully discussed and a final conclusion together with suggestion for the structure is given.

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2. Modeling

2.1. Description

The bridge Pond D. Luis I. is the biggest tourist attraction of Porto, located in the middle of the centre connecting the city centre of Porto and Vila Nova de Gaia. The bridge was realized from 1881 to 1886 and became famous with the name ‘Eiffel Bridge’, meanwhile the bridge was designed by the Belgian engineer Théophile Seyrig.

The bridge consist out of 3 parts, the upper deck, the arch and the lower deck. On the upper part a metro-‐connection between the two cities is realized, as well as a relatively safe sidewalk for pedestriants to cross from one city to the other. The lower part is designed for the traffic of motor vehicles to pass between the two cities. Also a narrow footpath is provided which enables the pedestrians to cross the bridge by the bottom deck. Eventhough this facility for pedestriants is less safe, in comparison to the option on the upper deck, it is frequently used, especially by tourists.

The structural part of the lower deck amounts 174m in length and 8.4m in wide. The deck is provided with two car-‐lanes (8.14m), each with a different direction, and on each side of the deck a sidewalk (1.2m). The structure is suspended from 4 tie-‐trusses, which are each connected to the arch. The two outer tie-‐trusses have a height of 25.5 m. The two in the middle have a height of 41.3m. The bridge is a pony truss bridge, symmetrical around its middle.

To invert this structure into a model several sources are used. The basic model is created from the original plans. The plans are old, hard to read and are missing essential dimensions. In order to complete and define the model, extra dimensions were necessary. These dimensions were obtained by measurements, done by professor Santos and student An Dejongh. Multiple visits to the bridge were necessary in order to check the dimensions, define the orientations and the composition of sections, establish the types of connections and supports and to eliminate doubts and contradiction between the different plans.

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2.2. Measurements

The extra measurements done by professor Santos and An Dejongh, with ‘HILTI PD32 laser range meter’, are mostly related to the tie-‐trusses. The plans give no specification of them, assign were only related to the dimensions of the autoCAD-‐file PONTE001.DWG. The autoCAD-‐file on the other hand had no information about the chord truss at the bottom of the tie-‐trusses. This elements contributes to the stiffness of the structure and also prevents the pillars from buckling, they are seen as important structural elements, which need to be taken into account.

The method applied for these measurements is based on simple trigonometry, as shown in figure 1.

The unknown dimension d, is calculated out of the other measured dimension by applying the Pythagorean triple. The measurements can be found in the annex 1. The general height of the first element is also measured in order to compare with the dimension given in autoCAD-‐file PONTE001.DWG. This measurement was very close to the dimension given in the autoCAD file. The other dimensions for the tie-‐trusses, are also taken out of this file, since it was not possible to do this measurements our self, with the available material.

Figuur 1: measurement of tietruss

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2.3. Sections The model is a simplified version of the reality, which only contains structural elements and pinned and rigid connections. The simplifications are made in such a way that the reality is approached as accurate as possible. The following important specifications are made regarding the sections and connections. Where possible the sections are substitute by commercial sections. This choice is based on common area and moments of inertia, taken into account is the formula for tension:

σ = !!+ !! ∗ !!!

+ !! ∗ !!!

There were the construction is mostly subjected to deflection, the moment of inertia is the most important parameter to admit. The deck is designed in a way that the beams are principally burdened on their strongest axis. For finding a suitable commercial section, the moment of inertia of the strongest axis needed to be correct. The diagonals on the side of the bridge, are mostly subjected to axial forces and buckling. Therefore the area is the most important parameter. When choosing a replaceable profile, the area must be identical. About the tie-‐trusses no information was available, the cross-‐sections are defined based on the pictures.

2.4. S implif ications and contradit ions The deck is modeled flat, in reality it has an arch form, see figure 2. This simplification has affects on the load transmission. Under the arch form a small part of the transversal load will be transferred to an axial load. The arch is orientated in a way that it will reduce the deflection. Due to this signification in the SCIA model the value of axial forces will be a little bit too low and the deformation of the deck will be higher than in reality.

Figuur 2: arch formed deck, picture (left) – flat formed deck, 3D model (right)

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Sometimes there were contradictions between the plans and reality. For example an important error on the plans, is the cross section of the beams, indicated by the red arrow, on figure 3. In reality this is no I beam, see figure 3 and 4.

Figuur 3: section of the deck by the original plans (left) – photo of the deck from bottom view (right)

Figuur 4: bottom view deck, 3D model

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2.5. Properties The model is very detailed, and consist out of an wrought iron structure with concrete slabs. An overview of the properties of the project data is given in table 2 , the material properties of wrought iron are given in table 1 1.

Wrought iron

!!" 157,5N

mm!

!!" 105 N

mm!

E 192,7 !"!!!

νs 0,2554

ρ 7700 kg/m!

ρ with paint and rivets included

8400 kg/m³

!2 0,00001 m/°C

To obtain the design yield stress the following formula is applied:

f!" =!!"γ! with fyk : the characteristic yield stress

γs: reduction factor for materials As reduction factor for materials, γs = 1,5 is used instead of the normal γs = 1,15. This is done in order to take into account the not as homogeneous structure of wrought iron. , γs = 1,15

The big amount of nodes and beams indicates the complexity of the model. The following figures 5,6,7 and 8 compare the structure with the model.

1 Jan Daniëls, Sara Pieters, Modeling and structural analysis of Ponte D. Luis I, 2013

Tabel 1: projcet data (left) – wrought iron properties (right)

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Figuur 5: bridge – picture (up) – model (down)

Figuur 6: detail pillar bottom view – picture (left) – model (right)

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Figuur 8: pillar, bottom view –picture (left) – model (right)

Figuur 7: structural reinforcement-‐ picture (left) – model (right)

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3. Loads

3.1. Self weight

Dead loads of the structure and the deck are the same as applied in the previous model.

Gk =5,15 kN/m².

Total permanent dead load: 2,02 + 3,13 = 5,15kN/m²

3.2. Vertical loads

For the vertical loads, 4 different load models are defined in EN 1991-‐23:

• A main load model (LM1), including concentrated loads (tandem systems, called TS) and uniformly distributed loads (called UDS) and applicable to all bridges

• A model consisting of a single axle with two wheels (LM2), in addition to the previous one (LM1) for the verification of short structural members (3-‐7m)

• A model made up by a set of special vehicles intended to take into account the effects of exceptional convoys (LM3)

• A model corresponding to the loading of the surface of the bridge with a uniformly distributed load of 5 kN/m², corresponding to the effects (dynamic amplification included) of a crowd (LM4).

For this study load model 1 and 4 are taken into account.

Load model 1 is the most general model, the model mimics the common traffic effects. It takes in account the uniformly distributed loads and the point forces caused by the vehicles. The model gives information about local and general effects. Load model 4, the model that generates the effect of a crowd has to be checked since the bridge is used for events, such as sport events. Load model 3 doesn’t need to be generated, since the bridge is limited for vehicles up till 30 ton, special vehicles are redirected through another route.

3 J. Calgaro, M. Tschumi, H Gulvanessian, “Designers’ guide to Eurocode 1: Actions on bridges: EN 1991-‐2, EN 1991-‐1-‐1,-‐1-‐3 to -‐1-‐7 and EN 1990 Annex A2”, Thomas Telford, London, UK, 2010, ISBN 978-‐0-‐7277-‐3158-‐6, p. 84 – 97; p. 216 -‐ 221

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3.2.1. Load model 1

This model is based on the guidelines given in NBN EN 1991-‐2, version of 2013, in combination with the national annex and the book ‘Designers’ guide to Eurocode 1: Actions on bridges EN1991-‐2, EN1991-‐1-‐1, -‐1-‐3 to -‐1-‐7 and EN1990 Annex A2.

Figuur 9 Application of TS and UDL along the longitudinal axis -‐Application of LM1 on the notional lanes-‐ Location of tandem system for the verification of short structural members

To use a various load model, the carriageway needs to be divided into notional lanes. Depending on the width w, the carriageway has to be divided into the greatest possible integer number of notional lanes, n1. The width w, is measured between the kerbs, situation (b) on figure 10. Table 2 is used to find the number of notional lanes.

Figuur 10: carriageway widths

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w = 5,79 m

5,4 m ≤ w < 6m

nl = 2

w! = !2= 5,79 !2

= 2, 895 !

with

w : width of the deck measured between kerbs n1 : greatest number of notional lanes w1 : width of a notional lane

The lanes are numbered in a way that the lane with the most negative effect gets the lowest number. Taken into account rule 5 of 4.2.4 Location and numbering of the lanes for design, NBN EN 1991-‐24 : “Where the carriageway consists of two separate parts on the same deck, only one numbering should be used for the whole carriageway.” So both lanes are considered as one single lane with lane number 1. The vertical loads on the lane can be divided in 2 different categories. The loads due to a tandem system including two axles TS and the uniformly distributed loads (UDL). For TS only complete tandem systems are taken into account. The UDL is only placed when it generates a global unfavorable effect. The characteristic values are found back in table3.

4 NBN EN 1991-‐2, p 33

Tabel 2: number and width of notional lanes

Tabel 3: load model1, characteristic values

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The adjustment factors take into account the various types of traffic, are given by table 4. The Luis I bridge is considered as a 2nd class road bridges.

Tabel 4: load classes for road bridges

This gives the following results: TS: αQ x Q1k = 0,9 x 300 kN =270 kN

Wheel load: 0,5 x αQ x Q1k = 0,5 x 0,9 x 300 kN = 135 kN/wheel

UDL: αq x q1k = 0,7 x 9 kN/m² = 6,3 kN/m²

αq, pedes x qfk, pedes = 0,7 x 2,6 = 1,82 kN/m²

The wheel loads of the tandem system are divided loads on a surface of 0.4m by 0.4m . Inputting them as point loads is not allowed because this would created very high local effects. The loads on the sidewalk are the same as these for footbridges, where the divded load is obtained with the following siplificated formula:

!!" = 2.0 + 120! + 30

!"!!

= 2 +120

174 + 30!"!!

= 2,6!"!!

with L: the loaded length in meter

qfk must be 2,5 !"/!² ≤ !!" ≤ 5 !"/!², so the value 2,6 !"/!² is accepted.

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3.2.2. Load model 4

The distributed load that represents the effect of a crowd is 5 kN/m².

3.3. Horizontal loads

Longitudinal forces due to breaking and acceleration, take on the surfacing level of the carriageway. The characteristic value is limited to 900 kN and is calculated as a fraction of the total maximum vertical loads due to LM1 and applied on lane number 1.

!!" = 0,6 !!! 2!!! + 0,1 !!! !!!!! !

180 !!! !" ≤ !!" ≤ 900 (!")

with L: length of deck or the part of it that under 2!!!: weight of the two tandem systems (TS), applied on lane No. 1 !!!: density of the uniformly distributed load on lane No. 1 !!: width of lane No. 1 !!!: An adjustment factor depending on the load class !!!: An adjustment factor depending on the load

For the bridge, the following values are applied; L = 174 m

2!!!= 2*300 = 600 kN !!! = 9 kN/m ² !! = 2, 895 !

!!!= 0,9 !!!= 0.7

!!" = 0,6 ∗ 0,9 600 + 0,1 ∗ 0,7 ∗ 9 ∗ 2.895 ∗ 174 = 641.35 !"

!!" is accepted 162 !" ≤ 641.35 ≤ 900 !"

This load is distributed over the deck as a surface load !"#.!"!"#∗!,!"

= 0,45 kN/m!.

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3.4. Wind

3.4.1. Wind Load on the deck

The calculation of the wind load is based on EN 1991-‐1-‐4 in combination with the Portuguese national annex.

For the wind only 1 of the 4 wind directions is calculated. In the most common situations the wind blows over the bridge originating from the west or the east. Which corresponds with wind coming from the ocean towards the land or the other way around, from the land towards the ocean. More unusual situations when the wind is blowing from the north or the south are not taken into account. The stiffness of the bridge along with the longitudinal length is high enough to carry this force. The western wind comes in from the Atlantic Ocean and is stronger than the eastern wind. Therefore only the wind originated in the west is calculated.

• Calculations of horizontal forces on the deck

In the situation where the wind comes from the west, the wind loads will be attached on the x-‐direction of the deck, see fig11 . This x-‐direction corresponds with the width of the deck, perpendicular to the span.

Figuur 10 directions of the axes of the wind on bridges

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There is no dynamic response calculation required, this allows the use of a simplified method with the following formula:

!! = !!∗ ! ∗ !!! ∗ ! ∗ !!"#,!

!!!!"#,!

= !!∗ ! ∗ !!! ∗ !

With:

ρ : air density vb : basic wind speed C : global wind factor Aref,x : reference area in the x direction ( = A1)

sum of the projected surfaces of the profiles perpendicular to the most unfavourable wind direction.

Important remark:

1) when the reference area is used in combination with traffic loads, in the case where traffic loads are the leading action an additional height d* should be taken into account. For road bridges d*= 2 m from the level of the carriageway to the most unfavourable length on the deck, independently from the position of the vertical components of the traffic load.

Since this bridge is one of the rare cases where the fences are higher than the actual vehicles riding on the bridge, d* will be smaller than d. For this reason no distinction is made between traffic loads as leading or non-‐leading action, and height d is applied in both cases.

• vb

The basic wind value vb is calculated using the following formula;

vb = cdir x cseason x vb,0 = 1 x 1 x 30 = 30 m/s

with vb : basic wind speed cdir : wind direction factor cseason : season factor vb,0 : fundamental factor of the basic wind speed

cdir = 1 cseason = 1

vb,0 depends on the zone in which the construction is situated. Zone A: the majority of territories, except for those who belong to zone B. Zone B: the Azores, Madeira and mainland regions located in a coastal zone off 5 km wide or at altitudes above 600 m.

The bridge is situated in zone B, which corresponds to a value vb,0 = 30 m/s.

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• wind load factor C = ce * cf,x = 1.95 * 1.3 = 2.536

with ce : exposure factor cf,x : drag coefficient for wind actions

c!,! = c!,!" = 1.30

with cf,x0 : force coefficient without free-‐end flow

!! ! = !!(!)!!

= !"#$.!"!"#.!

= 1.95

with q p(z) : extreme thrust on high z q b: basic thrush

!! = !!∗ ! ∗ !!! =

!!∗ 1,25 ∗ 30 !

= 562.5 !!!

with ! = density !! = basic wind speed

! = 1,25 !"!!

!! = 30 !!

q!(!) = 1 + 7 ∗ I! z ∗ !!∗ ρ ∗ v!! z = (1 + 7 *0.18) ∗ !

!∗ 1 .25 ∗ 27.87!

= 1097.14 !

!!

with I! z : turbulence intensity at height z ρ : density v!: mean wind speed

28

ρ = 1,25 !"!!

v! z = c! z ∗ c! z ∗ v! = 0.929 ∗ 1 ∗ 30 = 27.87 !

!

with c! z : roughness factor c! z : orography factor

v!: basic wind speed

c! z = 1

c! z = k! ∗ lnzz!= 0,170 ∗ ln

11.80,05

= 0.929

with k!: roughness factor depending on the roughness length z0

Values for z0 and z0,II, terrain category I are erased from figure 6

k! = 0,19 ∗z!z!,!!

!,!"

= 0,19 ∗0,010,05

!,!"= 0,170

I! z = σ!

v!(z)=

4.6825.57

= 0,18

with σ!: standard deviation of the turbulence

v!(!): mean wind speed at height z

σ! = k! ∗ v! ∗ k! = 0.156 ∗ 30 ∗ 1 = 4.68!

!

with k!: roughness factor

v!: basic wind speed k!: turbulence factor

k! = 0.156

vb = 30 !!

k! = 1

29

=> F!A!"#,!

= 12∗ 1,25 ∗ 30! ∗ 2.536 = 1426.5

Nm! = !.!"#

!"!!

with Aref,x = 322.78 m

A2 = 563.76 m2 Since Aref,x is not practical to work with in the SCIA model, the loads will be put on A2. With A2 : sum of the whole projected surface of the profiles perpendicular to the most unfavourable wind direction.

F!A!

= F!A!"#,!

∗ A!"#,!A!

= 1.427 ∗ 322.78 563.76

= !.!"#!"!!

This force is put on the first side of the bridge. For the second side of the solidity ratio is taken into account. The solidity ratio is described in the following expression:

ϕ = A!!= !"".!" !

!

!"#.!" !! = 0,57

with A: sum of the projected areas of the members Ac : the overall envelope area

A = 322.78 m2

Ac = 563.76 m2

The wind force on the second side is:

ϕ !!!!

= 0.57 * 0.817 = 0.465 !"!!

Figuur 12 terrain category and terrain parameters

Figuur 11: solidity ratio

30

3.4.2. Wind effect on the pi l lars

In normal circumstances the pillars are subjected to tensile as an effect of the self-‐weight of the bridge. When the pillars are also subjected to wind, the situation of complete tensile will shift to a situation of tensile and pressure. For the stability the maximal pressure allowed within the pillars should be checked, along with the different buckling-‐controls. In order to do this a separate study should be done. An option for this would be to do a simulation on the model in a wind tunnel. Another option could be a separate study of the deck, where the effect of the wind, only on the deck is visualized. Therefore the forces on the deck, due to the wind, can be placed on the pillars. Both of these options, are left aside for this study.

3.5. Thermal actions

The deck of the bridge, corresponds to type II: composed deck, given in EN 199-‐1-‐5. The thermal effects in the bridge decks are represented by 4 different terms all resulting from the sun. Figure 13 gives an overview of the 4 terms.

With (a) component of the uniform temperature (b) and (c) components of the temperature linearly variable according to the two axes in the plan of the section (d) residual component

Figuur 13: diagrammatic representation of constituent components of a temperature profile

31

• uniform component

The extreme values of the uniform temperature components are taken out of the national temperature map, figure 15.

For the minimum and maximum temperature Porto is situated in zone B, this gives Tmin = 0°C and Tmax = 40°C. These values are based on a return period of 50 years, derived from a Gumbel law.

The effective minimum and maximum temperature is taken out of figure 16 and is in function of the shade air temperature Tmin , Tmax and the type of deck.

Figuur 14: Thermal map of Portugal for Tmin (left) and Tmax (right)

Figuur 15: effective mimimum and maximum temperature

32

Next values are found: Te,min = -‐3 °C

Te,max = 56 °C

The variation of temperature is considered around an average effective value, T0. For the design of expansion joints and bearings the following formula is applied:

Te,max – Te,min + 2S

Since no information is available over T0, the value of S can be set on 20°C. This gives the following value for the total opening for expansions joint, or total movement for bearings:

56 -‐ (-‐3) + 2*20 = 96

The average value ΔT of the temperature difference in case of contraction and expansion, ΔTN,con and ΔTN,exp is found with the following formula:

!!!,!"# + !!!,!"# 2

with ΔTN,con = T0 − Te,min ΔTN,exp = Te,max – T0

= T!,!"# – T!,!"#

2

= 56 – (−3)

2

= 29.5

The thermal expansion coefficient of wrought iron is 10x10-‐6 per °C. The change in length is now calculated.

Δ! = ! ∗ Δ! ∗ ! Δ! = 10 ∗ 10!! ∗ 29.5 ∗ 174 ! = 0.05 ! = 5 !"

The change in length is acceptable, and for this reason not considered in the SCIA model. Expansion in the bridge is allowed and expansion joints are present, figure 16.

Figuur 16: possibility too expand

33

• Thermal gradient

The uniform linear gradient over the vertical direction is given in table 5.

For type II this corresponds with △ !!,!!"# = 15°C for when the top is warmer than the bottom. In the other case were the bottom is warmer than the top △ !!,!""# = 18°C.

Tabel 5: uniform linear gradient over a vertical direction

34

4. Combination of action

The fundamental combination of actions for ultimate limit states, is applied as prescribed in the Eurocode:

!!,!!!,! + !!,!!!,! + !!,!!!,!!!,!!!!!!!

with !!,! = partial factor for permanent action j !!,! = characteristic value of the leading variable action 1 !!,! = characteristic value of the other variable actions !!,! and !!,! = partial factors !!,! = partial factor for combinations

For permanent actions a partial factor is used depending on the effect; favourable or unfavourable. Also for the characteristic value of the leading variable the partial factor depends on the effect it generates. For the other variable actions an extra combination factor is used, depending on the type of load.

For bridges different partial factors are used, for example some variable actions have a partial factor of 1.35.

The design values of actions are given in table 6.

Tabel 6: desing valeus of actions (STR/GEO) (Set B)

35

For further calculations every combination should be taken into account, every load should be checked on his effect, favourable or unfavourable. When more variable loads take place at the same time, a combination factor is applied. A separate set of combination factors for bridges is found in the Eurocode, see table 7.

For the models in SCIA, the partial and combination factors are automatically generated with the combination settings set on eurocode-‐ultimate limit state.

Tabel 7: recommended values of ψ factor for road bridges

36

5. Control l ing the model

5.1. Classif ication

Local buckling within profiles is even possible before the stress reaches the level of yield stress. In order to determine how much the resistance and the rotationcapacity of the profile has diminished,due to local buckling, a classificationsystem is developed. The classification is based on 2 aspects the whidth-‐tickhness proportion and the development of the stress in the different elements.

Four different classes are defined according to EN 1993-‐1-‐15:

Class 1: cross-‐sections are those which can form a plastic hinge with the rotation capacity required from plastic analysis without reduction of the resistance.

Class 2 :cross-‐sections are those which can develop their plastic moment resistance, but have limited rotation capacity because of local buckling.

Class 3: cross-‐sections are those in which the stress in the extreme compression fibre of the steel member assuming an elastic distribution of stresses can reach the yield strength, but local buckling is liable to prevent development of the plastic moment resistance.

Class 4 : cross-‐sections are those in which local buckling will occur before the attainment of yield stress in one or more parts of the cross-‐section.

A classification is done for the cross bars and beams in the deck, since these are the elements under the highest compression. Compression created the most unfavourable effect, so the profiles are only checked under pure compression. The small bending moment that takes place is not taken into account. Figure 17 and 18 are applied for the classification.

Cross beams IPE 300

• Web

Class 1:

!!< 33ε with ! = !"#

!"#.! = 1.22

!"#.!!.!

< 33 ∗ 1.22

35.01 < 40.26

The web confirms class 1

5 NBN EN 1993-‐1-‐1, october 2005, p40

37

• Flange

Class 1:

!!< 9 ∗ ε with ! = !"#

!"#.! = 1.22

!".!"!".!

< 9 ∗ 1.22

6.68 < 10.98

The flanges conform with class 1.

The whole profile is defined as a class 1 profile.

Cross beams IPE 500 0

• Web

Class 1:

!!< 33ε with ! = !"#

!"#.! = 1.22

!"#.!!".!

< 33 ∗ 1.22

36.82 < 40.26

The web confirms class 1

• Flange

Class 1:

!!< 9 ∗ ε with ! = !"#

!"#.! = 1.22

!".!!".!

< 9 ∗ 1.22

4.75 < 10.98

The flanges conform with class 1.

The whole profile is defined as a class 1 profile.

Both profiles belong to class 1, calculations can be made plasticly , without reducing the resistance.

Since the best class is found for the profiles under the highest compression, a further manual check for the other elements is not done.

38

Figuur 17: classification of the web

Figuur 18: classification of the flanges

39

5.2. Buckling

While setting the buckling settings some problems were found in SCIA. Therefore a manual buckling control is done based on EN 1993-‐1-‐1. The controlling is done for the element with the highest chance of buckling. Extra buckling control is not done on the pillars, which also have a big critical length for buckling out of plane, since they are mostly under tension and not under compression.

The controlled beam is an alleged T-‐shape, on the side, where the diagonals come to getter, shown on figure 19 and 20.

The shape of the flanges changes over distance. Reinforcements are done, which corresponds to 5 different T-‐beams in the model. The height increase when approaching the pillars while the width remains the same. The control is done on the T member with the smallest dimensions, because this is the least safe. In order not to acquire the chance of buckling the following criteria should be respected:

!!"!!,!"

≤ 1

with NEd : design value of the compression force Nb;Rd : design buckling resistance of the compression member

!!,!" = !" !!!!!

for Class 1,2 and 3 cross sections

with ! : reduction factor for the buckling mode A: area

f! : yield strength !!! : particular partial factor

Figuur 19: bridge, buckling control Figuur 20: cross section T1

40

The value of the reduction factor ! can be in the buckling curves, figure 21, and is in function of the non-‐dimensional slenderness ƛ.

ƛ = ! !!!!"

for class 1, 2 and 3

with A : area f! : yield strength N!": the elastic critical force for the relevant buckling mode

based on the gross cross sectional properties A: 1.06 *104 mm2

fy : 105 !

!!!

!!" = !!!"!!"!

with E = 192,7*106 kN/m! Iy = 1,0694*10-‐4 m4 for buckling in plane Iz = 9.6028*10-‐5 m4 for buckling out of plane Lcr = 3m for buckling in plane Lcr = 3m for buckling out of plane

Buckling in plane:

!!" = !!!"!!"!

= !!∗!"#.!∗!"!∗!,!"#$∗!"!!

!! = 22598.5 kN

= 22598.5 *103 N Buckling out plane:

!!" = !!!"!!"!

= !!∗!"#.!∗!"!∗!.!"#$∗!"!!

!! = 20292,6 kN

= 20292,6 *103 N

For buckling in plane the non-‐dimensional slenderness is:

ƛ = !.!"∗!"! ∗!"#!!"#$.!∗!"!

= 0.22

For buckling out of plane the non-‐dimensional slenderness is:

ƛ = !.!"∗!"! ∗!"#!"!#!,! ∗!"!

=0.23

The buckling curve that must be applied depend on the shape of the element. Table 8 gives an overview.

41

Tabel 8: selection of buckling curve for a cross-‐section

For a T-‐ shape, buckling curve c is applied.

Figuur 21: buckling curves

42

For buckling in plane this gives ! = 1, or buckling out of plane ! = 0.98 .

This gives the following results for design buckling resistance.

In plane :

!!,!" = 1 ∗ 1.06 ∗ 10! ∗ 105

1= 1113000 !

=1113 kN Out plane :

!!,!" = 0.98 ∗ 1.06 ∗ 10! ∗ 105

1= 1090740 !

= 1090.74 kN

The values for NEd are calculated in SCIA. For the design value of compression force in plane, a combination of all possible load cases is made. For the one out of plane only the wind is considered, because this is the only force that will make buckling out of plane possible. This gives the following results:

In plane: NEd =-‐1603,21 kN

Nb,Rd = 1113 kN

!!"!!,!"

= !"#$,!"!!!"

= 1.44 > 1

The compression force which takes place in the beam is higher than the buckling resistance compression force. If we check the member we see that it is a reinforced T-‐beam, see figure 4. A new calculation for this cross section will be done.

Out plane: NEd =-‐1603,21 kN

Nb,Rd = 1090.74 kN

!!"!!,!"

= !"#.!"!"#".!"

= 1.47 > 1

This doesn’t correspond with the buckling criteria. A check of the beam is done within the construction, the beam corresponds with a reinforced T-‐ beam, see figure 22.

Figuur 22: cross section, T2

43

For the non-‐dimensional slenderness ƛ, the following value is found:

ƛ = ! !!!!"

A: 1.38 *104 mm2

fy : 105 !

!!!

!!" = !!!"!!"!

with E = 192,7*106 kN/m!

Iy = 1.1915*10-‐4 m4 for buckling in plane Iz = 1.387*10-‐4 m4 for buckling out of plane Lcr = 3m for buckling out of plane

In plane:

!!" = !!!"!!"!

= !!∗!"#.!∗!"!∗!.!"!# ∗!"!!

!! = 25178.7 kN

= 25178.7 *103 N Out plane:

!!" = !!!"!!"!

= !!∗!"#.!∗!"!∗!,!"#∗!"!!

!! = 29304kN

= 29304 *103 N This gives the following non-‐dimensional slenderness :

in plane:

ƛ = !.!"∗!"! ∗!"#!"#$%.! ∗!"!

= 0.24

out plane:

ƛ = !.!"∗!"! ∗!"#!"#$%∗!"!

= 0.22

44

In figure 21 we find that this corresponds with ! =0.98 for buckling in plane and ! = 1 for buckling out plane.

This following design buckling resistance are found:

in plane:

!!,!" = 0.98 ∗ 1.38 ∗ 10! ∗ 105

1= 1420020 !

= 1420.02 kN

out plane:

!!,!" = 1 ∗ 1.38 ∗ 10! ∗ 105

1= 1449000 !

= 1449 kN The buckling check gives the following result:

in plane

NEd =-‐1603.21 kN

Nb,Rd =1420.02 kN

!!"!!,!"

= !"#$.!"!"#$.!"

= 1.13 > 1

The axial force in the beam is higher than the maximum allowed axial forces, so buckling in plane is possible. The too high forces in the construction will cause deformation of the construction, and generate second-‐ order-‐ effects . These are effects due to the fact that the axis can be out of its original position. Since the construction is considered non-‐sway the second-‐order-‐effects should not be taken into account.

out plane

NEd = 135.56 kN

Nb,Rd = 1449 kN

!!"!!,!"

= !"#.!"!""!

= 0.094 ≤ 1

The profile will not buckle.

45

5.3. Lateral torsional buckl ing

A lot of beams are only subjected to axial forces, for those beams no lateral torsional buckling controle is needed. The beams under bending moment should be taken into account for lateral torsional buckling check. No seperate manual control is done.

5.4. Relative deformation

The deck is submitted to a maximum relative deformation of 356.5mm in the middle of the bridge. This effect is in reality reduced because of the slightly curved shape of the deck. No information is available over the exact bend degree, so the effect is not taking into account for the calculations. The slightly curved shape will generate a higher axial force in the deck because a part of the transversal forces taking place on the deck are converted to axial forces. The bending moment will be reduced by the arch shape of the bridge.

46

5.5. Model 0

The controlling of the model is done with the structure subjected only to the dead loads. These values are taken out of the study 'Estudo da viabilidade de utilização da Pont Luiz I pelo Metro ligeiro do Porto”, chapter 2.1.1. These study gives the values for the dead loads of the deck itself but not for the tie-‐trusses. A different model with only the suspension tie-‐trusses in it is made in SCIA, the self-‐weight is automatically generated by SCIA.

For the dead load of the structure6 17kN/m is found, divided over a structural wide of 8.4m, gives 2.02 kN/m². The deck itself has a dead load of 3,13 kN/m², so the total permanent dead load is :

2,02 + 3,13 = 5,15kN/m² This simplified model with only dead and live loads on it gives the following reaction forces;

First outer tie-‐trusses:

left node : 821.63 kN right node : 819.32 kN

average per node : 820.48 kN

Second outer tie-‐trusses:

left node: 823.41 kN right node: 818.36 kN


First middle tie-‐trusses:

left node : 759.77 kN right node : 759.69 kN


Second middle tie-‐trusses:

left node :734.28 kN right node : 783.50 kN


6 A. G. Coelho, J. A. M. Freitas, M.de Azeredo, A. Costa, J .S. Cruz, N. E. Santos, D. B. Lopes, “Estudo da viabilidade de utilização da Pont Luiz I pelo Metro ligeiro do Porto”, Revista Portuguesa de Engenharia de Estruturas (RPEE) N.° 43, Porto, Portugal, 1997, p 2.1.1

47

Table 9 gives an overview of the different values.

Different results are found between the two outer tie-‐trusses and between the two middle tie-‐trusses . Normally these should be equal, since the bridge is symmetrical and the dead loads

too. The difference is small, only 1 -‐ !"#.!" !"#.!!

= 0.05 % for the outer tie-‐trusses and 1 -‐ !"#.!" !"#.!"

=

3.03 % for the middle tie-‐trusses. These values are acceptable.

Tabel 9: Reaction forces of the deck subjected to dead loads

48

The following results of reaction forces due to the self-‐weight of the tie-‐trusses were obtained.

Two outer tie-‐trusses: 50.98 kN per node

Two middle tie-‐trusses: 83.43 kN per node

Table 10 gives an overview of the different values.

These reaction forces are put as forces on the supports of the tie-‐trusses of the global model, like this the self-‐weight of the tie-‐trusses is taken into account in the following calculations.

A general check of the model, only subjected to its self-‐weight, is done by controling the maximum tension in all the beams. Here for the Von Mises criteria is applied. The criteria states that the Von-‐Mises-‐stress must be lower than the yield stress, with f!" 105 N/mm² for wrought iron. Members with a Von Mises stress higher than 105 N/mm² are found in and around the tie-‐trusses, see fig 23.

Figuur 23: Von Mises stress, self weight

Tabel 10: dead load of the suspension trusses

49

Around the pillars the dimension of the diagonals changes, they becomes bigger when approaching them. Although there are bigger dimensions, it is not enough for all of these elements. Local reinforcements are done but not taken into account in this model , since there was no clear pattern. Having to control every single beam to determine whether it is reinforced or not, would take up too much time and would make the model very complex.

Also in the wind bracing, the maximum allowed tension is also sometimes exceeded . There was a lack of information regarding the pillars, they weren’t described in the original plan. The profile of wind bracing might be slightly different. Also local reinforcement is not taken into account in the model, because there was no information on this topic available. A detailed

analysis should be made on this reinforcement since it affects the way of force distribution, which also has an effect on the stress. Figure 24 shows the difference between the bridge itself and the model.

6. Calculations

6.1. General s implif ications For the models the following simplifications are made:

1) Snow is not generated in the models, this simplification is made in consultation with Prof. Santos. This decision is based on the following facts; the chance of having a big snow load in the South is very low and the Eurocode provides a rule which says that snow loads should not be combined with any group of traffic loads.

Figuur 24: differences between the bridge and the 2D/3D model

50

2) 3) Rainwater is not calculated in as a load. The structure of the bridge is open enough to

let the water flow away, there isn’t any chance that the water will collect on top of the deck.

4) Seismic actions are not taken into account for the reason that its effects and

corresponding calculation methods are not taught to Belgian engineering students. The author strongly recommends for further study addressing this matter.

6.2. Calculation Load model 1 The different loads placed on the first model in SCIA, are given in table 11

For a description of the loads and how the values were obtained see chapter 3. An overview from the settings in SCIA ; load cases and load groups can be find in annex 2.

Load case Description

Dead load Dead load

Wind Wind from the west

Thermal up-‐ down Terminal linear heating of the deck, with the top warmer than the bottom

Thermal down -‐ up Terminal linear heating of the deck, with the bottom warmer than the top

Traffic UDL Uniformly distributed load for traffic

Horizontal Horizontal forces

TS0 Tandem system including two axles, on position 0m, first lane

TS16 Tandem system including two axles, on position 16.5m, first lane








TS157 Tandem system including two axles, on position 157.5m, first lane

51


TS0 A Tandem system including two axles, on position 0m,second lane

TS16 A Tandem system including two axles, on position 16.5m, second lane

TS33 A Tandem system including two axles, on position 33m, second lane







TS157 A Tandem system including two axles, on position 157.5m, second lane


Tabel 11: description of the different loadcases, LM1

52

6.2.1. Posit ion of the tandem system

The different positions for the tandem system, as shown in table 12, they are chosen in a way that they will generate the most unfavorable effect. The bridge therefore is divided into 5 parts, two parts of 33m from the beginning and ending at the outer tie-‐trusses, and 3 parts of 36 m from tie-‐trusses to tie-‐trusses. The tandem systems are placed in the beginning, middle and end of every part. For the beginning and the end a maximal transversal force is expected in the supports and in the tie-‐trusses. For the middle a maximal bending moment is expected. The maximum internal forces on beams, are displayed in figure 25. Figure 26 shows the most important combinations, out of more than 1000 combinations.

Figuur 25: maximum internal forces on beams, LM1

53

Figuur 26: combination key

The two maximal axial forces are found in beam s14367 and s16149. The maximal values for the bending moments around the y-‐axes are found in beam s10783 and s15017. These four beams are explored into further detail. This corresponds with the most unfavourable effect for the tandem system on positioned on 16.5m and 157.5m.

54

Beam s14367

Beam s14367 is located on the outer side of the deck, right under the outer tie-‐truss as shown in figure 27.

The axial force reaches a maximum value of -‐7229,07 kN for combination 3:

1.35*dead loads + 1.5*wind + 1.35*traffic UDL + 1.35*TS157 + 1.35*TS157 A

The Von Mises stress is given in figure 28. The highest value found in the beam is 522.2 !!!!,

and mostly comes from the axial stress, this is logical since it is kind of a truss structure. This

value is greater than the critical value of 105 !!!!. Reinforcement is needed.

Figuur 27: representation of beam s14367

Figuur 28: Stress in beam s14367, model1

55

Beam s16149

Beam s16149 is located on the outer side of the deck, in the middle of the last slab as shown in figure 29.

The axial force is 6155.04 kN for combination 4:

1.35*dead loads + 1.5*wind + 1.35*traffic UDL + 1.35*TS16 + 1.35*TS16 A

The Von Mises stress is given in figure 30, with a maximum value of 369,6 !!!! coming solely

from axial tension. This value is greater than the acceptable stress of 105 !!!!.


Figuur 29: Stress in beam s16149

56

Beam s10783

Beam s10783 is located in the first span, on the outer side of the deck , as shown in figure 31.

The moment is -‐3329.07 kNm for combination 7 :

1.35*dead loads + 0.9*thermal down-‐up + 1.35 *traffic UDL+ 1.35*TS16 +1.35*TS16 A


from the axial tension. This value is greater than the acceptable stress of 105 !!!!.

Figuur 31: representation of beam s 10783

Figuur 32: Stres in beam s10783

57

Beam s15017

Beam s15017 is located at the on the outer side of the deck, right under the outer tie-‐truss as shown in figure 33.

The bending moment is 4777, 45 kNm for combination9:

1.35*dead loads + 0.9 thermal up-‐down + 1.35*traffic UDL + 1.35*TS16 + 1.35*TS16 A


from the axial tension. This value is greater than the acceptable stress of 105 !!!!.

Since all these beams have a stress greater than the design yield stress, reinforcement for these beams is definitely recommended. A lot of local reinforcements have been done, so there is the probability that the beams are already reinforced. The dimensions of the reinforcement necessary to obtain a safe situation, are highly related to used steal quality.


Figuur 34: stress in beam s15017

58

6.2.2. Percentage dead load

Comparing the reaction forces due to the dead load and the reaction forces obtained out of the combination, gives an idea about the importance of the dead loads in the total structure. The reaction forces for the combinations, can be found in figure 20. Reaction forces for the dead load are given in figure 35.

Figuur 35: Reactionforces

59

If we compare the reaction forces in the z direction, a total reaction force of 30374.69 kN is found as maximum value for the combination. The bridge only subjected to dead loads gives a reaction force of 7453.2 kN. The dead load represent the following percentage:

!"#$.!!"!#$.!"

∗ 100 = 24.54 %

Only 24.54 % of the reaction force is due to the self-‐weight of the structure. This value is very low for such a large bridge. The light structure is a truly good design of structure, where the material is used at its best. Since the percentage of dead loads is low other loads will have a bigger influence on the structure.

6.2.3. General

A general Von Misses check is done on the construction. Figure 36 illustrates the values higher or equal to the design yield stress fyd=105 kN/mm² in blue. For the tie-‐trusses no fixed conclusion can be made, since the dimensions of the profiles were assumed due to a lack of information about them. A large amount of elements does not confirm the design yield stress. The loads where the bridge originally was designed for are lower than these to which it is submitted today. The loads nowadays are multiplied with a partial factor so their influence increases . Reinforcement is recommended, in order to find out which element should be reinforced, a local study should be done.

Figuur 36: Von Mises stress in LM1

60

6.3. Calculation Load model 4

The effect of the crowd is generated on the bridge as a uniform divided load of 5 !"!!

An overview of the different loads placed on the model is given in table 12. The different load cases with their corresponding load groups can be find in the annex 3.

Load case Description

Dead load Dead load

Wind Wind from the west

Thermal up-‐ down Terminal linear heating of the deck, with the top warmer than the bottom

Thermal down -‐ up Terminal linear heating of the deck, with the bottom warmer than the top

Crowd Load from a crowd

Tabel 12: description of the different load cases, LM4

The maximum internal forces on beams, are shown in figure 37, figure 38 shows combination key.

Figuur 37: maximal internal froces, LM4

61

Figuur 38: Combinationkey, LM4

For the two maximal axial forces and the two maximal bending moments around the y-‐axes an individual stress control is done. The two maximal axial forces are found in beam s15017 and s10783. The maximal values for the bending moments around the y-‐axes are found in beam s15017 and s10783. A closer look is taken to these 2 beams.

62

Beam s15017

Beam s15017 is located at the on the outer side of the deck, right under the outer tie-‐truss as shown in figure 39.

The axial force of -‐4521.53 kN for beam s15017 is found in combination 2:

1.35*dead loads +1.35*crowd

The bending moment of 2973.32 kNm is found in beam s15017 for combination 7:

1.35*dead loads +1.5*wind

The Von Mises stress is given in figure 40, with a maximum value of 380,6 !!!! coming

completely from the axial tension. This value is higher than the acceptable stress of 105 !!!!.



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Beam s10783

The second highest axial force is located in beam s10783 and is located on the outer side of

the deck, right under the outer tie-‐truss as shown in figure 26.

The axial force reaches a maximum value of 3169.23 kN for combination 1:

1.35*dead loads + 0.9*thermal down-‐up + 1.35*crowd

The bending moment reaches a value of -‐1542.28 kNm for combination 2:

1.35*dead loads +1.35*crowd

The Von Mises stress is given in figure 42. The highest value found in the beam is 198.3 !!!!,

and comes completely from the axial stress. This value is bigger than the allowed value of 105 !

!!!. Reinforcement is needed.


Figuur 42: stress in beam s10783, LM4

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Since all these beams have a stress higher than the yield stress, reinforcement of this beam is definitely recommended. A lot of local reinforcements have already been done, so there is the probably that the beam has also been already reinforced. The steel quality used for the reinforcement has a big impact on the dimensions necessary to obtain a save situation.

6.3.1. Percentage dead load

The total reaction force in the z direction, found due to the self-‐weight of the structure is 7453.2 kN. The total reaction forces of load model4 in the z direction is 18416.7 kN and is given in table 13. The dead load represent the following percentage:

!"#$.!!"#!$.!

∗ 100 = 40.46 %

Only 40,46% of the reaction forces in the z-‐direction are due to the dead loads, the position and value of the other loads putted on the model still has a big influence.

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Tabel 13: Reaction forces LM4

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6.3.2. General

A general Von Mises check is done. Figure 44 gives an illustration of this, the tensions higher or equal to design yield value fyd=105 kN/mm² are marked in blue. We see a concentration of high values in and around the tie-‐trusses. About the tie-‐trusses no information was available, so the choice for the profiles is based on pictures. Since the profiles are determinative for the stress development, more information about them is necessary in order to have a correct view of the stress in the tie-‐trusses themselves. A lot of the elements that transcend the design value are still below the characteristic value of the yield stress. For this calculation a very high material reduction factor of γs = 1,5 is taken into account to cover the possible heterogeneity that might be in the old wrought iron. The loads on the bridge are also subjected to high partial factors imposed by the Eurocode. All these factors together lead to a big amount of bars that exceed the design yield factor. In reality this does not mean that all these elements will yield , although reinforcement is recommended.

Figuur 43: von mises stress, LM4

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7. Conclusion

The structure of the bridge itself is designed in a way that the material is used at its best. The self-‐weight is very low compared with the size of the bridge. This makes the design of the structure a engineering masterpiece.

For the resistance checks several problems are obtained. First of all the buckling conditions are not fulfilled. The axial forces are too high on certain places and buckling can occur according to the calculation rules of the Eurocode. Originally the structure was not designed for such a high axial forces and less margin was taken on the calculations. The model used for the calculations is very accurate and the loads are higher. Different explanations can be given to this higher load. Firstly the traffic volumes and density increased greatly over the past century, which automatically goes together with a change of loads. Secondly the Eurocode is very strict and provides high partial factors on the loads. The Von Mises-‐criterium is not fulfilled either. Already problems occur when the bridge is submitted only to its self-‐weight. The design yield stress is exceeded in several elements in-‐ and around the tie-‐trusses. The highest tensions take place in the tie-‐trusses since they are responsible for the force transmission between the supports and the deck. The cross-‐sections of the tie-‐trusses was not known and the choice of these profiles is based on pictures. An under sizing of the profiles can be a cause of the too high stress values. For model 1 more than half of the bars exceeds the design yield stress, for model 2 this is less than half. Frequently the stress in the bars is still under the characteristic yield factor. A very high reduction factor of material is applied on the construction. The reduction factor reduces the design yield factor to 2/3 of the characteristic yield factor, in order to take into account the less homogeneous of wrought iron, compared to modern steel. This very strict reduction factor is another cause of the high amount of bars, exceeding the design yield stress. On the other hand the loads are again multiplied with the partial factor, as provided by the Eurocode, which will also increase the stress in the structure. Another cause is that a lot of local reinforcements were done, but not taking into account in the model. These local reinforcements have an influence on the load transmission and stress development in the beams. In general the structure is not in conformity with the current standard of the Eurocode. This does not automatically means that the structure is unstable, since the requirements of the Eurocode are very high. There are some local problems and reinforcement is sincerely recommended. A more detailed study on these local elements is also recommended.

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Bibl iography

Eurocode 0: Grondslagen van het constructief ontwerp -‐ Bijlage A2 :Toepassing voor bruggen (+ AC:2010) (juli 2013)

Eurocode 1: EN 1991-‐1 Belastingen op constructies (januari 2004) EN 1991-‐5 Belastingen op constructies -‐ Deel 2 : Verkeersbelasting op bruggen Nationale bijlage (oktober 2010) Eurocode 3: EN 1993-‐1 Ontwerp en berekening van staalconstructies -‐ Deel 1-‐1: Algemene

regels voor gebouwen (+ AC:2009) Eurocode 4: EN 1994-‐2:2005 Ontwerp en berekening van staal-‐betonconstructies -‐ Deel 2:

Bruggen (+ AC:2008) (april 2011) Jean-‐Armand Calgaro, Marcel Tschumi and Haig Gulvanessian, Designers’ guid to Eurocode 1: Actions on bridges, E,1991-‐2, EN,1991-‐1-‐1,-‐1-‐3 to -‐1-‐7 and EN 1990 Annex A2, Thomas Telford,2010

Gerard Parker, Nigel Hewson, ICE manual of bridge engineering, Thomas Telford,2008

Jan Daniëls, Sara Pieters, Modeling and structural analysis of Ponte D. Luis I, 2013

Luc Vanhooymissen, Marc Spegelaere, Ann Van Gysel, Willy De Vylder, Gewapend beton, Berekening volgens NBN 15-‐002 ( 1999),Academia Press, 2002

Stef Pillaert, Staalbouw deel I en II, leidraad bij EC2, 2009

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Annexes

Nacalculatie van een StaalBeton Brug

Documents

Transcript of Nacalculatie van een StaalBeton Brug