Final design of the overhead line equipment in the extension of a tramway system in UK

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MASTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

FINAL DESIGN OF THE OVERHEAD LINE

EQUIPMENT IN THE EXTENSION OF A

TRAMWAY SYSTEM IN UK

AUTHOR: RUBÉN SANZ ABAJO

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ESCUELA

TÉCNICA

SUPERIOR DE

INGENIERÍA

(ICAI)

MÁSTERUNIVERSITARIO ENSISTEMASFERROVIARIOS

Proyecto realizado por el alumno: Rubén Sanz Abajo

Fdo:………. Fecha: 05/07/2016

Autoriza la entrega del proyecto cuya información no es de carácter confidencial

EL DIRECTOR DEL PROYECTO

Joaquín Ramos Rodríguez

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ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)

MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

FICHA TÉCNICA

TRABAJO FIN DE MASTER (6 ECTS)

Y

AMPLIACIÓN DEL TRABAJO FIN DE MASTER (6 ECTS)

MASTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS CURSO 2015 – 2016

Datos del alumno:

Apellidos: Sanz Abajo

Nombre: Rubén NIF: 03139096-X

Datos del director del Trabajo:

Apellidos: Ramos Rodríguez

Nombre: Joaquín NIF: 09027143B

Título del Trabajo Fin de Máster:

Diseño final de la catenaria en la prolongación de un sistema tranviario en Wolverhampton, Reino Unido.

Descripción breve:

El proyecto consiste en el diseño de detalle final para el sistema de captación de energía de una red tranviaria que transcurre por un entorno urbano, teniendo en cuenta los requerimientos y requisitos

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ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)

MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

soporte del hilo de contacto de la catenaria; para proceder después con el diseño mecánico y geométrico de las distintas soluciones particulares aplicadas a cada punto de sustentación: pórticos funiculares, atirantados horizontales y ménsulas.

El proceso de dimensionado de estos elementos se retroalimentará en varias etapas, con el objetivo de optimizar su diseño y ajustarlo a un óptimo desde el punto de vista mecánico y de seguridad en la operación de la línea.

Finalmente, se proporcionará la información necesaria para llevar a cabo la fase constructiva del proyecto: detalles de instalación, secciones particulares y especificaciones para los elementos principales para efectuar un presupuesto.

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I

NDEX

ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)

MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

Index

Chapter 1: Introduction ...4

1.1 Summary ...4

1.2 Objectives ...5

1.3 Tasks and planning ...6

Chapter 2: Design

...8

2.1 Project Overview ...8

2.2 Scope of works ... 10

2.2.1 Differences and Implications ... 10

2.3 Referenced standards and documents ... 12

2.4 System description ... 14

2.5 Inputs, Assumptions and requirements ... 19

2.6 New OLE Layout Development ... 26

Chapter 3: Calculations ... 27

3.1 Loads ... 27

3.1.1 Loads on wires... 28

3.1.2 Loads on poles ... 34

3.2 Studied cases ... 37

3.3 Geometrical and Mechanical calculations ... 40

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I

NDEX

ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)

MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

4.3 Loads and Tensile efforts ... 64

4.3.1 Case A ... 64

4.3.2 Case B1 ... 68

4.3.3 Case B2 ... 71

4.3.4 Case C ... 74

Chapter 5: Conclusions ... 79

Chapter 6: Contributions ... 80

Annex A: Old and New tramway line layouts ... 81

Annex B: Cross Sections ... 82

Annex C: Bill of Quantities ... 83

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I

NDEX OF FIGURES

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MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

Index of figures

Figure 1: Wolverhampton City Centre Extension. New Tramway section. ... 8

Figure 2: weight per unit length (catenary) against weight per horizontal length (parabola) .. 24

Figure 3. Radial force due to contact wire stagger. ... 29

Figure 4. Vertical tensile force due to different contact wire heights. ... 31

Figure 5. Forces applied to poles. ... 34

Figure 6. Cross-span wire supporting the contact wire by using a steady arm with delta

suspension ... 41

Figure 7. Main actions applied to the elements of a head-span arrangement ... 42

Figure 8. Single Pull-off horizontal arrangement... 47

Figure 9. Single Pull-off arrangement. Main acting forces ... 47

Figure 10. Single Pull-off arrangement. X, Y plan sketch with applied forces. ... 49

Figure 11. Single Pull-off arrangement. Three acceptable configurations for the same

solution. ... 51

Figure 12. Double Pull-off arrangement. Main acting forces. ... 52

Figure 13. Double Pull-off arrangement. X, Y plan sketch with applied forces... 54

Figure 14. Double Pull-off arrangement. Vertical section sketch with applied forces. ... 56

Figure 15. Double Pull-off arrangement. Two acceptable configurations for the same

solution. ... 59

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Introduction

ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)

MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

Chapter 1:

INTRODUCTION

1.1

S

UMMARY

The present document is the final mechanical final design of the catenary system equipment for a tramway line. This line runs across the city centre of Wolverhampton, precisely concerning a new section for an existing line.

Due to the particularities of the line and the urban environment, the mechanical design has taken into account a range of inputs and interfaces with other engineering disciplines, as electrical design, civil works and track alignment. The development of the final design shall comply with all these factors, as well as international applied standards for the designing of overhead line equipment.

It is important to highlight that this project comes from a previous delivery stage, in which a basic design was carried out. The final detailed design has suffered several changes in the approach and the scope of the proposed solution. These issues have been implemented within the calculations of the different structures, and the results have been updated and developed to a complete detailed level.

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Introduction

ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)

MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

1.2

O

BJECTIVES

The objectives of the Master’s thesis are:

- Develop the mechanical and geometrical design for the catenary system of a tramway line extension, considering the specific client requirements, the applied design international standards and the integration with urban environment and the existing line.

- Check the calculations in order to fulfill with good mechanical engineering criteria, safety and interoperability, as key points of the design.

- Create a tool to study and calculate different catenary arrangements for tramway lines. The tool shall be capable of carrying out the mechanical calculations and sizing a precise arrangement for a single point.

The objectives of the extension of the Master’s thesis are:

- Provide the detailed information for the constructive project: Final overhead line equipment layout with the allocation of the main structures, installation details, contact wire adjustments, cross sections, bill of quantities and equipment specifications.

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Introduction

ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)

MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

1.3

T

ASKS AND PLANNING

TASK

CODE TASK (MASTER THESIS)

ESTIMATED

TIME

T.1 Study of delivered basic design: track alignment, proposed layout

for OLE structures, design parameters and criteria. 2 weeks

T.2 Study of the applied standards and project requirements. Learning

mechanical design methodology for catenary systems 2 weeks

T.3 Design changes impact assessment. Track alignment, allocation of

supporting structures, safety parameters. 4 weeks

T.4 New layout proposal for the OLE equipment. 2 weeks

T.5 Calculation and sizing of the supporting systems. Check the results

and optimization of the design. 10 weeks

TASK

CODE TASK (EXTENSION OF MASTER THESIS)

ESTIMATED

TIME

T.6 Implementation the results within appropriate drawings: layout and

cross sections. 4 weeks

T.7 Development of detailed information for the project: Bill of

quantities, equipment specifications. 2 weeks

T.8 Check the requirements and defined modifications are fulfilled.

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Introduction

ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)

MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

TASK DEC 15 JAN 16 FEB 16 MAR 16 APR 16 MAY 16 JUN 16

T.1 T.2 T.3 T.4 T.5 T.6 T.7 T.8 report

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Design

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MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

Chapter 2:

DESIGN

2.1

P

ROJECT

O

VERVIEW

Wolverhampton City Centre Metro Extension (WCCE) consist in a new phase of development for the existing Tramway line of the city, being a relevant stage of the intermodal transport planning for the city of Wolverhampton (Wolverhampton Interchange Project).

The main project aim is provide people different options in terms of public transport systems; it will include metro rail, bus and tramway sub-systems. Moreover, one of the key points of the infrastructure development is the interoperability of the different transport means, for example, the bus station and rail station will be redeveloped in order to connect each other by means of pedestrian walkways. New tram stops will be also allocated in order to connect the new tramway line with both the bus station and rail station, allowing easy movement for people between tram and buses or trains.

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Design

ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)

MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

Several companies have been working in cooperation with Wolverhampton City Council since 2009 to develop, assess and refine a preferred route for the Wolverhampton City Centre Metro Extension (WCCE) proposed scheme links the train station to the recently constructed Wolverhampton bus station as an extension to the existing Metro network. The tram currently ends at St George’s and the proposal includes a 700 m extension of twin track from Bilston Street along Pipers Row and across Railway Drive, terminating at Wolverhampton Railway Station. As a result of this extension, an improved bus, rail and tram interchange will be provided to the city.

A wider extension of Midland Metro known as the 5W’s route is planned to extend from Wolverhampton to Walsall and join the existing Line 1 at Wednesbury. WCCE is also a part of this extension.

This document describes the overhead line equipment (OLE) detailed Design of the 200m electrified section of the WCCE between Bilston Street and Wolverhampton bus station to support a Transport and Works Act Order.

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Design

ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)

MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

2.2

S

COPE OF WORKS

The existing tramway Line 1 that currently connects Birmingham Snow Hill and St George´s at Wolverhampton is to be extended in order to link the tramway line to Wolverhampton bus station and Wolverhampton train Station. The proposed extension comprises 700m of which the first 200m are to be electrified. The scope of works consist in producing the OLE detailed design of the proposed Midland Metro – Wolverhampton City Centre Extension (WCCE) tram system between Bilston Street and Wolverhampton Bus station to support the Transport and Works Act Order application.

The OLE design is based on the proposed track alignment fully compliant with the design specifications and is integrated with the existing systems in operation. A description of the existing Line 1 OLE system is included in this document for information.

OLE works include the installation of cross-span wires and pull-offs supported by building fixings, dedicated OLE poles or combined OLE and street lighting poles where needed. It is also required to justify the adopted decisions by appropriate calculations based on accepted standards, client requirements and best mechanical engineering criteria.

2.2.1

D

IFFERENCES AND

I

MPLICATIONS

A previous preliminary OLE design was developed and delivered to the client. Due to the amount of unknown information about the existing OLE system and the constraints that the urban streetscape imposes and limits the design, a new track alignment layout was proposed (See Annex A for differences between old and new track layout) considering the comments of the client and implementing provided data about these issues.

The new track layout makes necessary to redesign all the OLE system, including the client preferences and adjusting it to the existing tramway line configuration.

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Design

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The main implications within the mechanical design of the WCCE are summarized below:

- New layout considers just 200 meters to be electrified with overhead equipment of the 700 m of the whole section. The 500 meters left will be design with a no-catenary system for the trams. Limitations about installing poles in the surroundings of the bus station as well as the high difficulty of anchoring supporting wiring system to the canopy and the facade of the station have lead the decision of modifying the scope of the OLE design.

- Although the client preferences are attaching supporting wires of the OLE equipment of the tramway running new line to the nearest buildings, it has been necessary to check both the

availability of space and the safety of the installation. In some cases buildings have been proved to not be suitable for that purpose, so then the installation of OLE poles have been required.

- The crossing section with the existing line between Bilson Street and Piper’s row has been studied carefully, in order to integrate the new design with the existing one. This has implied the re-arrangement of some OLE poles and the installation of new ones, as well as the use of side bridle arrangements because of the track small radius. The use of existing lighting poles for anchoring OLE structures has to be also checked, in terms of the capacity of those structures to withstand the efforts of the contact wire and auxiliary elements.

All these issues have had an impact in the OLE design, so they have been taken into consideration in the development of the detailed design for the new tramway line.

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Design

ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)

MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

2.3

R

EFERENCED STANDARDS AND DOCUMENTS

The following reference documents have been used, in whole or in part, for the development of this project:

1. Guidance on Tramways

ORR Railway Safety Publication 2, November 2006

2. Developers Technical Guide for Metro Extensions 2002 Revision CENTRO, Issue 1 – April 2002

3. Midland Metro Line One – Overhead Contact Line System – Technical Requirements

Brecknell, Willis & Co Ltd, Issue E – 08 August 1998

4. Contact Lines for Electric Railways – Planning – Design – Implementation Kiessling, Puschmann, Schmieder, Schneider.

5. Design parameters of Line 1 OLE CENTRO Health & Safety files

6. Midland Metro – Code of Practice – Stray Current Corrosion Control Mott MacDonald, June 1989

7. Midland Metro Line One – Voltage Limiting Devices/Earthing & Bonding Design Statement, Kennedy & Donkin Ltd, 10 November 1997

8. Midland Metro Phase II – Technical Report for Outline Business Case – Volume 5 – Five Ws

CENTRO/Atkins, Revision 4 – 28 May 2006

9. Railway Safety Principles and Guidance part 2 section C – Guidance on Electric Traction Systems

ORR Railway Safety Publication 10. Pedestrian Safety

ORR Tramway Technical Guidance Note 2, October 2008 11. Design Standards Stray Current Management

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Design

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MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

ORR Tramway Technical Guidance Note 3, November 2008

12. BS EN 50119:2010 Railway Applications – Fixed Installations – Electric traction overhead contact lines

13. BS EN 50149:2002 Railway Applications – Fixed Installations – Electric traction – Copper and copper alloy grooved contact wires

14. BS EN 50122-1:1998 Railway applications. Fixed installations. Protective provisions relating safety and earthing.

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Design

ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)

MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

2.4

S

YSTEM DESCRIPTION

The 20.7 km long Line 1 from Birmingham Snow hill to Wolverhampton St George’s was opened in 1999. The former Great Western Railway alignment was followed in most of the route.

The line is electrified at 750V DC. There are six substations along the route plus one substation to feed the depot at Wednesbury.

Twin contact wires with an aerial parallel feeder supported by poles and cantilever arms are used for the off-street route.

A single contact wire is used for the street-running and depot. There is no parallel feeder. The OLE is generally supported by span wires. Fixings to buildings and street lighting poles are allocated where possible. When not possible, dedicated poles are used.

For planned and emergency isolation of the OLE, isolating switches are provided along the route, in the substations and at intermediate points in trackside cabinets.

The electric traction current returns to the substations through the running rails.

Main and Ancillary Conductors

- Contact wire type

 150 mm2 hard drawn copper.

- Contact wire tension

 For the off-street route, constant tension controlled by gas tensioners.

 For street-running and the depot, variable tension (fixed termination) with only one continuous tension length.

- Parallel feeder type

 Stranded 19/2.8 mm (120 mm2) hard drawn copper, insulated with polyethylene coating.

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Design

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- Traction feeder cable type

 240 mm2 copper (EPR-CSP insulated) for the positive cable between substation and OLE.

 150 mm2 copper (EPR-CSP insulated) for the negative cable between substation negative busbar and track.

Support and Suspension System Components

- Poles

 Stepped circular hollow section. Steel tube galvanized inside and out. Painted in ‘Midnight Blue’. Top finial is molded rubber ‘Strawberry Red’. They will be attached to foundation via bolt cluster and flange plate.

 Four types, type A for single track cantilevers, type C for single and double track cantilevers, for cross spans and for anchors, type D for feeders at substations and trackside isolators and type E for combined street lighting with cross-spans.

- Cantilevered arms

 Four types: short, medium, long and overlap. Galvanized steel.

- Insulators

 Insulators are used as secondary insulation, between the cantilever arm and pole for cantilever supports, and are of composite material.

 Primary insulation is provided by an insulated steady arm.

- Parafil rope

 Flexible insulated rope used for many of the support assemblies. It provides secondary insulation.

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Design

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 Two types: three meter and street running. The three-meter is used for auto-tensioned equipment and the street running is used for variable tension

equipment. 6 mm parafil rope is used with aluminum bronze clamps and stainless steel terminations.

- Steady arms

 Two types, plain and insulated. Where a plain arm is used, insulation is gained via a parafil sling connecting the arm to the cantilever or span wire. For variable tension areas where radial loads are small, steady arms are not used. Instead the delta suspension provides the registration.

- Span wires

 8 mm diameter parafil rope with aluminum bronze parafil terminals, aluminum anchor clamp, and nylon Cardan suspension clamps.

- Section insulators

 Lightweight construction fitted with overlapping runners for uninterrupted current collection, bi-directional.

Foundations

- Concrete piles with three arrangements of bolt cluster. Four types of pile with varying load capacities.

 Type A: 64 kNm bending load; 10.2 kN vertical load; 14.3 kN lateral load.

 Type C: 133 kNm bending load; 20 kN vertical load; 19.9 kN lateral load.

 Type D: 145 kNm bending load; 25 kN vertical load; 19.9 kN lateral load.

 Type E: 145 kNm bending load; 25 kN vertical load; 19.9 kN lateral load.

Conductor Profile

- Contact Wire Height Off-Street Running

 Nominal design height is 4800 mm.

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Design

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- Contact Wire Height Street Running

 Normal minimum is 5800 mm.

 Normal maximum is 6300 mm.

 The contact wire is at a minimum height of 6550 mm across the junction of Bilston Street with the Ring Road at Wolverhampton.

 In a failure of one pole or support the contact wire should not sag below 5200 mm above the highway. Spacing of existing poles suggests this criteria is not met and instead an alarm system may have been implemented whereby the Operator is alerted to an incident before safety of pedestrians is compromised.

- Contact Wire Grading

 Maximum grading is 1:40 for all areas.

- Contact Wire Stagger

 Nominal stagger is ± 200 mm.

- Parallel Feeder Height

 At least 5200 mm above platform or rail level, whichever is the higher.

- Span Length

 Optimum span length is 45 m with a 3 m stitch.

Environmental

- Maximum design temperature is 35 ⁰C. - Minimum design temperature is -15 ⁰C. - Maximum design wind speed is 25 m/s. - Maximum gust speed is 42 m/s.

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Design

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- Clearance to People

 Minimum vertical clearance between public or restricted access standing surfaces and live parts of the OLE is 5200 mm.

- Electrical and Mechanical Clearances

 Minimum static electrical clearance between live parts of the OLE, including any non-insulated 750 V DC conductor, and earth or any metal at earth potential is 100 mm.

 Minimum passing electrical clearance between live parts of the OLE, including pantograph, and structures or earth is 50 mm. Allowance is made for contact wire uplift, pantograph sway, track tolerances and wear of the pantograph and contact wire.

 Minimum mechanical clearance between pantograph and common live metal is 80 mm, except to steady arms attached to the contact wire, where it is 15 mm. These clearances allow for contact wire uplift, pantograph sway, track tolerances and wear of the pantograph and contact wire.

Earthing and Bonding

Accessible voltages should not exceed 60 V and the design of electric traction power supply should ensure it. The study of the earthing and bonding design criteria is out of the scope of the present document.

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Design

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2.5

I

NPUTS

,

A

SSUMPTIONS AND REQUIREMENTS

In this section of the present document it is going to be defined the main considerations and constraints to be taken into account in the design process for the overhead line equipment.

The first step in the design process has been determining the location of the supporting elements for the different OLE arrangements, like poles and building fixings. In order to develop the layout with the position of these elements, the inputs that have been considered are:

- Information provided from the client based upon the proposed new line drawings.

- Design parameters for the existing OLE equipment in existing tramway line within the city centre. - Track alignment design.

Design Parameters

The following parameters have been used to develop this design: - Single contact wire of 150 mm2 section.

- Variable tension (fixed termination) contact wire with only one continuous tension length. The contact wire nominal tension for the designed OLE system will be 15 kN.

- Contact wire height

 Normal minimum 5200 mm.

 Nominal 6000 mm.

 Normal maximum 6300 mm. - Contact wire stagger

 200 mm tangent track – nominal.

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Design

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Support Poles

16 new support poles are required as part of this design. Proposed poles are allocated following the recommendations of selected manufacturer design range. The poles are to be painted ‘Midnight Blue’ and finished with a top finial which is moulded rubber in ‘Strawberry Red’ colour to match the existing poles.

Poles shall be attached to the foundation through a bolt cluster and flange plate buried below ground in the street areas to provide a smooth surface around the pole.

All OLE poles that are liable to high vehicle traffic areas are to be protected. This issue shall be taken into consideration, but the design of the protection for the new poles is out of the scope of the current project.

There are four types of poles that are included in the design for the current project to withstand the OLE equipment. The specific data of the type of poles used for this project can be found in Annex D:

POLE WORST OTM

(KNm)

POLE

TYPE

Pole 1 45.28 F273

Pole 2 101.47 F406

Pole 3 123.41 F406

Pole 0 126.99 F406

Pole 4 68.42 F323

Pole 5 75.16 F323

Pole 6 50.54 F273

Pole 7 37.85 F273

Pole 8 10.12 F219

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Design

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Pole 10 102.98 F406

Pole 11 28.38 F273

Pole 12 41.79 F273

Pole 13 16.98 F273

Pole 14 42.50 F273

Foundations

Foundation works are required as part of this design. The load cases provided by the OLE calculations will provide foundation requirements so that a suitable foundation will be allocated for each proposed pole. Civils team requires the acting forces at the base of each considered pole in order to design the precise foundation. Mechanical team is in charge of providing these data for developing the foundations of the project.

The preferred type of foundations is the gravity slab type. The following types of foundations have been considered in the design. They are associated to each type of pole.

FOUNDATION

TYPE

POLE TYPE DIMMENSIONS NUMBER OF

STUDS

1 F219 1.8 x 1.8 x 0.9 6

2 F273 2.3 x 2.3 x 0.9 6

3 F323 2.6 x 2.6 x 1.0 6

4A F406 3.0 x 3.0 x 1.0 8

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Design

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BUILDING

FIXING

RESIN ANCHOR

TYPE

BF 1 M20 6 kN

BF 2 M20 6 kN

BF 3 M20 6 kN

BF 4 20 kN

BF 5a 20 kN

BF 5b M20 12 kN

Span wires

Where span wires are to be allocated, they will be of 8mm stainless steel with made of ends.

Registrations

All registration and support equipment associated with this design is to be allocated from selected manufacturer design range.

Insulators

All insulators required as part of this design are to be of the polymeric type. Insulators are to be allocated from selected manufacturer design range.

Earthing and bonding

Accessible voltages should not exceed 60 V and the design of electric traction power supply should ensure it.

- Poles

 Poles are not bonded directlty to running rails as this would create an undesirable path for stray DC cuurrent to return to the substation, i.e via the ground and any buried services. Poles are protected from fault currents by double insulation between the pole and the contact wire.

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Design

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- Cross Bonding

 To reduce stray currents to an acceptable level, the traction return rails are cross bonded at regular intervals to provide a parallel low resistance path for return currents to the substation.

- Structures at risk to fault currents

Confirmation is sought as to whether this arrangement was installed:

 Where poles may be at risk to fault currents resulting in unacceptable touch voltages, e.g a feeder pole that carries traction feeder cables, a voltage limiting device is connected between the pole and the running rails.

 Metallic structures inside the contact wire de-wirement zone are not bonded directly to running rails but instead are connected to them using a voltage limiting device, to mitigate against dangerous touch voltages.

Mechanical calculations: parabolic approximation of catenary wires

Prior to proceed with the mechanical methodology for calculating the forces that the proposed OLE system shall withstand, and conditioning the preferred arrangement solution for each case, it is necessary to define the assumptions that have been considered for the calcs.

The contact wire between two supporting points has a catenary wire shape, and the material and type of the contact wire has been specified with a weight per unit length of 1,334 kg/m. The equation that defines a typical catenary shape wire is:

𝐻 ∙

𝜕

2

𝑧

𝜕𝑥

2

= −𝑚𝑔

𝜕𝑠

𝜕𝑥

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Design

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Figure 2: weight per unit length (catenary) against weight per horizontal length (parabola)

𝜕𝑠 ≈ 𝜕𝑥 → 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑖𝑐 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛

2

𝑧

𝜕𝑥

2

= −

𝑚𝑔

𝐻

With this assumption, the calculation of vertical loads of the contact wire due to its self-weight turn easier, but without losing accuracy in the results. In the next table there is represented a comparative between the results of considering either the catenary expression or the parabolic approximation.

mg (N/m) H (kN) fparab (m) fcat (m)

10 15 0,3 0,30001

20 15 0,6 0,60008

10000 15 30 41,23

It is observable that with the parameters of the contact wire used in the project: P’= 13 N/m and H= 15 kN, the error that the parabolic approximation introduces in the calculations are less than 0,01%

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regarding the maximum vertical distance that the contact wire reach in a span length. As a conclusion, the parabolic approximation for the contact wire is a good assumption for the mechanical design process.

Finally, it is also considered that the wire flexural rigidity is nearly 0, which implies that all the conductors considered in the mechanical design will work only with axial traction efforts. This is also a common practice within OLE design and it doesn’t introduce inaccuracies in the results.

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2.6

N

EW

OLE

L

AYOUT

D

EVELOPMENT

In order to accommodate the new Permanent Way of the WCCE at Bilston Street Junction, significant alterations to the existing OLE arrangement are required. The design of the junction has been based on the running lines at Manchester Piccadilly Gardens, where dwell lines and offset tracks converge. The best way to integrate the existing line with the new supporting infrastructure has been carefully considered due to the complexities of the area. The following considerations have been taken into account for the design of the Junction:

- Ground conditions, affecting in the contact wire height adjustment. - Accommodation and protection of poles within footways and islands. - Building fixing compatibility (if appropriate).

- Suitability of existing support infrastructure to accommodate additional pull off loads, supporting wires from horizontal arrangements or head-spans, etc.

- Rationalisation of existing and new support structures (overview check).

Existing poles W172 and W144 have been used to allocate new cantilever arms to span the WCCE line. These poles are now named Pole 1 and Pole 3 respectively. The following reasons have determined the use of a side-bridle arrangement structure with pull-off arrangement (backbone arrangement).

- The small track radius, which originates sufficiently high radial loads to allow steady arms to be supported only from one side of the track, instead of being supported by cross-span wires.

- The desire to use the minimum poles within footways and islands to reduce the risk of being hit by traffic and also to avoid the necessity of protecting the poles from vehicles.

It is to be noted that the anchors of the wire runs at the beginning of the extension have not been defined due to lack of existing information. In this case assumptions will be made to anchoring points.

In the rest of the line, the allocation of building fixings has been proposed, where possible, in order to minimise the use of dedicated OLE poles and reduce the visual effect of OLE.

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Chapter 3:

CALCULATIONS

In this section of the document it will be defined the process of determining the mechanical and geometrical design parameters for the overhead line equipment.

3.1

L

OADS

Contact wires and contact lines are subjected to vertical and horizontal forces that stress and displace them. These forces are called “Loads”.

Loads group the forces that the supporting elements of the contact wire are supposed to withstand. The mechanical design of the overhead line shall be ensure that loads and stresses are within specific limits as per referenced standards. Depending on the studied case, different loads caused by different phenomena will be applied on specified points as well as with precise module and direction.

Loads can be classified as permanent and variable actions, in accordance with civil engineering principles. Dead weights and constant tensile forces for the contact wire are the main permanent forces considered in the design; wind and ice loads are part of the variable actions.

It is also important to separate loads acting on wires and conductors with those that are applied on structural elements such poles and foundations.

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3.1.1

L

OADS ON WIRES

Self-weight

It is a vertical load resulting from the dead weight of wires. They can be calculated with the weight per unit length of the contact wire value, which is defined by the type and material of the contact wire used in the project. The following equation calculates this value:

𝐺𝑐𝑤[𝑁] = 𝑚′𝑐𝑤 ⋅ 𝑔 ⋅ 𝑙

Where:

- m’CW = 1,334 kg/m is the mass per unit length of the contact wire.

- g = 9,81 m/s2 is the gravity value.

- l= is the span length between continuous supporting points in meters (m).

Considering the arrangement of the overhead line equipment, in which each supporting point is set between two spans, the total vertical load due to the self-weight of the contact wire that will be applied in the supporting points shall be:

𝐺𝑖,𝐶𝑊 [𝑁] = 𝑚′

𝑐𝑤 ⋅ 𝑔 ⋅ (

𝑙𝑖−1

2 + 𝑙𝑖+1

2 )

Where li-1 , li+1 are the lengths of the previous and following spans between the supporting element

“i”.

Constant tensile forces

These are loads due to the contact wire tensile stress. They can be divided into two components: Radial forces and Vertical forces because of difference of height between successive supporting points.

Radial forces:

Radial forces occur as a result of the change of direction of the contact wire, which is produced by the supporting elements of the overhead line equipment. The change of direction for the contact wire is

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also called stagger. It is intentionally considered in the design for two main reasons: adapt the contact wire running path to the curves of the track and ensure an homogeneous wear of the pantograph surface due to the friction between that and the contact wire. The stagger range is limited by the characteristics of the line and it is also an input value for the design.

Radial forces appear in a plane defined by the contact wire and the applied stagger to it, and with the direction from/to the supporting element of the contact wire (poles or pull-offs normally), depending of the stagger. These loads can also change with the span length.

Along straight stretches of the track, the radial/horizontal forces are the result of the stagger and the anchoring geometry. Otherwise, in curved track sections the radial forces appear because the pull-off arrangements that adapt the contact wire to the track layout.

Figure 3. Radial force due to contact wire stagger.

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- α = is the angle between contact wire in the spans between the supporting element. It is variable with the stagger and with the running line of the track.

If the contact wire angle due to the stagger cannot be easy determined, the radial force can also be calculated with the next equation:

𝑇𝑅 [𝑁] = 𝑁 ⋅ 𝑇𝐶𝑊⋅ [

𝐷 − 𝐷0 𝐿0 +

𝐷 − 𝐷1 𝐿1 ]

Where:

- D, D0 and D1: are the lateral positions of the contact wire at the supports in m

- L0 and L1: are the span lengths in m.

Vertical tensile forces:

The tensile stress of the contact wire not only causes a radial force parallel to the track plane but it can also appear a vertical force due to the different elevation of the supporting points across the line path. This force has to be considered because if the neighboring supports have higher elevation than the studied one, it counteracts the self-weight load, the contact wire could be uplifted when the train pantograph force appears and as a result the contact between contact wire and pantograph could disappear. This is one of the main goals of the OLE design: avoiding that contact wire supporting points to have positive vertical reaction forces. It is mandatory to check this issue and adjust the heights of the supporting arrangements where required.

On the other hand, if the adjacent supports have lower elevation, the vertical reaction would be increased and risk of contact wire uplift is minimized. This can be observable in the next sketch.

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Figure 4. Vertical tensile force due to different contact wire heights.

Vertical tensile force can be calculated as the projection of the contact wire tensile stress with the Z axe with the following expression:

𝑇(𝑖)𝑅,𝑍 [𝑁] = 𝑇𝐶𝑊 ⋅ [𝑍𝑖− 𝑍𝑖−1

𝐿0

𝑍𝑖− 𝑍𝑖+1 𝐿1 ]

Where:

- Zi-1, Zi, Zi+1: are the track elevations at previous, precise and continued points where the

supporting elements are established, in meters.

- L0 and L1: are the span lengths between three consecutive supporting points, in meters.

- TCW = 15 kN, is the nominal tensile force for the contact wire.

If the dead load of the contact wire is added, the total vertical reaction force at support “i” can be completely defined:

𝑉𝑖[𝑁] = 𝑚′

𝑐𝑤 ⋅ 𝑔 ⋅ (

𝑙𝑖−1

2 + 𝑙𝑖+1

2 ) + 𝑇𝐶𝑊 ⋅ [

𝑍𝑖− 𝑍𝑖−1

𝐿0

𝑍𝑖− 𝑍𝑖+1

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F′W [𝑁 𝑚⁄ ] = qz· GC· CC· d

Where

- GC = 0,75 is the structural response factor for conductors taking into account the

response of movable conductors to wind loads. - CC = 1,0 is the drag factor of the conductor.

- d = 0,014 m is the diameter of the contact wire.

qZ is the characteristic dynamic wind pressure acting on elements of overhead contact lines, in N/m2

according to:

qZ= ρ/2 · Gq· Gt· V2

Where:

- ρ = 1,225 kg/m3 is the air density (15ºC and 600 m height).

- Gq = 2,05 is the gust response factor.

- Gt = 1,0 is the terrain factor taking into account the protection of lines, e.g. in cuts, cities or forests. In open terrain Gt is considered as 1,0.

- V = 42 m/s is the survival gust speed in m/s.

Therefore, the wind load per unit length in exposed conditions will be F′W= 23,26 N/m.

Ice load

Snow and ice load shall be taken into account at temperatures up to + 5 º C, where applicable. They are considered as an uniform, increasing dead load in the wires. The load should be specified as nominated in the following, which contains values that are valid for conductors in the usual diameters, i.e. 10 mm and 20 mm.

Class

Ice load

(N/m)

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Class

Ice load

(N/m)

I1 (low) 3,5

I2 (medium) 7

I3 (heavy) 15

Due to the environmental conditions, a medium load (I2) has been taken into account. Then, the ice load per unit length on the contact wire will be F′I= 7 N/m.

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3.1.2

L

OADS ON POLES

Basically, the applied loads on poles have the same origin as those applied on wires and conductors. The purpose of the OLE mechanical design is to define the forces and moments that apply to the base of the pole completely, due to the fact that these structures shall withstand all the loads applied in both the contact wire and supporting wires. With these data, civil engineers are able to design the foundations for the whole supporting system.

The next picture shows a summary of the principal forces acting on a single OLE pole.

Figure 5. Forces applied to poles.

Self-weight

The load per unit length of a pole (G’P) is calculated as before:

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Where:

- m’P = 30 kg/m is the mass per unit length of the pole (assuming an average pole

diameter of 250 mm).

- g = 9,81 m/s2 is the acceleration because of gravity.

For this project, the poles have been selected from a specific manufacturer catalogue, so the total weights for every considered pole are defined (See ANNEX D: Equipment specifications)

Vertical forces

As well as the self-weight of the pole, there is also a vertical force produced by the related reaction of the cross span wire that is transmitted by supporting wires and shall considered to be assumed by the pole.

Horizontal forces

Horizontal forces acting on poles are due to the radial forces of head-span and cross-span wires. The value of these forces should be evaluated for each considered arrangement, and where poles are considered as supporting elements for the head span wires.

Wind load

According to EN 50119, wind forces on structures can be calculated with the following equation:

𝑄𝑊𝑠𝑡𝑟= 𝑞𝑍· 𝐺𝑠𝑡𝑟· 𝐶𝑠𝑡𝑟· 𝐴𝑠𝑡𝑟

Where

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𝐴𝑠𝑡𝑟 = ℎ · 𝑑

Where

- h = 8 m is the height of the pole and

- d = 0,25 m is the diameter of the pole.

Ice load

The ice effects on structural elements shall also be taken into account. It is required to consider a 10 % increase in self-weight of both the pole and the cantilever due to ice loads.

Reactions at the base of the pole:

As a result of the forces applied to the pole, some reactions will appear at the base of it, and they will be the input data for the appropriate design of the foundation:

𝑅𝑍 [𝑁] = 𝐺′𝑃 + 𝑉𝐶𝑊

𝑅𝑌[𝑁] = 𝐻𝐶𝑊+ 𝑄𝑊

𝑀𝑋𝑌[𝑁𝑚] = 𝐻𝐶𝑊∙ ℎ𝑝+ 𝑄𝑊∙ ℎ𝑔

Where:

- G’P is the weight of the pole (with the considered increase of the ice load, where applied)

- VCW is the vertical component of the force that the supporting wires apply to the pole structure.

- HCW is the horizontal component of the force that the supporting wires apply to the pole structure.

- QW is the wind load applied to the pole.

- hp is the maximum height where the supporting wires are attached to the poles. It is a conservative

criteria for calculating the moment at the base of the pole.

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3.2

S

TUDIED CASES

The load cases that shall be considered for each type of structure are summarized in the following table from the referenced standard EN 50119:

Type of structure

Load case to be considered

A Minimum temperature f

B Wind g

C Ice e, h

D Wind and

ice e, h

E Construction

and maintenance e, h

F Accidental g

1a Cantilevers hinged X X X X X -

1b Cantilevers rigid X X X X X X

2 Head spans X X X X X X a

3 Portals / Crossbeams X X X X X X

4 Suspension

structures X X X X X -

5 Pull-off structures X X X X X -

6 Midpoint anchor X X X X X X

7 Midpoint structures X X X X X -

8

Structures for flexible and rigid cross supports

X X X X X X a

9

Structures for horizontal catenary wire arrangements

X X X X - X

10 Tensioning

structures X X X X X X

c

11

Structures with feeder and parallel reinforcing lines

X X X X X X e

12

OCS structures carrying additional power lines b

X X X X X X

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h

Temperature to be assumed with ice loads and combined wind and ice load, where relevant. See 6.2.7 of EN 50119.

In particular, the types of structures applicable for the present project, according to EN 50119, are:

- Head spans: Head spans carry overhead contact lines by means of rope elements and insulators under tensile load only. The relevant load cases are A, B, C, if necessary D, if necessary E and F. The latter is applicable only for head spans with midpoints.

- Structures for flexible and rigid cross-supporting structures: Structures designed to resist the forces resulting from any kind of cross-supporting structures such as head spans, cross beams and cross spans. The relevant load cases are A, B, C, if necessary D and if necessary E. Load case F shall be considered if a midpoint is arranged there.

- Structures for horizontal catenary wire arrangements: At structures for horizontal catenary wire arrangements forces act in several directions and at different heights simultaneously. The relevant load cases are A, B, C, if necessary D and F.

NOTE: A horizontal catenary wire is an arrangement where the contact wires are supported from wires that are mainly in a horizontal position. This arrangement is mainly used within urban areas. The masts or buildings where the horizontal wires are fixed can be relatively far from the tracks.

- Anchor supports: Anchor supports are structural elements to resist the tensile forces of stay wires supporting structures of contact lines. The load cases shall be selected according to the type of anchored structure.

- Support foundations: Foundations shall be considered in accordance with 6.5 of EN 50119. The calculation of the foundations is out of the scope of the present project. Civils team requires the resulting forces in the poles for developing the design of the foundations.

On the basis of the abovementioned information, the designed overhead line equipment shall be calculated to withstand three different scenarios:

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- Case A: Permanent Loads. Constant tensile forces and self-weight considered as main actions in the supporting arrangements.

- Case B: Permanent Loads + Wind. Conductor tensile forces increased by the action of wind and wind loads acting on each element in the most critical direction.

- Case C: Permanent Loads + Ice. Conductor tensile forces increased by the ice loads and also ice load in structures shall be considered.

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3.3

G

EOMETRICAL AND

M

ECHANICAL CALCULATIONS

The calculation process for the different OLE arrangements used in the design process is explained in this section. Starting with the input data of the project, the calculated loads for each studied case and some assumptions made; the main geometric parameters as well as the forces in wires and other structural elements can be perfectly determined.

Depending on the arrangement used for the supporting of the contact wire, the calculation process will be applied for each one.

3.3.1

F

LEXIBLE CROSS

-

SPAN ARRANGEMENTS

Flexible cross-supporting structures, called head-spans, carry the vertical loads due to the self-weight forces and vertical component of contact wire tensile, by means of a stressed head span wires. They also assumed the radial forces because of curved tracks and stagger.

The head-span wire geometry is determined by the position and defined load state imposed from each supported contact wire. The purpose of rating the head span wire is defining their lengths and tensile forces (cables only can perform axial traction forces) in order to determine the reactions on the supporting poles.

Head-span arrangements are able to support two or more contact wires without the required space between tracks needed to install individual poles for each track. Due to the fact that the designed line goes throughout a urban environment, the anchorage points of the head-span wires have been fixed to the surrounding buildings where possible by means of building fixings, so the installation of dedicated poles for the head-span arrangements are not needed. This issue avoids the impact in the urban streetscape that the poles could have because of the lack of space in the streets, as well as it provides more flexibility to adjust the overhead line to an existing environment.

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Figure 6. Cross-span wire supporting the contact wire by using a steady arm with delta suspension

Head span arrangements for tramway lines without additional feeder wire have one supporting wire that is directly joined to the contact wire by means of either steady arm or delta suspension of both. This configuration provides enough stiffness to the arrangement as well as anchoring points of the required wires; and structural elements such as poles and building fixings are reduced.

Rating of head-span arrangements

The first step to calculate a precise head-span arrangement shall be known the loads that the supported contact wires cause in the head-span wire. This process has been widely explained in section 3.1. Once the Loads acting on contact wires are known, the geometry can be calculated as well as the tensile forces of the head-span wires. In the next sketch, the main parameters of a typical head-span arrangement are represented:

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Figure 7. Main actions applied to the elements of a head-span arrangement

The distances between the main elements of the cross-span arrangement can be obtained for each case. Values a, b and c depend on either the proposed layout for the line or the available space on the street for establishing dedicated poles or the possibility of anchoring the head-span wires to adjacent buildings. These values are considered as inputs for the calculation process.

Heights of points 1 and 2 are also defined with the proposed track elevation according to the track line configuration. Because the height of the contact wire above the track is perfectly specified in the project requirements (see section 2.5 of this document), the difference of height between contact wire points 1 and 2, ΔZ12 relies on the difference of track elevation and the imposed variation of height

made at the beginning of the design process, so ΔZ12 is also a value defined prior the rating process is

developed.

Once the data from inputs and related information have been determined, the values that have to be obtained shall be counted:

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- Tensile forces of the three head-span wires: V1 and H1 from cable 1, V2 and H2 from

cable 2 and V3 and H3 from cable 3. There are 6 unknown values to be determined.

- Geometry for the heights of wires 1 and 2. Relative height of wires 1 and 2 define the point on the pole they should be anchored to. ΔZA1 and ΔZB2 are also unknown values

to define.

As a result, there are 8 unknown values. Then, balance equations for horizontal and vertical actions at points 1 and 2 are outlined. It is required that all the acting forces on both points be balanced, in order to achieve mechanical and geometrical balance.

∑ 𝐹𝑥 = 0

∑ 𝐹𝑌 = 0

Point 1:

𝑇

𝑅1

+ 𝐻

2

= 𝐻

1

(1)

𝑉

1

+ 𝑉

2

= 𝐹

𝑉1

(2)

Point 2:

𝐻

3

= 𝐻

2

+ 𝑇

𝑅2

(3)

𝑉

3

= 𝐹

𝑉2

+ 𝑉

2

(4)

In addition to forces balance equations on points 1 and 2, there are three expressions that establish equivalences between forces and geometry, due to the assumptions made for the supporting wire

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Wire 2:

∆𝑍12

𝑐

=

𝑉2

𝐻2

(6)

Wire 3:

∆𝑍𝐵2

𝑏

=

𝑉3

𝐻3

(7)

It is not possible to define the geometry and the mechanical actions for the head-span arrangement with 7 equations and 8 unknown parameters,. In order to solve this, the gradient of either cable 1 or cable 3 is set previously. Good engineering practices for OLE design establish that this value can range from 1:10 to 1:15, but this range could be exceeded if the surrounding restrictions don’t allow it, with proper justification for each case. With this assumption there is an extra equation, so the system is perfectly defined.

Depending on the restrictions for anchoring the wires 1 and 3 to adjacent buildings or poles, the pre-set of the gradient will be made for one or another wire. In this case it is considered that head span wire 1 gradient is defined. This gradient can also be modified in order to obtain homogeneous forces and similar geometry for the head-span wires, within an iterative designing process.

∆𝑍𝐴1

𝑎 = 𝑘 ∈ [ 1 10,

1 15]

After defining the gradient of wire 1, the rest of the unknown values can be determined from equations (1), (2), (3), (4), (5), (6) and (7). In the next table the unknown values are isolated with related expressions.

VALUE

EXPRESSION

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H

1

𝐹

𝑉1

− ∆𝑍

12

𝑐 ∙ 𝐹

𝑅1

∆𝑍

𝐴1

𝑎 +

∆𝑍

𝑐

12

V

1

∆𝑍

𝐴1

𝑎

∙ 𝐻

1

H

2

𝐻

1

+ 𝐹

𝑅1

V

2

∆𝑍

12

𝑐

∙ 𝐻

2

H

3

𝐻

2

− 𝐹

𝑅2

V

3

𝐹

𝑉2

+ 𝑉

2

∆𝑍

𝐵2

𝐻

𝑉

3

3

∙ 𝑏

With these parameters defined, the total tensile stress for each head-span wire can be determined, as well as the required length of cable for the three supporting wires, using the following equations:

For head-span wire tensile:

𝑇𝑖 [𝑁] = √𝑉𝑖2+ 𝐻𝑖2 𝑖 = 1,2,3

For the length of wires:

𝐿1 [𝑚] = √∆𝑍𝐴12+ 𝑎2

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3.3.2

H

ORIZONTAL REGISTRATION ARRANGEMENTS

Horizontal registration arrangements are also called side-bridle arrangements; they can be found in the central urban areas of mass transit installations. They are designed to permit the support of trolley wire systems from either walls of adjacent buildings or dedicated poles that can be set relatively far from the rail line track.

These arrangements can also adapt the contact wire lines above crossing and branching tracks at large squares. The poles where the different supporting wires are attached can be installed at locations where they do not interfere with the road traffic as well as with the urban streetscape.

For this project, it has already been mentioned that the tramway line runs across the city centre of Wolverhampton, so these types of OLE solutions are recommended and they have been implemented in the design. The application of these solutions is suitable within sections with small curve radius, in order to adapt the contact wire running path to the track

There are two different configurations of horizontal registration arrangements calculated for this project, which can be differentiated with the number of pull-offs that the supporting wires withstand: single and double pull-off horizontal arrangements. While in single arrangement there are three supporting wires that shall be calculated, in the double one there are five. That is the reason why the rating process of these two configurations is slightly different so they are explained in separated sections:

3.3.2.1

Single pull-off horizontal registration arrangements

As well as in the case of head span arrangements, it is necessary to know the loads that are acting to the single pull-off arrangement prior to calculate the tensile forces and the geometry of the supporting wires. The loads can be defined following guidelines of section 2.5.

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Calculations

ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)

MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

Figure 8. Single Pull-off horizontal arrangement.

As it can be observed in the

Figure 8

, in a horizontal single pull-off arrangement there are three main conductors that support the loads and keep the contact wire in their precise position to allow the pantograph collect the current in the best way. All the wires are supposed to be balanced so related equations and expressions shall be considered to define all the geometrical parameters and tensile efforts.

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Calculations

ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)

MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

Contact wire vertical (FV) and horizontal (TR) loads are applied to point O. Both combined result in a

tensile force FT (x,y,z) that has the direction of the supporting wire OP. The tensile stress for cable OP

is defined with the combination of loads applied to it:

𝐹𝑇 [𝑁] = √𝐹𝑉2+ 𝑇𝑅2 = 𝑇𝑂𝑃 [𝑁]

The wire OP transmits this tensile stress TOP until point P, in which two other wires (BP and AP) join

with it. In this point P is where the balance equations shall be studied. Before that, the height above the track of point P is defined, considering that the height of point O is previously known due to the proposed track layout and the input data for contact wire height range, with the following expression:

𝑍𝑃 [𝑚𝑚] = 𝑍𝑂+ ∆𝑍𝑂𝑃

∆𝑍𝑂𝑃 [𝑚𝑚] = 𝐹𝑉 𝑇𝑅 ∙ 𝑙𝑂𝑃

Where:

- ZP and ZO are the total height above the track of respectively points O and P.

- lOP is the horizontal distance between points O and P, measured in the layout plan (x,

y).

Once the point P is defined in terms of applied forces and relative position (x,y,z coordinates are known), the balance equations in the three main axes can be set. Looking at

Figure 9

, the balance equation in the direction of Z axe is obtained with the vertical components of the tensile forces of wires AP and BP as well as the vertical load applied in O and transmitted to P by means of wire OP (FV):

∑ 𝐹𝑍 = 0;

𝐹𝑉= 𝑇𝐴∙ cos 𝛽𝐴+ 𝑇𝐵∙ cos 𝛽𝐵

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Calculations

ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)

MÁSTER UNIVERSITARIO EN SISTEMAS FERROVIARIOS

- βA and βB are the angles between wires AP and BP and the horizontal plan (x,y)

- TA and TB are the tensile efforts of wires AP and BP in N.

It is helpful to see the horizontal projection of the system for setting the balance equations in x and y axes, as it can be observable in the

Figure 10

:

Figure 10. Single Pull-off arrangement. X, Y plan sketch with applied forces.

∑ 𝐹𝑋 = 0 ;

𝐹𝑇∙ cos 𝛼𝑃= 𝑇𝐴𝐻∙ cos 𝛼𝐴+ 𝑇𝐵𝐻∙ cos 𝛼𝐵 = 𝑇𝐴∙ cos 𝛽𝐴∙ cos 𝛼𝐴+ 𝑇𝐵𝐻∙ cos 𝛽𝐵∙ cos 𝛼𝐵

Figure

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