Structural properties and out-of-plane stability of roller bent steel arches

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Structural properties and out-of-plane stability of roller bent

steel arches

Citation for published version (APA):

Spoorenberg, R. C. (2011). Structural properties and out-of-plane stability of roller bent steel arches. Technische

Universiteit Eindhoven. https://doi.org/10.6100/IR716581

DOI:

10.6100/IR716581

Document status and date:

Published: 01/01/2011

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Structural properties and out-of-plane stability of roller

bent steel arches

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Structural properties and out-of-plane stability of roller bent steel arches

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de

Technische Universiteit Eindhoven, op gezag van de

rector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor

Promoties in het openbaar te verdedigen

op dinsdag 13 september 2011 om 16.00 uur

door

Roeland Christiaan Spoorenberg

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Dit proefschrift is goedgekeurd door de promotoren: prof.ir. H.H. Snijder en prof.dr. D. Beg Copromotor: dr.ir. J.C.D. Hoenderkamp

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Samenstelling van de Promotiecommissie:

prof.ir. E.S.M. Nelissen (voorzitter) Technische Universiteit Eindhoven

prof.ir. H.H. Snijder Technische Universiteit Eindhoven

prof.dr. D. Beg University of Ljubljana

dr.ir. J.C.D. Hoenderkamp Technische Universiteit Eindhoven

prof.ir. F. Soetens Technische Universiteit Eindhoven

prof.dr.ir. L.J. Sluys Technische Universiteit Delft

prof.dr.ir. Ph. Van Bogaert Universiteit Gent

dr.ir. J. Maljaars TNO

ISBN 978-90-77172-76-6 First printing July 2011

Keywords: Arch, Out-of-plane stability, Roller bending process, Residual stresses, Mechanical properties, Finite element analyses.

This thesis was prepared in MS-Word by the author and printed by Ipskamp Drukkers B.V. Cover design: Roel Spoorenberg

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the copyright holder.

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This research was carried out under the project number MC1.06262 in the framework of the Research Program of the Materials innovation institute (M2i)

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Summary vii

Summary

Structural properties and out-of-plane stability of roller bent steel arches

In contemporary architecture the use of steel arches has seen a significant increase. They are applied in buildings and large span bridges, combining structural design with architectural merits. For arches lacking lateral support (or freestanding arches) the out-of-plane structural stability behavior is the decisive design criterion. However, suitable methods or design rules to assess the out-of-plane structural stability resistance of arches are lacking and the collapse behavior is often unknown. Nowadays engineers have to perform laborious calculations which can lead either to conservative or nonconservative arch designs.

This Ph.D. project is aimed at studying the out-of-plane structural stability behavior of steel arches, and developing design rules for these arches. The out-of-plane structural stability behavior was studied by means of geometrical and material non-linear finite element analyses including structural imperfections with ANSYS v. 11.0. The investigation was confined to wide flange circular freestanding arches which are subjected to in-plane vertical loads and manufactured by the roller bending process. The roller bending process is a manufacturing technique by which steel members are bent at ambient temperature into circular arches. It was expected that the residual stresses and mechanical properties (e.g. yield stress, ultimate tensile stress) are altered due to roller bending. Since the alteration of residual stresses and mechanical properties (imperfections) can affect the out-of-plane structural stability of freestanding steel arches, the influence of the roller bending process was studied first. Residual stress measurements and tensile tests were conducted on both straight and roller bent members to assess the influence of roller bending. In addition to the experiments, finite element simulations of the roller bending process were performed in the ANSYS v. 11.0 environment to estimate the residual stress distribution in the arches. Good agreement between the experimentally and numerically obtained residual stresses in roller bent arches was observed.

Based on the experimental and numerical studies of the imperfections in roller bent arches a residual stress model and distribution of mechanical properties across the steel bent section were proposed which serve as the initial state of a roller bent arch when assessing its structural performance by means of non-linear finite element simulations. Numerical analyses showed that the residual stresses in roller bent arches have a minor influence on the load carrying capacity. However, the alterations of the mechanical properties can result in a significant reduction of the arch strength.

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The existing column curve formulations as given in EC3 were adapted to include out-of-plane buckling of arches by altering the form of the imperfection parameter. The column curves can give an accurate prediction of the out-of-plane buckling load provided an appropriate imperfection parameter is selected and the non-dimensional slenderness is known. Based on numerous finite element calculations an imperfection parameter curve was derived which was substituted into the column curve formulation rendering a column curve for roller bent arches failing by out-of-plane buckling. Finite element results showed that multiple column curves were necessary to capture the out-of-plane buckling response of arches for various load cases and steel grades. The column curves require the determination of the non-dimensional slenderness represented by the in-plane plastic capacity and out-of-plane elastic buckling load. Since any closed form equations are lacking to approximate these buckling parameters finite element techniques were adopted. For future research it is recommended that closed-form solutions or design graphs should be derived with mechanical models to obtain the elastic-plastic buckling load of freestanding roller bent arches without using finite element analyses.

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Samenvatting ix

Samenvatting

Structural properties and out-of-plane stability of roller bent steel arches

Het gebruik van stalen bogen in de hedendaagse architectuur heeft een enorme vlucht genomen. Stalen bogen worden vooral toegepast in gebouwen en bruggen met grote overspanningen, waarin de constructieve meerwaarde van de stalen boog wordt gecombineerd met de architectonische verschijningsvorm. Voor bogen die geen zijdelingse steunen bezitten is knik uit het vlak het belangrijkste ontwerpcriterium. Geschikte methoden of rekenregels voor het bepalen van de kniklast uit het vlak zijn nog niet beschikbaar voor ontwerpende ingenieurs en het bezwijkgedrag van deze bogen is vaak onbekend.

Dit promotieonderzoek is gericht op het bestuderen van knik van stalen bogen uit het vlak en het ontwikkelen van rekenregels voor deze bogen. Geometrische en materiaal niet-lineaire imperfecte analyses met behulp van het commercieel beschikbare eindige elementen pakket ANSYS v. 11.0 werden uitgevoerd voor het bestuderen van het knikgedrag van deze bogen. Het onderzoeksgebied was beperkt tot cirkelvormige vrijstaande bogen, onderworpen aan verticale belastingen in het vlak van de boog. De bogen werden gefabriceerd uit rechte breedflensprofielen door middel van profielbuigen, een fabricageproces dat plaatsvindt bij kamertemperatuur. Het lag in de lijn der verwachting dat de restspanningen en mechanische eigenschappen (vloeispanning, treksterkte, etc.) worden beïnvloed door het profielbuigen. Omdat deze verandering weer van invloed kan zijn op de stabiliteit van vrijstaande bogen werd de invloed van het profielbuigen eerst bestudeerd. Restspanningen en trekproeven werden uitgevoerd op rechte en gebogen profielen om de imperfecties ten gevolge van het profielbuigen vast te stellen. De restspanningen werden daarnaast ook bepaald door middel van eindige elementen berekeningen met het pakket ANSYS v. 11.0. Goede overeenkomst werd gevonden tussen experimenteel en numeriek bepaalde restspanningen. Op basis van het experimentele en numerieke werk van de imperfecties werden een restspanningsmodel en verdelingsmodel van de mechanische eigenschappen gesuggereerd, die dienen voor de eindige-elementen-berekeningen voor de zijdelingse stabiliteit van bogen. Eindige-elementen-berekeningen lieten zien dat de restspanningen in gebogen profielen maar weinig invloed hebben op het stabiliteitsgedrag uit het vlak van stalen bogen. De mechanische eigenschappen in gebogen profielen hebben een duidelijke invloed op het knikgedrag en de bijbehorende kniklast.

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De bestaande knikkrommen van EC3 werden aangepast voor het knikgedrag uit het vlak van bogen door de imperfectieparameter te veranderen. Knikkrommen kunnen een goede benadering van de elasto-plastische kniklast geven wanneer een goede imperfectieparameter is geselecteerd en de slankheid bekend is. Op basis van een groot aantal eindige-elementen-berekeningen werd een imperfectieparameterkromme afgeleid die in de huidige knikkrommen werd gesubstitueerd om tot een knikromme te komen voor bogen. Op basis van de eindige-elementen-berekeningen zijn meerdere knikkrommen voorgesteld om een nauwkeurige bepaling van de kniklast uit het vlak mogelijk te maken. De slankheid, gevormd door de plastische capaciteit in het vlak en de elastische kniklast uit het vlak, is nodig voor het bepalen van de elasto-plastische kniklast. Vandaag de dag zijn eenvoudige formules voor het bepalen van deze parameters niet aanwezig, waardoor eindige elementen analyses nodig zijn voor het bepalen van de slankheid. Voor vervolgonderzoek is daarom aanbevolen om eenvoudige formules af te leiden om de slankheid van vrijstaande bogen te bepalen, waardoor de kniklast van bogen kan worden bepaald zonder het gebruik van eindige-elementen-analyses.

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Preface xi

Preface

This thesis is in the first place devoted to God in heaven, who is forgiving, tenderhearted and beyond imagination. Without Him nothing would have been possible in the first place. All glory to Him, forever and ever.

I would like to express my sincere appreciation to my supervisors in Eindhoven: Prof. Snijder and Dr. Hoenderkamp, for the pleasant and successful period of research during the last 4 years. Their guidance and support has proven to be essential. My second supervisor, Prof. Beg from the University of Ljubljana is highly acknowledged for supplying me abundant information about residual stress measurements during the initial stage of the research and later carefully reading the manuscript of my thesis. I would like to thank the members of the doctoral committee, Prof. Sluys, Prof. Soetens, Prof. van Bogaert and Dr. Maljaars, for reviewing the manuscript and their suggestions for improvements

Dr. Monique Bakker was my co-supervisor during the initial stage of my Ph.D. and provided me with great help. Although she only supervised my Ph.D. from May 2007 until the summer of 2009, her input was of great importance for the later phases.

I like to thank Theo van de Loo and Eric Wijen of the laboratory of the group of Structural Design and Construction Technology at Eindhoven University of Technology. The residual stress measurements and tensile tests would have been impossible without their help. They assembled the measurement equipment and arranged test set-up and can be largely credited for the successful experimental results. Hans Lamers and Martien Ceelen ordered the strain gauges and helped me with the measurement planning. Harrie de Laat and Mariële Dirks – Smit of the GTD (Gemeenschappelijke Technische Dienst) are acknowledged for removing the test coupons from the steel members with the Electric Discharging Machining (EDM) technique and allowing me to conduct several residual stress measurements in their production hall using the EDM technique in the autumn of 2008.

The steel members for the experiments were delivered by Deltastaal BV and bent by Kersten Europe BV free of charge, for which I am thankful. Special thanks to Bart Simonse of Kersten Europe BV who gave me extra information about the roller bending process.

I would like to express my gratitude for the (former) MSc-undergraduates Paulien Hanckmann, Maartje Dijk, Eeuwe Bloemberg, Michael van Telgen, Linh Sa Lê and Rianne Luimes who helped me during the first 1.5 year of my Ph.D. by working on various research topics.

The help of Dr. Leroy Gardner of Imperial College in London during my research is greatly appreciated. Prof. F.M. Mazzolani of the University of Naples is acknowledged for sending his residual stress measurement reports. Dr. Dagowin La Poutré, whom I met on the Eurosteel Conference in Graz, Austria in September 2008, provided extensive information about his Ph.D. research on arch buckling.

I would like to acknowledge my (former) Ph.D. colleagues at the Department of Architecture, Building and Planning - unit of Structural Design in Eindhoven: Johan Maljaars, Ernst Klamer, Natalia Kutanova, Paul Teeuwen, Edwin Huveners, Dennis Schoenmakers, Frank Huijben, Lex van der Meer, Sander Zegers, Sarmediran Silitonga, Ronald van der Meulen and Juan Manuel Davila Delgado for the pleasant working atmosphere, great help and advice. I like to thank the

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secretary of the unit of structural design and design systems for their support and help throughout my Ph.D: Bianca Magielse, Marlyn Aretz, Litania van de Venne and Nathalie Rombley.

My research was partly funded by the Materials innovations institute (M2i) in Delft. Although I was stationed in Eindhoven and was not able to visit M2i that frequently, I would like to acknowledge the complete staff of M2i in Delft for their support and in particular: Alice Sosef, Monica Reulink, Gitty Bouman, Pia Legerstee, Margo Poelman-van Os and Irina Bruckner. The initiative for research on arch buckling was made by the Dutch organization for constructional steelwork: Bouwen met Staal and the Dutch federation for steel work: Staalfederatie Nederland. I was given the ample opportunity to give presentations for the technical committee of Bouwen met Staal „BmS/TC8‟ about the Ph.D. process and research results. The complete committee of „TC8‟ is greatly acknowledged for their remarks and suggestions during all presentations. I also had frequently contact with my friends from college who gave me useful advice during my research. I like to mention Wouter ten Napel, Rick Bruins, Caspar Breman, Wim de Groot and Inge Schouwenaars.

The extensive football matches during my whole Ph.D. proved to be invaluable. It is for me a great pleasure to thank my friends of the football squad of SV Orion 7 Nijmegen (and later SV Orion 8): Sweder Scholtz, Remco van Rooijen, Jimmy Knubben, Joost Rooijakkers, Philip Mendels, Timo Brits, Tijn Frik, Joris van Halder, Wouter Schoot, Jorn van Dorst, Daan Sutmuller, Joep Rooijakkers, Jochem van Halen, Michel Freriks, Rens Zwakenberg, Gijs Graste, Tom Frenken, Thijs Smarius, Rikkert Heydendael, Marc Hesselink, Stijn Vissers and Arjan Zoet for the good times on almost every Saturday! In addition I would like to thank my friends from the Republic of Moldova for their hospitality and the great times in the summers of 2008, 2009 and during the winter of 2009: Rita Postica, Irina Postica, Cris Petrimari, Natasha Groza and the Dutch volunteers Dirk Willem Klos and Welmoed van der Veen. The weekly fitness courses together my friends in Eindhoven: Jop Courage, Robbert Lieven, Eeuwe Bloemberg and Wouter Schoot gave me a good insight in the world of iron and motivation next to the world of steel. I consider myself lucky since both of my grandma‟s are living in (the neighborhood of) Eindhoven and I enjoyed visiting them during my research. Also my aunt Stella, my uncle Rob and their daughters are acknowledged for the pleasant moments in the last 4.5 years.

I would like to acknowledge the great support of my parents, Cees and Marjon Spoorenberg, and my twin brother Bram. Thank you very much for the necessary support and great laughter. I owe a great debt to my girlfriend Yuzhong Lin. She made the last year of my Ph.D. very comforting for me. Your support throughout the final stages of my Ph.D. was priceless. Thank you for the nice times thus far: boat trip in Giethoorn, train travel in Ireland, and so forth. The trip to the People‟s Republic of China and visiting your parents and family was superb! I hope we can have great times for the nearby future and beyond as well!

Roel Spoorenberg Eindhoven, July, 2011

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Nomenclature xiii

Nomenclature

Abbreviation Unit Latin capitals A Section area [mm2_] E Young‟s modulus [N/mm2_]

E0.2 Modulus of elasticity at 0.2 % proof stress [N/mm2]

It Torsional moment of inertia [mm4]

Iw Warping moment of inertia [mm6]

Iy Major moment of inertia [mm4]

Iz Minor moment of inertia [mm4]

G Shear modulus [N/mm2_] F Concentrated force [N] L Span of arch [m] M Bending moment [Nm] R Arch radius [m] S Arch length [m]

Wpl plastic section modulus [mm3]

Latin lower case

b width of section [mm]

f Rise of arch [mm]

h height of section [mm]

n hardening exponent [-]

m hardening exponent [-]

q Uniformly distributed load [N/m]

tf flange thickness [mm]

tw web thickness [mm]

fp proportional limit (0.01 % offset proof stress) [N/mm2]

fy yield stress (0.2 % offset proof stress) [N/mm2]

ft ultimate tensile stress [N/mm2]

u,v,w displacements in x,y,z respectively [mm]

x,y,z coordinates [mm]

Subscripts

s straight

r roller bent

imp geometric imperfection perm permanent deformation

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Greek lower case

α shape factor / subtended angle / imperfection factor / [-] / [º] / [-] /

load factor [N]/[N/m]/[Nm]

αcr critical load amplifier [N]/[N/m]/[Nm]

αpl plastic collapse load amplifier [N]/[N/m]/[Nm]

αult ultimate load amplifier [N]/[N/m]/[Nm]

γ half of subtended angle [º]

ε0.01 strain at proportional limit: fp/E + 0.0001 [-]

ε0.2 strain at 0.2 % proof stress: fy/E + 0.002 [-]

εy yield strain: fy/E [-]

εt strain at ultimate tensile stress [-]

εu strain at fracture [-]

δ,ε,ζ rotations about x,y,z-axis respectively [º]

ε imperfection parameter [-]

 non-dimensional slenderness [-]

0

 plateau length column curve [-]

ζ stress [N/mm2_]

ζrc compressive residual stress [N/mm2]

ζrt tensile residual stress [N/mm2]

ζfrc compressive residual stress in flanges [N/mm2]

ζfrt tensile residual stress in flanges [N/mm2]

ζwrc compressive residual stress in web [N/mm2]

ζwrt tensile residual stress in web [N/mm2]

ϕ variable angle [º]

χ reduction factor [-]

Greek capital

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1 Introduction

This thesis describes the properties and out-of-plane structural stability behavior of roller bent steel arches. In this chapter, the applications, classification and scope of research are introduced and the problem statement and objectives of the research are described. Introductions on stability and the roller bending process are given as well. Finally the methodology and contents of this thesis are outlined.

1.1 Arches

1.1.1 Application

Arches have a widespread application and continue to find new applications in many different fields. The application of arches seems to have started around two thousand years ago. The Romans were the first to use arch structures of a significant size and some of these structures can be seen even today. One example of the application of arch structures in Roman building engineering is the Pont du Gard (France). This engineering achievement was not exceeded until medieval times by the construction of the Pont d‟Avignon (France). The first arches were executed in stone or masonry construction, the only materials available to bridge spans with considerable length. During the industrial revolution the first major advances in arch engineering were made with the use of cast iron.

Bridges

Nowadays the application of braced arches in bridges is quite common up to 500 m. Arch bridges may comprise concrete, steel or hybrid structures. Table 1 lists the longest span steel arched bridges. These bridges all comprise two parallel arch-ribs.

Table 1 Longest span steel braced bridges

Name Span

(m) Location completion Year of

New River Gorge

Bridge 518 Fayetteville, West Virginia, USA 1977

Bayonne Bridge 504 Kill van Kull, New Jersey, New York, USA

1931 Sydney Harbour

Bridge

503 Sydney, Australia 1932

Wushan Bridge 460 Chongqing, China 2005

Caiyanba Bridge 420 Chongqing, China 2007

Freestanding arches are also applied in bridge design although to a lesser degree and with smaller spans in comparison to non-freestanding arches. The Svinesund bridge in Norway/Sweden1

(Steiner and Wagner [127], Jordet and Jakobsen [51]) is an example of a freestanding arch. The arch comprises a concrete section which is fixed at the supports. Forces from the bridge deck are transmitted to the arch by means of cables. A total span of nearly 275 m was achieved. An freestanding arch with a smaller span is the Yarra River Bridge in Melbourne, Australia2_(Figure

2).

1_{Lund & Slaatto Arkitekter A/S} 2_{Architect: engineers Whilybird}

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Figure 1 Svinesund bridge, Norway/Sweden

Figure 2 Yarra River Bridge, Melbourne Australia

In some arch bridges the plane of the arch is inclined with respect to the horizontal plane as can be seen in Figure 3, York Millennium bridge, York, UK3_{(Mairs [71]). This bridge has a total}

length of approximately 150 m with a 4 m wide, 80 m long main span. The hollow arch is 600 mm by 200 mm in cross-section made from four plates of high strength stainless steel. Another example for an inclined freestanding arch can be found in the Gateshead millennium bridge in Newcastle, UK4_{(Figure 4) (Clark and Eyre [23], Curran [28]). The hollow parabolic arch of the}

Gateshead millennium bridge is made of internally stiffened steel plates. The span equals 100m.

Figure 3 York, Millenium Bridge, York, UK Figure 4 Gateshead Millennium Bridge, Newcastle, UK Buildings

Besides bridges, arches are also used in large span roofs or coverings. Their application is mainly in sports stadiums, halls and railway stations. Stadiums supported by arches are the Olympic Stadium of Athens, Greece5_{(Figure 5) (Anon. [5]), the Olympic Stadium or Telstra Stadium of}

Sydney Australia6_{(Bennett [14]) and the Alfred McAlpine Stadium, Huddersfield, UK}7_(Wilson

[142]). One of the most recent applications is the new Wembley Stadium in London, UK8_(Figure

6) (Anon. [6] and Woertman [143]).

3_{Architect: Wilkinson Eyre Architects} 4_{Architect: Cocks Carmichael Whitford} 5_{Architect: Santiago Calatrava}

6_{Architect: Bligh Lobb Sports Architects} 7_{Architect: HOK & LOBB}

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Introduction 3

Figure 5 Olympic Stadium, Athens, Greece Figure 6 Wembley Stadium, London, UK Arches can also be found in office buildings. Examples can be found in the London Exchange House, London, UK9_{(Boks [18]) and the Ludwig Erhard House, Berlin, Germany}10_{(Anon. [4]).}

In both examples the arch serves as a superstructure; see Figure 7 and Figure 8.

Figure 7 London Exchange House, London,

UK Figure 8 Ludwig Erhard House, Berlin, Germany

1.1.2 Types of arches

Arches can be classified in several ways. In this thesis a distinct difference is made between arches and curved beams. An arch is supported in such a way that outward spreading of the arch is prevented which induces major compressive actions in the arch-rib in addition to bending. So the structure is classified by the end conditions. As outward spreading is prevented, an arch can be pinned supported or two-hinged, Figure 9(a), or fixed, Figure 9(b). In case outward spreading is not prohibited the acting loads are primarily resisted through bending action (Figure 10). A mixed support combination (e.g. a hinge for the left support and a fixation for the right support) is also possible. The orientation of the supports is presented in accordance with the local axis of the arch Figure 11(a), instead of the global coordinates of the system Figure 11(b).

(a) Pinned supported (b) Fixed

Figure 9 In-plane support conditions arches. Figure 10 Curved beam

The efficiency of an arch as a load-carrying structure is dependent on the extent with which the thrust-line follows the arch-rib. An optimum design is achieved when the shape of the arch matches the thrust-line. In that case the arch-rib experiences compressive stresses and no bending stresses.

9_{Architect: SOM}

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Z X Mx z x MT Z X MZ z x MT MX MZ MX

(a) Local coordinates (b) Global coordinates

Mx

Figure 11 Orientation of supports.

The thrust-line is determined by the acting load. A circular arch is most efficient if it carries radial load uniformly distributed around the centroidal axis of the arch, Figure 12(a), a parabolic arch when carrying a uniformly distributed load, Figure 12(b), and an inverted catenary arch carrying self-weight only, Figure 12(c).

(a) Circular arch (b) Parabolic arch (c) Inverted catenary arch

Self-weight

Figure 12 Arch shapes with matching thrust-lines.

For practical design loadings, however, the arch acts in combined bending and compression. Whether compression or bending is the major action depends on the loading conditions, support conditions, subtended angle and arch length. When an arch is subjected to load uniformly distributed along the horizontal projection of the entire arch, the compression is relatively high and the bending moment is relatively low. In contrary, in case an arch is subjected to a central point load, the bending moment is relatively high and the compression is relatively low. Arches used in roofings are subjected to forces, for example, induced by purlins. These forces maintain their original direction in the deformed shape and are denoted as conservative forces, Figure 13(a). When an arch is applied in bridge design, the forces are often transmitted through cables connected to the arch-rib and to the bridge deck, see for example Figure 6. The direction of force changes with deformation of the arch which is better known as a non-conservative force; see Figure 13(b).

(a) Conservative force (b) Non-conservative force

F F

Figure 13 Conservative force and non-conservative force in deformed shape

The support conditions are featured by the in-plane and out-of-plane supports characteristics. For a freestanding, in-plane, pinned supported and out-of-plane pinned arch the torsional rotation at the support should be prevented to avoid rigid body motion. A fork support is sufficient to enforce no torsional rotation, see Figure 14. However, when the subtended angle is 180 degrees lateral stability is no longer ensured and a rigid body rotation about the line through both supports occurs. An out-of-plane fixed arch is featured by a fork support where the out-of-plane

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Introduction 5

rotations are prevented, see Figure 15. The effect of restrained warping is an additional degree of freedom for the out-of-plane support conditions.

(a) Beam x z y z x y

(b) Arch (a) Beam (b) Arch

u = v = w = δ = 0 u = v= w = δ = ζ = 0 y x z u w v δ ζ ε

Postive local coordinate system with deformation abbreviations

x z y z x y

Figure 14 Fork support Figure 15 Fork support with out-of-plane restraint An out-of-plane support can be provided by lateral bracing, Figure 16(a). However, when lateral bracing is absent, in which case the arch is considered freestanding, lateral support must be provided by the out-of-plane bending stiffness, torsional stiffness and warping rigidity of the arch-rib, Figure 16(b). The boundary conditions also contribute to the lateral stability of freestanding arches.

(a) Laterally braced Arch

Bracing

(b) Freestanding Figure 16 Out-of-plane support.

Arches are made from different materials. Early arches were made of masonry or timber. Nowadays most arches are made of steel, reinforced concrete or a combination of both. The use of material determines the cross section. In general it can be stated that steel arches applied for relative small spans are made from roller bent steel sections, whereas for larger spans welded box-sections made out of plates are used. A full overview on the arch dimensional parameters is presented in Figure 17. L S f R γ α h L = Span S = Arch length R = Radius f = Rise h = Section height α = Subtended angle

γ = Half of subtended angle = Angular coordinate 

 Figure 17 Dimensional arch parameters

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1.1.3 Scope

This thesis is confined to the study of freestanding circular arches subjected to two different loads: a central point load and a uniformly distributed load (UDL) over the entire span see Figure 18.

F _q

(a) Central load (b) Full span UDL

L

Figure 18 Investigated loadcases for in-plane pinned supported arches.

Commonly arches in bridge and roof structures have in-plane pinned or fixed connections. This study is limited to in-plane pinned support conditions, since fixation of the supports in the plane of the arch will have only a minor contribution to the structural performance. The supports out-of-plane are completely fixed: rotations and warping deformations are restrained, to give additional out-of-plane stiffness. An overview of the support conditions is presented in Table 2. The loads are conservative, i.e. no directional change of the force vector during loading is taken into account.

Table 2 Investigated support conditions

In-plane Out-of-plane Warping at support

Pinned Fixed Restrained

Only arches made from roller bent wide flange steel sections are investigated. The roller bending process with a three-roller bending machine is a widely used method to arch a straight hot-rolled sections and this study is limited to this method of arch production. It is mentioned that the definition of “wide flange section” applies to hot-rolled I-shaped sections for which the width of the top flange approximately equals the height, whereas sections featured by smaller height-to-width ratios are annotated as “I-sections”. In this thesis however, the definition of wide flange sections applies to all hot-rolled I-shaped sections.

1.2 Stability

In this section the phenomenon of structural stability is treated. First the most common failure modes for beams, columns and arches with respect to stability are outlined. Subsequently the different theories for structural analysis are presented and their relationship with structural stability.

1.2.1 Description

Columns

Columns are prone to three different stability phenomena: flexural buckling, torsional buckling and flexural- torsional buckling as illustrated in Figure 19. Flexural buckling (Figure 19(a)) occurs when a concentrically loaded column changes from axial deformation to a flexural deformation. Torsional buckling (Figure 19(b)) is a dominant mode of buckling for double-symmetrical open sections subjected to compression. Due to rotation of the column around its shear centre, which coincides with the centre of gravity, instability occurs. Flexural-torsional buckling (Figure 19(c)) is a combination of flexure and torsion occurring when shear centre and centre of gravity do not coincide. Flexural-torsional buckling occurs when the load application does not coincide with the shear centre. For all three phenomena the strength of slender columns is lower than the squash load or plastic capacity of the cross section.

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Introduction 7

(a) flexural buckling (b) torsional buckling (c) flexural-torsional buckling Figure 19 Stability phenomena of columns.

Beams

The global instability of beams is denoted by lateral-torsional buckling. Lateral-torsional buckling arises in slender beams subjected to general loading Figure 20(a). This is due to a compressive action in the upper section of the beam. The deformation of the beam changes from in-plane deformation to a combination of in-plane deformation, twist and lateral deflection Figure 20(b). Slender beams fail by lateral-torsional buckling before reaching their in-plane plastic capacity.

v

w

(a) Loading (b) Deformation of cross section

z y

x

δ

Figure 20 Lateral-torsional buckling Arches

Stability of freestanding arches can be subdivided into three stability phenomena: snap-through instability, Figure 21(a), in-plane instability, Figure 21(b) and out-of-plane instability, Figure 21(c). This study is focused on the out-of-plane stability of arches.

(a) Snap-through (b) In-plane buckling (c) Out-of-plane buckling

F F F F

Figure 21 Stability phenomena

Snap-trough instability occurs in shallow arches which are restrained against out-of-plane displacements. Due to axial shortening, the arch is capable of moving „through‟ the span and will subsequently act in tension. In-plane instability is a dominant mode of failure for non-shallow arches prevented from out-of-plane buckling. In-plane instability can be featured by a symmetric or (more common) asymmetric buckling mode. Out-of-plane instability of arches is a combination of out-of-plane flexural and lateral-torsional buckling; therefore the out-of-plane stability of arches is sometimes denoted as flexural-torsional buckling. Similarly to columns and beams, slender arches may fail by out-of-plane buckling prior to the attainment of the in-plane plastic capacity. Arches, made from thin plates can also fail in a local buckling mode as featured by flange buckling or web crippling. This buckling mode, however, is not investigated in this thesis.

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1.2.2 Stability and Stability analyses

In this section a small overview on the issue of stability is given. Consider a pin-ended steel column with flexural stiffness EI and length L subjected to a vertical load F at the top and a smaller - but proportional - horizontal load H applied at mid height (Figure 22(a)). This column can be analyzed according different theories, each featuring different degrees in output and structural phenomena taken into account. The analysis types available to assess the structural response can be subdivided into 2 categories: 1st_{order analysis and 2}nd_{order analysis. For each}

category a further distinction between elastic (Figure 22(b)) or plastic material (Figure 22(c)) behavior can be made, rendering a total of four structural analysis types.

1st_{order analysis}

In a 1st_{order analysis it is assumed that the deformations do not influence the equilibrium}

equations. When the column is analyzed by the 1st_{order elastic theory, loads and deformations}

are linearly related; indicating that any increase of the force will induce a proportional increase in deformations. Hence, plotting the vertical load F on the ordinate and horizontal deflection at mid-height w on the abscissa in a conventional load-deflection graph gives a straight line as shown in Figure 22(d). Incorporating the yielding behavior of the material into the structural analysis by performing a 1st_{order plastic analysis will induce a change in the load-deflection graph. At}

relative low load levels the column behaves according to the 1st_{order elastic theory. As soon as F}

approaches the plastic collapse load Fpl* the load-deflection graph starts to deviate from the

straight line from 1st_{order elastic analysis (Figure 22(f)). As the load approaches F}

pl* any small

increase in load will induce a larger increase in deformations, represented by a decreasing slope of the load-deflection graph. At Fpl* the load carrying capacity is exhausted and a plastic collapse

mechanism has formed. Due to the presence of the horizontal force H the plastic collapse load is smaller than the squash load Fpl of the cross-section. Although a 1st order analysis has proven to

be an economic basis for the design of structures when loss of stability is not an issue, it may be inaccurate when confronted with structures under large compressive forces. For these structures recourse has to be taken to 2nd_{order analyses.}

2nd_{order analysis}

In a 2nd_{order analysis the equilibrium equations are formulated in the deformed state of the}

structure, and hence can be classified as non-linear. It is obvious that the horizontal force (resembling actual steel imperfections) induces a small deflection and bending moment. Since the column has deflected in horizontal direction, this in turn will induce an extra bending moment due to emerged misalignment between the centroid of the column and force F. The bending moments will induce additional deformations, which will increase the bending moments even more, and this process will continue until equilibrium between external and internal forces is achieved. The load-deflection graphs of a 2nd_{order elastic analysis deviates from the straight line}

from the 1st_{order analysis at the onset of loading and moves asymptotically towards the elastic}

buckling load Fcr as shown in Figure 22(e). Provided the column is not susceptible to

lateral-torsional buckling due to the presence of the horizontal force at mid-height the elastic buckling load for this column can be computed by the well-known equation of Euler: Fcr=π2EI/L2. From

the load-deflection graph it can be seen that at relative low load levels of F the line follows the 1st

order elastic theory closely since the column is predominately under axial compression and bending actions are relative small. As the load increases, the column experiences an increasing amount of flexure, featured by a decreasing slope of the load-deflection graph, while the axial force remains equal to the vertical load F. In the vicinity of the elastic buckling load the column experiences large bending actions. Due to the presence of a horizontal force at the onset of loading the axial compression and bending action are coupled to certain extent. When the horizontal force H is taken smaller, axial forces and bending actions become decreasingly coupled. In the limiting case (i.e. when the horizontal force is taken to be infinitely small) there is

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Introduction 9

no coupling between axial forces and bending actions. In this case the load-deflection graph is manifested by two straight lines or load paths, intersecting at a bifurcation point or the elastic buckling load Fcr. The first branch represents the axial load path and the second branch the

bending load path.

A 2nd_{order elastic-plastic analysis takes into account both the geometrical and material}

non-linearities. The 2nd_{order elastic-plastic load-deflection graph is identical to the line from the 2}nd

order elastic analysis, until the yield stress in the cross-section is reached (Figure 22(g)). After the onset of yielding the column is able to sustain a small increase in load until the limit load, featured by a maximum in the load-deflection graph. The limit load from a 2nd_order

elastic-plastic analyses is also known as elastic-elastic-plastic buckling load or ultimate load. Due to the complexity inherent to the analysis type the 2nd_{order elastic-plastic response is often investigated}

with finite element analyses or similar numerical procedures.

F

w F

(d) 1st_{order elastic} _{(e) 2}nd_{order elastic} F (a) Column F w F Limit load (g) 2nd_{order elastic-plastic} Commencement yielding Fpl* Fcr Fcr Fult H Bifurcation point Decreasing H L EI 3 48EIF HL 1 w w ζ ε E 1 (b) elastic material law ζ ε E 1 w 1st_order ₂nd_order (c) elastic-plastic material law fy εy Fpl* (f) 1st_{order elastic-plastic}

Figure 22 Stability behavior of a column. Stability essentials

From the preceding text it becomes clear that the subject of stability can only be studied sufficiently with 2nd_{order elastic-plastic analyses. Furthermore, the elastic buckling load,}

featuring the bifurcation point, shows little resemblance to a structure failing in an elastic-plastic buckling mode. Moreover, the elastic buckling load is insensitive to the horizontal force H or equivalent imperfection. Hence, performing an elastic buckling analysis to find Fcr is only a small

step in a stability check. The horizontal load does influence the load-deflection curve for the 2nd

order analysis and thus the limit load. The magnitude of the horizontal force indicates whether gradual or more explosive buckling behavior is will take place. From this it can be concluded that imperfections have a great influence on the stability behavior of steel structures. Imperfections can be subdivided into three categories: geometrical imperfections (or deviations from the ideal geometry), residual stresses and non-uniform distributions of mechanical properties. The first imperfection category applies to all 2nd_{order analyses, whereas the latter two only apply to 2}nd

order elastic-plastic analyses. Stability and design rules

It would be complete madness to perform a 2nd_{order elastic-plastic analysis to check the stability}

of every steel column or beam susceptible to buckling. Therefore, in order to check the stability resistance, design rules have been proposed, which make proper allowance for the buckling phenomena. A structural engineer can check the buckling resistance by conducting a 1st_order

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elastic analysis to find the internal force distribution and adopt the design codes to make proper allowance for any buckling issues. However, design rules are not available for every structural element.

1.3 Roller bending process

Steel arches in wide flange sections are manufactured in many cases by cold bending of steel wide flange sections with the use of rolls, also known as the roller bending process. In the roller bending process a member is placed in the machine and curved between three rolls at ambient temperature (Figure 23(a)). Because of the three main rolls‟ pyramid arrangement the roller bending process is sometimes called pyramidal rolling. Permanent curvature in the member is achieved due to movement of the right roller along a prescribed path (Figure 23(b)) and subsequent rolling of all rolls (Figure 23(c)) inducing a process of continuous plastic deformations. The member is rolled back and forth on multiple passes until the desired radius is achieved. Due to placement requirements only a part of the total beam length can be bent, leaving straight material on either end of the curved section. At the inside of the top flange (subjected to elongation) a flange support roller is utilized to provide additional restraint and thereby preventing the web from crippling. The roller bending process can be applied about the weak or strong axis of wide flange sections and also allows the forming of non-circular curved beams. However, this investigation is confined to roller bent sections bent about the strong axis into a circular geometry. A detailed description of the roller bending process has been presented by Bjorhovde [17], Weisenberger [138], and Alwood [3] . The elongated flange and shortened flange are denoted in this thesis as top flange and bottom flange respectively and in the subsequent sections results will be presented according to this notation.

(a) initial-state (b) movement right hand roller (c) rolling Center roller Flange support rollers Outer rollers I-section Beam movement waste Fixed path Top flange Bottom flange

Figure 23 Roller bending process.

Since the member is curved at ambient temperature it is most likely that the distribution of mechanical properties, residual stresses and geometric imperfections are altered. In similar studies on cold-formed steel sheeting, it was found that the residual stresses and mechanical properties in the cold-formed area are different compared to other areas in the sheeting.

1.4 Problem statement and objectives

In paragraph 1.1 it was shown that arches are used in various structures. It was also shown that arches are prone to various instability phenomena as described in paragraph 1.2. Out-of-plane instability is an issue for freestanding arches to which this thesis is confined. In addition it was noted that the stability resistance for beams and columns can be verified through the use of design rules, thereby avoiding laborious finite element calculations. The roller bending process to

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Introduction 11

arch straight beams into the desired curvature alters the imperfections of the member section in terms of cross-sectional shape, residual stresses and variation in mechanical properties as outlined in paragraph 1.3. From this it follows that the manufacturing process may influence the elastic-plastic buckling behavior of roller-bent arches.

Currently no design rules are available which take into account the change in imperfections arising from the roller bending process and their influence on the out-of-plane structural stability behavior of steel arches. Nowadays, when confronted with freestanding roller bent arches, engineers have to take recourse to time-consuming finite element calculations, without even being aware of the influence of imperfections or failure modes inherent to out-of-plane buckling. On the one hand this situation can lead over-dimensioned arches and hence inefficient structures. On the other hand the lack of knowledge on out-of-plane buckling of freestanding arches can compromise the safety of the arch structure.

Based on the preceding text, the problem statement of the research can be summarized as follows: For freestanding arches, subjected to in-plane loading, design rules do not exist which give insight into the structural behavior and take into account the influence of the roller bending process.

The objective of this research is to gain insight into the out-of-plane buckling behavior of freestanding roller bent circular steel arches and to derive design rules for the limit load, in which the influence of the roller bending process is taken into account.

The field of application for freestanding roller bent arches is small. Freestanding arches are often manufactured with a different method than roller bending. Roller bent arches, in contrary, are often applied in roofings for which the purlins provide lateral stability and are therefore not considered freestanding but laterally braced. The design rules for roller bent freestanding arches will therefore serve as a first step towards design rules for (a) laterally braced arches and (b) freestanding arches manufactured with other manufacturing processes prone to out-of-plane instability. The derived design rules for freestanding arches can also be applied to arches subjected to non-conservative forces as a safe approximation, since these non-conservative forces result in a higher failure load when compared to conservative forces.

1.5 Methodology and outline of thesis

This thesis is divided into three distinct parts: (I) investigation of residual stresses in roller bent wide flange sections, (II) examination of influence of roller bending process on mechanical properties in wide flange sections and (III) examination of carrying capacity of freestanding arches and accompanying proposal for design rules for freestanding arches. Each part is subdivided into two or more chapters.

In chapter 2 an overview of earlier research on out-of-plane buckling of steel arches is given, in addition to earlier suggested design rules.

The residual stresses in roller bent arches are measured and predicted with a finite element model. The experimental results and finite element analyses are presented in chapter 3 and chapter 4, respectively. From these finite element results a residual stress model is proposed. The derivation of this model is presented in chapter 5.

The mechanical properties are measured by performing tensile tests on coupons taken from roller bent arches. The experimental results are given in chapter 6. Subsequently the experimental results are used to arrive at a prediction model by which the mechanical properties over the

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cross-section of a roller bent arch can be predicted (chapter 7). The residual stress model and the prediction model are used to define the initial state of a roller bent arch for elastic-plastic buckling computations.

In chapter 8 the finite element model with all implemented imperfections is outlined. The performance of the finite element model is examined through comparison with experimental results. The influence of the imperfections from the roller bending process on the out-of-plane elastic-plastic buckling response is investigated by means of sensitivity analyses. The apotheosis of the thesis, reflected by the design rules, is presented in chapter 9. The underlying procedure to arrive at the design rules is presented.

Conclusions and recommendations for further research are given in chapter 10. A full outline on the methodology and accompanying chapter numbers in the thesis is presented in Figure 24.

Literature Survey (2)

I Residual stresses in roller bent wide flange sections

III Out-of-plane structural stability of roller bent wide flange sections

Experiments (3) Finite element simulations (4)

Residual stress model (5)

Finite element analyses (8) Proposed design rules (9)

Conclusions and recommendations (10)

II Mechanical properties of roller bent wide flange sections

Experiments (6) Prediction model (7)

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2 Literature study on arch buckling

This chapter gives an overview of the theoretical and numerical studies on the structural performance of steel arches. The first section deals with the in-plane plastic collapse load of steel arches. The literature on the elastic buckling load is outlined in the subsequent section. Numerous research studies were devoted to obtain a closed-form equation to obtain the elastic buckling load for freestanding arches subjected to various loading conditions. The design rules emanating and non-linear finite element analyses for freestanding arches are presented in section 2.5 and 2.4, respectively.

2.1 Introduction

The literature on the structural response of arches and arch buckling is extensive. A large amount of text books, journal articles and conference papers was devoted to this subject. The overview of this chapter is limited to freestanding circular arches, loaded in-plane and predominantly failing by out-of-plane buckling. The first two sections deal with the in-plane plastic collapse load of steel arches and the elastic buckling load of arches. These subjects have no immediate relevance to the research subject, but are important topics in view of the design rule which will be presented in chapter 9. The literature on residual stresses and mechanical properties of roller bent sections will be treated in the corresponding chapters.

2.2 In-plane plastic collapse load

This paragraph gives a review of the literature on the in-plane plastic collapse load of (steel) arches. The earliest reported research on the plastic collapse analysis of steel arches was performed by Hendry [46]. A graphical method was verified with experiments. The investigation was confined to parabolic arches with square and I-sections. Axial thrust was measured in the experiments, however, not taken into account in the analyses. It was stated that the effect of the axial stress on the strength was in no case sufficiently large to reduce the plastic moments. The investigation of Onat and Prager [81] was based on the assumption that for shallow arches the influence of axial forces can no longer be neglected. The in-plane strength was evaluated for circular arches subjected to three loadcases: a single force at the crown, a uniformly distributed load over the entire span and a uniformly distributed load over the left-hand half of the span. In the first loadcase the plastic capacity was evaluated by employing two methods: a kinematic approach based on a collapse mechanism and an evaluation of the internal forces. For the other two loadcases only the kinematic approach was used. The investigation was confined to arches with a rectangular cross section. Stevens [128] also found that the application of the plastic-theory to arches requires a reduction of the bending capacity due to considerable axial forces associated which arch action. Secondly he found that small deflections can produce significant changes in the force and bending-moment systems. Theoretical investigations were compared with experimental results which included both small and large-scale arches with various loadings, shapes and end conditions. The theoretical investigations were based on graphical methods since the solution required a large computational effort. The analyses were limited to rectangular cross sections. A computer program was written by Cornforth and Childs [25] to compute the in-plane plastic collapse load based on an iterative approach. It was assumed that the compressive action in the arch-rib was not of any influence with respect to plastic moment capacity of the section. A step-by-step approach for the analysis of elasto-plastic arches under quasi-static loading was presented by Cohn and Abdel-Rohman [24]. Techniques for matrix analysis were extended to take into account both the effects of axial forces and deformations on the stress distribution. Four types of support conditions and loading were considered:

Structural properties and out-of-plane stability of roller bent steel arches

Structural properties and out-of-plane stability of roller bent

steel arches

Citation for published version (APA):

Spoorenberg, R. C. (2011). Structural properties and out-of-plane stability of roller bent steel arches. Technische

Universiteit Eindhoven. https://doi.org/10.6100/IR716581

DOI:

10.6100/IR716581

Document status and date:

Published: 01/01/2011

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Link to publication

Structural properties and out-of-plane stability of roller

bent steel arches

Structural properties and out-of-plane stability of roller bent steel arches

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de

Technische Universiteit Eindhoven, op gezag van de

rector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor

Promoties in het openbaar te verdedigen

op dinsdag 13 september 2011 om 16.00 uur

door

Roeland Christiaan Spoorenberg

Summary

Samenvatting

Preface

Nomenclature

Contents

1 Introduction

1.1 Arches

1.2 Stability

1.3 Roller bending process

1.4 Problem statement and objectives

1.5 Methodology and outline of thesis

2 Literature study on arch buckling

2.1 Introduction

2.2 In-plane plastic collapse load