Research Collection

Report

Load-bearing behaviour of composite beams in low degrees ofpartial shear connection

Author(s): Bärtschi, Roland

Publication Date: 2005

Permanent Link: https://doi.org/10.3929/ethz-a-004956129

Rights / License: In Copyright - Non-Commercial Use Permitted

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Load-Bearing Behaviour of Composite Beams in Low Degrees

of Partial Shear Connection

Roland Bärtschi

Institute of Structural Engineering

Swiss Federal Institute of Technology Zurich

February 2005

Preface

Thanks to partial shear connection the economical performance of composite beams in build-ings can be considerably improved. However, ductile load-slip behaviour of the shear connec-tors is required for partial shear connection. The ductility and deformation capacity of the shearconnectors is compared to the ductility required in the composite beam. The required ductilitydepends on the geometry, the support conditions and the load distribution of the compositebeam. For the frequently-used headed studs there are simplified design rules for the minimumdegree of partial shear connection. But for novel shear connectors, as investigated in several re-search projects at our institute, the appropriate data for a simple and economic application ismissing.The present research report concerning the load-bearing behaviour of composite beams withlow degrees of partial shear connection was developed into a PhD thesis and investigates thefundamental influence of the load-slip behaviour of the shear connectors on the load-bearing be-haviour of the composite beam. From these investigations, information is derived on the defini-tion of a minimum degree of partial shear connection depending on the shear connector‘s load-slip behaviour. The present work is part of our research efforts in the field of steel-concrete com-posite beams and the development of novel shear connectors for buildings. Extensive experi-mental investigations on this subject are reported in detail in report IBK 278.I would like to thank Mr. Roland Bärtschi for his great effort and his extensive work. With hisexperimental, numerical and theoretical investigations he has contributed considerably to ourunderstanding of the composite behaviour of steel-concrete composite beams with low degreesof partial shear connection. A special thanks also goes to Mr. Hp. Arm and P. Hefti for their sup-port during the experimental investigations as well as to Mr. Hermann Beck and the firm Hiltifor their financial and technical support during the development of the novel shear connectors.

Zurich, February 2005 Prof. Dr. Mario Fontana

Vorwort

Dank Teilverbund kann die Wirtschaftlichkeit von Verbundträgern im Hochbau markant verbes-sert werden. Voraussetzung für die teilweise Verdübelung ist ein duktiles Last-Verformungsver-halten der Dübel. Der vorhandenen Duktilität und Verformungsfähigkeit der Dübel steht die be-nötigte Duktilität des Verbundträgers gegenüber. Diese hängt stark von der Geometrie des Trä-gers, der Lagerungsart und der Belastungsanordnung ab. Für die häufig verwendeten Kopfbol-zendübel liegen vereinfachte Bemessungsregeln für den minimalen Verdübelungsgrad vor. Für neuartige Verbundmittel, wie sie im Rahmen verschiedener Forschungsprojekte am Institut un-tersucht wurden, fehlen entsprechende Angaben für eine einfache und wirtschaftliche Anwen-dung.Der vorliegende als Promotionsarbeit verfasste Forschungsbericht zum Last-Verformungsver-halten von Verbundträgern mit geringem Verdübelungsgrad untersucht den grundlegenden Ein-fluss des Last-Verformungsverhaltens der Dübel auf das Tragverhalten der Träger. Aus diesen Untersuchungen werden Angaben für die Festlegung eines minimalen Verdübelungsgrades in Abhängigkeit des Last-Verformungsverhaltens der Dübel erarbeitet. Die Arbeit ist Teil unserer Forschungsanstrengungen im Bereich der Stahl-Beton-Verbundträger und der Entwicklung neuer Verbundmittel für den Hochbau. Umfangreiche experimentelle Untersuchungen zu die-sem Thema sind im IBK Bericht Nr. 278 detailliert dokumentiert. Ich möchte an dieser Stelle Herrn Roland Bärtschi für seinen grossen Einsatz und seine umfang-reiche Arbeit herzlich danken. Er hat mit seinen experimentellen, numerischen und theoreti-schen Untersuchungen wesentlich zum Verständnis des Verbundverhaltens von Stahl-Beton-Verbundträgern mit geringem Verdübelungsgrad beigetragen. Danken möchte ich besonders auch den Herren Hp. Arm und P. Hefti für ihre Unterstützung bei den experimentellen Untersu-chungen, sowie Herrn Hermann Beck und der Firma Hilti für die finanzielle und fachliche Un-terstützung bei der Entwicklung der neuartigen Verbundmittel.

Zürich, Februar 2005 Prof. Dr. Mario Fontana

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Application range for the rigid-ideal plastic design method . . . . . . . . . . . . . . . . . . 21.2.2 Advanced design methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Composite Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Definitions for shear connectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Definitions for composite beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Behaviour and design of composite beams in buildings . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Behaviour with no shear connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Behaviour with rigid shear connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Discussion of typical load-slip characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 Analytical models for research and design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.2 Numerical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.3 Experimental investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Analytical Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Fundamental equations for composite beams in pure bending. . . . . . . . . . . . . . . . . . . . 16

3.2.1 Loads, forces and moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.2 Kinematic relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Differential equation for linear elastic composite action . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Differential equation for nonlinear composite action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4.1 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4.2 Section geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.5 Well known design models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.5.1 Rigid-ideal plastic design model in Eurocode 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 223.5.2 γ-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5.3 Design model for linear elastic chords and rigid-plastic connectors . . . . . . . . . . . 24

4 Experimental Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Experimental studies reported in [31] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Push tests with StripCon on IPE 270 steel profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Numerical Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.1 Two-dimensional beam models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.1.1 Model geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.1.2 Section geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.1.3 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.1.4 Nonlinear component behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.1.5 Simulation arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.1.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2 Two-dimensional plane models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.3 Three-dimensional volume models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.3.1 Description of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.3.2 Verification and first applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3.3 Parametric investigation series A200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3.4 Parametric investigations series C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.3.5 Parametric investigation series C3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.3.6 Parametric investigation series C4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.4 Evaluation: minimum degree of partial shear connection . . . . . . . . . . . . . . . . . . . . . . . . 605.5 Discussion of numerical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.5.1 Two-dimensional beam models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.5.2 Three-dimensional volume models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.5.3 Asymmetrical effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.2 Position of the critical cross-section in composite beams . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2.1 Position of the critical cross-section with concentrated loads . . . . . . . . . . . . . . . 646.2.2 Position of the critical cross-section with uniform load . . . . . . . . . . . . . . . . . . . . . 656.2.3 Position of the critical cross-section with large values of η . . . . . . . . . . . . . . . . . . . . 66

6.3 The degree of partial shear connection required for full shear connection . . . . . . . . . . . 676.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.3.2 Rigid-brittle shear connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.3.3 Flexible brittle shear connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.4 Application range for the rigid-ideal plastic design method . . . . . . . . . . . . . . . . . . . . . . . 696.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.4.2 Design criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.4.3 Deformation behaviour of a composite beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.4.4 The minimum degree of partial shear connection . . . . . . . . . . . . . . . . . . . . . . . . 76

6.5 Advanced design method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.6 Generalisation to continuous composite beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.7 Example: Application of the deformation-based method . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.7.1 Overview and design for construction state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.7.2 Design for final state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.7.3 Comparison: deformation-based method vs. Eurocode 4 formulas . . . . . . . . . . . 87

7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.1 Critical cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.2 Minimum degree of partial shear connection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.3 Shear connector database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.4 Serviceability limit state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.5 Research-oriented analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907.6 Application range of the deformation-based method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

8 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918.1 Application range of the deformation-based method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 918.2 Stability questions and local effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918.3 Connector behaviour and lift-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Index of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Index of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Abstract

Composite beams with partial shear connection are widely used for buildings. The type and the number of shear connectors between the concrete slab and the steel beam decisively influence the load-bearing behaviour and the economic performance of the composite beam. For reasons of economy, composite beams should if possible be designed using the plastic method. To be able to carry out plastic design of a composite beam the chords must exhibit sufficient rotational capacity, and the shear connectors greater ductility and deformation capacity than required by the composite beam. Otherwise, the bending resistance of a composite beam is less than that de-termined with the plastic design method. With shear connectors of insufficient ductility, the shear forces cannot be fully redistributed among all shear connectors. Therefore, the maximum axial force in the chords is smaller than predicted. With shear connectors of insufficient defor-mation capacity, the maximum axial force in the chords is reached before the chords exhibit suf-ficient rotation to develop their full plastic bending moment.The ductility and the deformation capacity required by a composite beam strongly depend on the resistance and the stiffness of the shear connection. Nevertheless, the shear connectors used in steel-concrete composite construction are usually rather stiff. Therefore, the influence of the shear connection‘s stiffness on the required ductility and deformation capacity in a composite beam is usually neglected and rigid behaviour is assumed. The effects of the shear connection‘s resistance on the required ductility and deformation capacity in a composite beam are consid-ered by defining a minimum degree of partial shear connection. In composite beams with a de-gree of partial shear connection lower than the minimum degree of partial shear connection, the required ductility and/or deformation capacity is greater than the ductility or deformation capac-ity, respectively, provided by the shear connectors used. The composite beam would thus fail at lower loads than predicted by the plastic design method. Therefore, the plastic design method must not be used with composite beams with degrees of partial shear connection lower than the minimum one.The values for the minimum degree of partial shear connection given in today’s relevant stand-ards are based on the load-bearing behaviour of the most commonly used welded headed studs. However, with alternative connector systems exhibiting load-bearing behaviours differing greatly from that of a headed stud, the minimum degree of partial shear connection of a com-posite beam can differ greatly from that of a composite beam with headed studs.

The present publication reports on comprehensive experimental, numerical and analytical in-vestigations performed in recent years at the Institute of Structural Engineering (IBK) of the Swiss Federal Institute of Technology (ETH) in Zurich in order to develop a novel design meth-od for composite beams which deals consistently with a wide variety of shear connectors. Apart from a novel, deformation-based analytical design method for the minimum degree of partial shear connection, a comprehensive numerical model suitable for the advanced analysis of composite beams is presented.

Zusammenfassung

Verbundträger mit teilweiser Verdübelung werden häufig im Hochbau eingesetzt. Die Art und Anzahl der Verbundmittel beeinflusst das Last-Verformungsverhalten und die Wirtschaftlich-keit eines Verbundträgers entscheidend. Aus wirtschaftlichen Gründen sollten Verbundträger wenn immer möglich mittels plastischen Methoden bemessen werden. Damit die plastische Be-messung eines Verbundträgers möglich ist, müssen die Gurte eine genügende Rotationsfähig-keit aufweisen, und die Verbundmittel müssen mehr Duktilität und Verformungsfähigkeit haben als der Verbundträger benötigt. Andernfalls ist der Biegewiderstand eines Verbundträgers gerin-ger als mit der plastischen Bemessungsmethode berechnet. Mit Verbundmitteln von ungenügen-der Duktilität können die Scherkräfte nicht vollständig zwischen allen Verbundmitteln umgela-gert werden. Dadurch ist die maximale Normalkraft in den Gurten kleiner als berechnet. Mit Verbundmitteln von ungenügender Verformungsfähigkeit wird die maximale Normalkraft in den Gurten erreicht, bevor die Gurte genügend Rotation aufweisen, um den vollen plastischen Biegewiderstand zu entwickeln.Die benötigte Duktilität und Verformungsfähigkeit in einem Verbundträger hängt stark von der Steifigkeit und vom Tragwiderstand der Verbundfuge ab. Da jedoch die im Stahl-Beton-Ver-bundbau üblicherweise verwendeten Verbundmittel im Allgemeinen ziemlich steif sind, wird der Einfluss der Steifigkeit der Verbundfuge auf die benötigte Duktilität und Verformungsfähig-keit in einem Verbundträger zumeist vernachlässigt und starrer Verbund angenommen. Die Aus-wirkungen des Tragwiderstandes der Verbundfuge auf die benötigte Duktilität und Verfor-mungsfähigkeit in einem Verbundträger werden durch die Definition eines minimalen Verdübe-lungsgrades berücksichtigt. In Verbundträgern, welche einen kleineren als den minimalen Ver-dübelungsgrad aufweisen, ist die benötigte Duktilität und/oder Verformungsfähigkeit grösser als die Duktilität bzw. die Verformungsfähigkeit des verwendeten Verbundmittels. Der Ver-bundträger würde somit bei geringerer Last versagen als mit der plastischen Methode berechnet wurde. Deshalb darf die plastische Bemessungsmethode nicht für Verbundträger verwendet werden, welche einen geringeren als den minimalen Verdübelungsgrad aufweisen.Die Werte für den minimalen Verdübelungsgrad, wie sie in den heutigen Normen angegeben sind, basieren auf dem Last-Verformungsverhalten des zumeist verwendeten Kopfbolzendü-bels. Mit alternativen Verbundmitteln, welche vom Kopfbolzendübel z. T. erheblich abweichen-de Last-Verformungscharakteristiken aufweisen, kann sich der minimale Verdübelungsgrad ei-nes Verbundträgers jedoch beträchtlich von demjenigen eines Verbundträgers mit Kopfbolzen-dübeln unterscheiden.

Der vorliegend Bericht enthält Angaben über die umfangreichen experimentellen, numerischen und analytischen Untersuchungen, die in den letzten Jahren am Institut für Baustatik und Kon-struktion (IBK) an der ETH Zürich (ETH) durchgeführt wurden, um eine neuartige Berech-nungsmethode für den minimalen Verdübelungsgrad in Verbundträgern zu entwickeln, welche eine weite Bandbreite von Verbundmitteln mit einer einzigen, konsistenten Theorie abdeckt. Nebst einer verformungsbasierten, analytischen Berechnungsmethode für den minimalen Ver-dübelungsgrad wird auch ein umfassendes numerisches Modell vorgestellt, welches für die wei-terführende Analyse von Verbundträgern geeignet ist.

Résumé

Les poutres mixtes avec connexion partielle sont fréquemment utilisées dans le bâtiment. Le type et le nombre de connecteurs entre la dalle en béton et la poutre en acier influencent de façon déterminante le comportement charge-déformation de la poutre mixte ainsi que sa rentabilité économique. Pour des raisons économiques, les poutres mixtes sont d‘ordinaire dimensionnées plastiquement. L’emploi de cette méthode de calcul requiert d’une part que les membrures pré-sentent une capacité de rotation suffisante et, d’autre part, que les connecteurs possèdent une ductilité et une capacité de déformation plus grandes que celles requises par la poutre mixte. Si ces conditions ne sont pas respectées, la résistance à la flexion de la poutre est inférieure à la valeur déterminée lors d’un dimensionnement selon la méthode plastique. Si les connecteurs sont insuffisamment ductiles, l’effort rasant ne peut pas être uniformément redistribué entre tous les connecteurs. Par conséquent, l’effort normal maximal dans les membrures est inférieur à la valeur prédite. Si les connecteurs ont une capacité de déformation insuffisante, l’effort normal maximal dans les membrures est atteint avant que celles-ci ne subissent la rotation nécessaire au développement du moment plastique.La ductilité et la capacité de déformation requises dans une poutre mixte dépendent largement de la résistance et de la rigidité de la connexion. Comme les connecteurs utilisés habituellement dans la construction mixte acier-béton sont en général très rigides, l’influence de leur rigidité sur la ductilité et la capacité de déformation requises de la poutre mixte est souvent négligée et un comportement infiniment rigide est admis. Les effets de la résistance des connecteurs sur la ductilité et la capacité de déformation requises sont considérés par l’intermédiaire d’un degré minimal de connexion partielle. Lorsque le degré de connexion partielle d’une poutre mixte est inférieur au degré minimal de connexion partielle, la ductilité et/ou la capacité de rotation re-quise est supérieure à celle disponible. La poutre mixte atteint ainsi l’état limite ultime pour un niveau de charge inférieur à celui prédit par un dimensionnement plastique. Par conséquent, la méthode de dimensionnement plastique ne doit pas être utilisée avec des poutres mixtes présen-tant un degré de connexion partielle inférieur au degré minimal.Les valeurs de degré minimal de connexion partielle contenues dans les normes actuelles sont basées sur le comportement charge-déformation des goujons à tête les plus fréquemment utili-sés. Ce degré minimal de connexion partielle peut être fort différent de celui d’une poutre mixte avec des goujons à tête si d’autres systèmes de connexion, présentant un comportement consi-dérablement différent de celui des goujons à tête, sont employés.

La présente thèse décrit les vastes recherches expérimentales, numériques et analytiques qui ont été effectuées ces dernières années à l’Institut de Structures (IBK) de l’Ecole Polytechnique Fé-dérale de Zurich (ETHZ), dans le but de développer un modèle de calcul novateur pour le di-mensionnement des poutres mixtes, qui tienne compte de manière consistante du comportement réel d’une large variété de connecteurs.En plus d’une méthode analytique de dimensionnement novatrice basée sur la déformation des connecteurs pour déterminer le degré minimal de connexion partielle, un modèle numérique complet adapté à l’étude détaillée des poutres mixtes est présenté.

Overview

1

1 Introduction

1.1 Overview

Composite construction has become more and more important in recent years. By means of an optimal use of materials resource-saving ecological floor systems can be erected. The charac-teristics of the connection between concrete slab and steel beam and the number of connector elements decisively influence the load-bearing behaviour and the economical performance of a composite member.When using composite beams in buildings the construction process often governs the size of the steel section. As a result, in many cases only little additional bending resistance from composite action is required for the final state. Therefore, only a small amount of shear force needs to be transferred through the shear interface. For economic reasons the number of shear connectors is then often reduced to the structural minimum. This situation is called partial shear connection. With partial shear connection the shear interface governs failure of the composite beam. If slip in the interface exceeds the connector’s deformation capacity the interface and thus the beam fails. This occurs before the predicted ultimate load (according to the rigid-ideal plastic design method in Eurocode 4 [26]) is reached if the degree of partial shear connection is too low.Usually, composite beams in buildings are designed by means of a rigid-ideal plastic design method. Below a certain minimum degree of partial shear connection this design method may not be used. Eurocode 4 [26], for example, gives simple formulas for the determination of this minimum degree of partial shear connection for composite beams with headed studs. However, these formulas neglect some parameters and therefore give only a rough approximation valid for a limited application range. In particular, they are not necessarily applicable to composite beams with alternative shear connectors exhibiting a different load-slip behaviour than headed studs. Typical examples for such alternative connector systems are the Hilti X-HVB, the Tecnaria shear connectors or the new types of shear connectors with powder-actuated fasteners recently developed at the Institute of Structural Engineering at ETH Zurich called „RibCon“ and „Strip-Con“ (cf. [31] and Figure 1.1).

Figure 1.1 Left: RibCon (left) and StripCon (right) shear connectors. Right: Tecnaria (left) and Hilti X-HVB (right) shear connectors.

Introduction

2

Today, alternative shear connectors are used in special cases only (e.g. when welding is not pos-sible). With a design model allowing the use of the rigid-ideal plastic design method with alter-native shear connectors at very low degrees of partial shear connection, a wide variety of shear connectors may be used in the future, which is of great economic interest in terms of safety, cost and construction time.

1.2 Scope

The goal of the presented work is to define a design method for composite beams in buildings with alternative shear connectors. This goal is achieved by two approaches:

1) Re-define the application range for the rigid-ideal plastic design method

2) Develop advanced design methods

1.2.1 Application range for the rigid-ideal plastic design method

Composite beams in buildings can easily be designed if the rigid-ideal plastic design method is used. Therefore, a first approach is to define the conditions under which composite beams with a wide variety of shear connectors may be designed using the rigid-ideal plastic design method as shown e.g. in [26] (cf. section 6.4 on page 69).

1.2.2 Advanced design methods

The rigid-ideal plastic design method is in many cases rather conservative due to its very nature. Therefore, advanced design methods are developed. These design methods require far more de-sign effort but can go somewhat beyond the limitations given by the rigid-ideal plastic design method. One advanced design method, a numerical model, is presented in section 5.3 on page 37 ff. In addition, an advanced analytical design method is presented in section 6.5 on page 78.

1.3 Limitations

• The present research is limited to composite beams in buildings made of a hot-rolled steel section of Class 1 according to Eurocode 3 [24] and a concrete slab, both of which are of con-stant sectional geometry over the whole length of the beam.

• The investigations reported below are limited to simply-supported composite beams subject-ed to symmetrical loads. However, the findings for simply-supported beams are generalised to continuous composite beams in section 6.6 on page 78. In continuous composite beams, sufficient rotational capacity is also assumed to be available at the inner supports.

• In the present research only symmetrical concentrated loads and a uniform load are consid-ered as actions. The effects of temperature, imposed deformations, shrinkage and further types of actions may be considered in an analogous manner.

• Slim-floor beams are not considered in the present research.• The reinforcement in the concrete slab of a composite beam is assumed to be of sufficient

ductility such that brittle failure of the longitudinal reinforcement in the parts of the concrete slab in tension (e.g. the inner supports of a continuous composite beam) can be excluded.

• Questions of stability such as local buckling or lateral-torsional buckling are excluded.

Limitations

3

• Local failure of the steel beam at the support due to transverse compression is not considered in the research presented in this report.

• The effects of vertical separation of steel beam and concrete slab (lift-off) are excluded.• Dynamic effects and time-history effects such as creep and shrinkage are not considered in

the work reported below.• The design models shown below are not valid for shear connectors whose ductility is very

small compared to the ductility required in the composite beam (cf. limitations given in sec-tion 6.4.4.2 on page 77, b) and c)) as e.g. glued connections used in composite bridges and reported in [62].

• The extension of the application range given in section 6.4 on page 69 for the rigid-ideal plas-tic method shown in [26] is limited to composite beams with a γ-value (cf. section 3.5.2 on page 23) greater than γ = 0.9. This implies that the shear connectors must exhibit small initial slip δi and thus large initial stiffness Si.

Introduction

4

Definitions

5

2 Composite Beams

2.1 Definitions

2.1.1 Definitions for shear connectors

For design, the real connector behaviour is simplified to a user-defined characteristic connector behaviour as shown in Figure 2.1. The simplified behaviour is described by the values PRk, δPu, δi and δu as described below. From these values, the ductility of a shear connector is determined as δu - δi and the initial stiffness as Si = PRk/δi.

2.1.1.1 Ultimate shear resistance of a shear connector Pu

The ultimate shear resistance Pu of a shear connector is the ultimate shear force which the shear connector can resist. Pu can e.g. be derived from the results of push tests by means of statistical methods as a lower fractile value.

2.1.1.2 Characteristic shear resistance of a shear connector PRk

PRk is the characteristic shear resistance of a shear connector. In Eurocode 4 [26] (Annex B) PRkis defined as PRk = 0.9 * Pu. However, with the method shown in section 6.4 on page 69, PRkcan be chosen freely by the engineer in a range between 0 and Pu with the corresponding defor-mation capacity δu and initial slip δi.

Figure 2.1 Real and 3 examples of characteristic load-slip behaviour of a shear connector. Characteristic behaviour #1 shows the greatest characteristic shear resistance PRk, but the lowest initial stiffness Si, deformation capacity δu and ductility δu - δi. Characteristic behaviours #2 and #3 show lower shear resistances, but higher in-itial stiffnesses Si, deformation capacities δu and ductilities δu - δi.

δPu

δ i,1δu,1

δu,2δi,2δu,3

δi,3

Slip [mm]

Shea

r for

ce [k

N]

Real behaviourCharacteristic behaviour 1Characteristic behaviour 2Characteristic behaviour 3

PRk,3

PRk,2

PRk,1

Pu

Si,1

1

Composite Beams

6

2.1.1.3 Initial slip of a shear connector δi

The initial slip δi of a shear connector is the slip when reaching the characteristic shear resist-ance PRk on the ascending branch of the load-slip curve (cf. Figure 2.1). This value must be de-termined e.g. from the results of push tests by means of statistical methods as an upper fractile value.

2.1.1.4 Deformation capacity of a shear connector δu

The deformation capacity δu of a shear connector is the slip when reaching the characteristic shear resistance PRk on the descending branch of the load-slip curve (cf. Figure 2.1). This value must be determined e.g. from the results of push tests by means of statistical methods as a lower fractile value.

2.1.1.5 Ductility of a shear connector δu - δi

The ductility of a shear connector is the width of the yield plateau of the characteristic load-slip behaviour of the shear connector (cf. Figure 2.1). The ductility of a shear connector is thus de-termined as the difference between the deformation capacity and the initial slip (δu - δi) of a shear connector. With shear connectors showing rigid-plastic behaviour (δi = 0) the ductility of a shear connector is equal to its deformation capacity. If the ductility of the shear connector amounts to zero, the shear connector is termed brittle.

2.1.1.6 Initial stiffness of a shear connector Si

The initial stiffness Si of a shear connector is the secant stiffness on the characteristic load-slip curve (cf. Figure 2.1) between a slip of zero and δi. The initial stiffness Si is used for servicea-bility limit state analysis e.g. with the γ-method (cf. section 3.5.2 on page 23).

2.1.1.7 Stiffness of a shear connector: rigid vs. flexible

A shear connector is flexible if the initial slip δi is greater than zero and thus the initial stiffness Si is not infinite. If the initial slip δi is equal to zero and thus the initial stiffness Si is infinite, the shear connector is termed rigid.

2.1.1.8 Slip at ultimate load of a shear connector δPu

The slip at ultimate load δPu of a shear connector is defined as the slip when reaching Pu (cf. Figure 2.1). Thus, δPu does not depend on PRk. This value can be determined by means of sta-tistical methods as an upper fractile value from the results of push tests.

2.1.2 Definitions for composite beams

2.1.2.1 Composite beam

Composite beams are flexural members made of two or more longitudinal members which are constrained in their relative longitudinal displacements at the interface by means of a shear con-

Definitions

7

nection of the members. In this report the term „composite beam“ is also used synonymously for steel-concrete composite beams.

2.1.2.2 Steel-concrete composite beam

Steel-concrete composite beams are flexural members made of a steel beam and a concrete slab. In between the two longitudinal components, shear connectors are placed to transmit the longi-tudinal shear forces between the steel beam and the concrete slab.

2.1.2.3 Components of a composite beam

A steel-concrete composite beam consists of three components: The steel beam, the concrete slab and the interface.

2.1.2.4 Chords of a composite beam

The chords of a composite beam are the two longitudinal components of the composite beam, namely the steel beam and the concrete slab.

2.1.2.5 Interface (shear connection) of a composite beam

The interface (or synonymously: shear connection) in a composite beam is a collective term comprising all shear connectors which connect two chords. It also stands for the region where the shear connectors are placed. In steel-concrete composite beams the interface is the area be-tween the upperside of the steel profile‘s upper flange and the underside of the concrete slab comprising all shear connectors placed between the two chords.

2.1.2.6 Slip in a composite beam d

The slip d is the difference in longitudinal displacement between the two connected faces of the chords at the interface, e.g. the underside of the concrete slab and the upperside of the steel beam.

2.1.2.7 Critical cross-section

A critical cross-section of a composite beam is a section in the beam where the bending moment first reaches the bending resistance at the respective longitudinal position of the composite beam.As the bending resistance of a composite beam is in general not constant over its length, critical cross-sections include, but are not limited to the sections in which the bending moment from actions is maximum. Critical cross-sections can be located at the following longitudinal posi-tions within a composite beam:• Positions with maximum bending moment• Positions subjected to concentrated loads or reactions (including supports)• Positions with a sudden change of cross-section (including the ends of a beam)• Positions explained in section 6.2 on page 63In composite beams two critical cross-sections occur per half-span: one in the field and one at the support. While at the support the critical cross-section is assumed to be located directly at

Composite Beams

8

the support, the location of the critical cross-section in the field is variable as determined in sec-tion 6.2 on page 63.

2.1.2.8 Shear length

The shear length is the longitudinal distance between two critical cross-sections with unequal bending moment and thus typically the distance between the critical cross-section in the field and the support. In Eurocode 4, the term „critical length“ is used for a similar definition.

2.1.2.9 Bending resistance of a composite beam

The bending resistance of a composite beam is the ultimate bending moment at a critical cross-section of the composite beam. The bending resistance of a composite beam can be determined using the rigid-ideal plastic de-sign method given in Eurocode 4 [26] if the rotational capacity of each chord is sufficient (cf. [26]) and the shear connection is ductile and exhibits sufficient deformation capacity. Other-wise, elastic or nonlinear methods need to be used.

2.1.2.10 Behaviour of a shear connector in a composite beam: brittle vs. ductile

A ductile shear connector is a shear connector whose ductility is greater than or equal to the duc-tility required in the composite beam to allow full plastic redistribution of the shear forces among all shear connectors located within the shear length with the corresponding degree of partial shear connection (cf. section 6.4 on page 69). Shear connectors exhibiting less ductility are non-ductile. If the ductility of a shear connector δu - δi is zero, the shear connector is termed brittle.

2.1.2.11 Resistance of the shear connection in composite beams: full vs. partial shear connection

If the shear interface governs failure of the composite beam, the beam is in partial shear con-nection, otherwise in full shear connection.

2.1.2.12 Degree of partial shear connection

The degree of partial shear connection is the ratio of the sum of the shear resistance of all shear connectors to the difference in axial force of the two connected chords over a shear length (typ-ically at the support and in the field of a composite beam) according to equation (2.1). In simply-supported beams the axial forces in the chords at the support are zero. Therefore, the term „min(|Na,pl|;|Nc,t|)“ in the denominator of equation (2.1) is zero for simply-supported beams.

with (2.1)

n = Number of shear connectors per shear lengthnf = Number of shear connectors per shear length required for full shear connection

(2.2)

(2.3)

η NNf------

n P⋅ Rkmin Na pl, Nc pl,;( ) min Na pl, Nc t,;( )+------------------------------------------------------------------------------------------------------ n

nf----≠= =

N n P⋅ Rk=

Nf min Na pl, Nc pl,;( ) min Na pl, Nc t,;( )+=

Behaviour and design of composite beams in buildings

9

In the nomenclature of Eurocode 4 [26] the designations „n“ and „N“ are both used synony-mously for the number of shear connectors. Consequently, the designations nf and Nf stand for the number of shear connectors required for full shear connection. However, this definition is only justified with ductile shear connectors (cf. section 2.1.2.10 on page 8). With non-ductile shear connectors a complete redistribution of shear forces among all the shear connectors in a shear length is not possible. Therefore, the required number of shear connectors for full shear connection is greater than Nf/PRk (cf. section 6.3 on page 67).

(2.4)

A much more stable approach is to define Nf as the difference in axial force in a chord between two neighbouring critical cross-sections with full shear connection (cf. equation (2.3)). N is then defined as shown in equation (2.2).With this definition it is obvious that the degree of partial shear connection required for full shear connection may be greater than η = 1.0 if complete redistribution of shear forces among the connectors is not possible due to a lack of ductility (cf. section 6.3 on page 67).

2.1.2.13 Stiffness of the shear connection in a composite beam: rigid vs. flexible

The shear connection may be rigid or flexible. With rigid shear connection, the composite beam exhibits full interaction, with flexible shear connection the composite beam exhibits partial in-teraction. Without shear connection the composite beam exhibits no interaction.

2.1.2.14 Degree of partial interaction

The degree of partial interaction is expressed in a simplified form by the γ-value suggested by Möhler [51] and explained in section 3.5.2 on page 23. The degree of partial interaction char-acterizes the stiffness of the shear connection and the axial stiffness of the chords compared to the sum of the bending stiffnesses of the single chords. A value of γ = 1.0 stands for full inter-action due to rigid shear connection, γ = 0.0 for no interaction, while values for γ between 0 and 1 stand for partial interaction due to flexible shear connection.

2.2 Behaviour and design of composite beams in buildings

The rigid-ideal plastic design method shown in Eurocode 4 [26] for composite beams with par-tial shear connection for buildings assumes that all shear connectors and the steel beam or the concrete slab are at ultimate limit state at the same time (cf. Figure 2.2, diagram „FE: δu = 6 mm, 2 concentrated loads“). In this case the bending resistance of a composite beam is determined as shown in equation (2.5). The degree of partial shear connection η is defined as shown in equation (2.1).

(2.5)

While the concrete slabs and steel sections used in buildings usually provide a more than ade-quate rotational capacity to remain at ultimate limit state over a large range of rotation, the shear interface often provides relatively poor deformation capacity. Composite beams should there-

nfNf

PRk--------->

MR n P⋅ Rk e⋅ Mpl N,+=

Composite Beams

10

fore be designed in such a way that either the steel beam or the concrete slab reaches ultimate limit state due to bending and axial force prior to failure of the shear interface.

At low degrees of partial shear connection large slip develops in the shear interface between the steel section and the concrete slab. This excessive slip may lead to premature failure of the com-posite beam (cf. [43]) if one of the two following conditions is met:a) The maximum of the sum of the shear forces of all shear connectors is less than the number

of shear connectors multiplied by the characteristic shear resistance of a connector. This hap-pens because not all shear connectors exhibit the same slip and redistribution of the shear forces among the shear connectors is limited. Some connectors have not yet reached their maximum load while others have already passed the ultimate limit state (cf. Figure 2.2, dia-gram „FE: δu = 2 mm, uniform load“). In the following this condition will be referred to as „ductility criterion“.

b) The sum of the shear forces of all connectors reaches a maximum before the chords develop their full plastic bending moment (cf. Figure 2.2, diagram „FE: δu = 2 mm, 2 concentrated loads“). This occurs if the deformation capacity of the shear connectors is less than the max-imum slip necessary for the chords to reach the rotation at which they show full plastic bend-ing resistance. In the following this condition will be referred to as the „deformation capacity criterion“.

Figure 2.2 Load-deformation behaviour of a simply-supported composite beam with partial shear connection. η = 0.3, L = 7.2 m, Steel section IPE 270, fy = 300 N/mm². Shear connectors with δu (0.9 * Pu) = 2 to 6 mm evenly-distributed over the whole length of the beam.

0

50

100

150

200

250

0 20 40 60 80 100 120Deflection [mm]

Ben

ding

Mom

ent [

kNm

]

Analytical: rigid shear connection

FE: δu=2mm, uniform load

FE: δu=2mm, 2 concentrated loads

FE: δu=6mm, 2 concentrated loads

FE: no shear connection

Analytical: Steel beam

=Mpl,N

<Mpl,N

=n *PRk*e

=n *PRk*e

<n *PRk*e

Analytical: η=0.3

Behaviour and design of composite beams in buildings

11

Therefore, with partial shear connection, application of the rigid-ideal plastic design model re-quires the shear connectors to exhibit sufficient ductility (δu – δi) and sufficient deformation ca-pacity δu. Further, the stiffness of the interface must be sufficient (cf. chapter 6 on page 63 ff).

In general, composite beams can be modelled as a pair of beams. Both members exhibit the same vertical displacement w(x). As vertical displacements are small compared to the system geometry, both members are assumed to exhibit the same rotations w’’(x) (cf. Figure 2.3). With flexible connectors these rotations cause strains which differ between the concrete slab’s under-side and the top of the steel beam. As a result, slip occurs in the shear interface. This slip occurs due to two influences, namely the rotation of the chords (dominant with a very soft interface) and shear (dominant with a rather stiff interface), and is a maximum if the shear connector’s stiffness equals zero.Each member has its own neutral axis. For rigid shear connection the two axes coincide (cf. Fig-ure 2.3, 2nd from the left). With flexible shear connection the vertical distance between the two neutral axes is smaller than with no shear connection, leading to slip movements between zero and the slip determined for the case with no shear connection.The actual vertical distance between the neutral axes in composite beams with flexible shear connection is difficult to determine as slip and shear forces interact. This interaction is account-ed for in the differential equation for elastic composite action (cf. e.g. [33]). However, solution of this equation is only possible for special cases. If nonlinear behaviour of the concrete slab, the shear interface and the steel beam needs to be considered, analytical description of the be-haviour is even more complicated. Therefore, many authors (e.g. [6], [15], [21] or [43]) have relied on a numerical analysis of the problem.

2.2.1 Behaviour with no shear connection

With no shear connection (η = 0) the shear forces in the interface and thus the axial forces in both members are zero. The steel beam’s neutral axis is located in the centroid of the steel sec-tion (cf. Figure 2.3, right). For the concrete slab the position of the neutral axis depends on the amount of tensile force which can be transmitted in the part of the slab in tension. The largest slip in the shear interface is found for unreinforced concrete with no tensile strength. In this case

Figure 2.3 Composite section, strain planes for rigid, flexible and no shear connection. For cracked concrete averaged strains are shown. For a given curvature w‘‘(x) the dif-ference in slip at the shear interface ∆ε(x) depends on the vertical distance z0,c - z0,a between the neutral axes of the two chords.

Composite Beams

12

the concrete slab cracks over its whole depth, and the „neutral axis“ is located on top of the con-crete slab. These cracks are modelled as longitudinally smeared, and „averaged“ strains are shown in Figure 2.3 (right).

2.2.2 Behaviour with rigid shear connection

With rigid shear connection no slip occurs and only one neutral axis is found. Under elastic con-ditions the shear forces are proportional to vertical shear and thus can be easily calculated using well-known beam theory. Inelastic deformation of the concrete slab (cracking and crushing of the concrete) or the steel beam (yielding) results in large, localised shear forces in the interface due to large local differences in strain between the underside of the concrete slab and the upper-side of the steel beam.

2.3 Discussion of typical load-slip characteristics

The shape of a shear connector’s load-slip curve has a decisive influence on the behaviour of a composite beam, particularly with partial shear connection. As an example, some typical load-slip characteristics and their effect on the behaviour of the composite beam given in Figure 2.2are discussed in the following (cf. Figure 2.4).The rigid-ideal plastic design model in Eurocode 4 [26] assumes connector behaviour c) with δu = 6mm. With large values of δu (e.g. δu = 15 mm) the ultimate loads predicted by the rigid-ideal plastic model were confirmed by finite element simulations with connector behaviour e). Behaviour e) is an approximation of a headed stud’s behaviour as shown in f). Therefore, with type f) similar results are obtained. Type d) may only be used with the rigid-ideal plastic Eurocode 4 model if δi is small. Otherwise, the connectors located in regions with small slip (d(x) < δi, near the centre of the beam or next to a plastic hinge) are not fully activated at ulti-mate limit state and the ductility required in the composite beam is greater than assumed for the rigid-ideal plastic model due to insufficient initial stiffness Si of the shear connector (cf. section 6.4 on page 69).

Figure 2.4 Typical connector characteristics (left) and their effect on the bending resistance of the same composite beam mentioned in Figure 2.2 (right). For composite beams with ductile shear connectors of sufficient deformation capacity the rigid-ideal plastic design method according to Eurocode 4 [26] may be used for degrees of partial shear connection . With brittle shear connectors the plastic bending resistance of a composite beam is only reached with η >> 1.0.

00.0 0.5 1.0 1.5 2.0

η [-]

e) δu=1mm, FE, uniform load

c) δu=oo (EC4)

Mpl,a

Mpl

a) (uniform load)

a) (2 concentrated loads)

e) δu=1mm, FE, 2 concentrated loads

e) FE, δu=15mm

MR,el

η 0≥

State of the art

13

With non-ductile shear connectors a reduction of bending resistance for partial shear connection compared to the rigid-ideal plastic design model with is observed (e.g. type e) with δu = 1 mm in Figure 2.4).With brittle shear connectors (e.g. a) or b)) the influence of the load distribution becomes crucial as plastic redistribution of the shear forces is not possible (nor elastic redistribution with type a)) and each connector must show enough resistance to take the full shear load in its respective place. With a uniform load the shear distribution has a triangular shape. Therefore, the maxi-mum shear force is double the mean shear force determined by . For this case elastic design of the composite beam with non-equidistantly arranged shear connectors is recommended. When the composite beam reaches elastic bending resistance MR,el the shear in-terface must transfer the elastic axial force at yield Ny. With evenly spaced shear connectors of type a) the degree of partial shear connection η must therefore amount to at least η = Ny/Npl in a beam subjected to just two concentrated loads or to η = 2*Ny/Npl with a uniform load in order to reach MR,el.If plastic bending resistance should be obtained, plastic hinges must develop. At these plastic hinges, large, localised shear forces develop in the interface with rigid shear connection. There-fore, brittle shear connectors either must be strong enough to resist these large peak forces or, alternatively, failure of the shear connectors near to the plastic hinge must be considered in de-sign. For brittle connectors with only very large values of η a bending resistance significantly greater than MR,el is obtained.Composite beams with shear connectors of type a) exhibit a linear decay of bending resistance as shown in Figure 2.4 (right) as η decreases. If η is too small the shear connection produces no additional beam resistance because the shear interface fails prematurely (cf. Figure 2.2).

2.4 State of the art

2.4.1 Analytical models for research and design

Eurocode 4 [26] is one of today’s well-known references for state-of-the-art analytical design of composite beams in Europe. The design model for partial shear connection given in Eurocode 4 is based on the properties of the welded headed stud. Alternative shear connection systems are classified either as „ductile“ when featuring a characteristic deformation capacity of at least 6 mm or as „non-ductile“ if the characteristic deformation capacity is less than 6 mm. This characteristic deformation capacity is defined as the slip at 90 % residual load. The limits given for the minimum degree of partial shear connection are based on the assumption of a char-acteristic slip of 6 mm. For shear connectors with increased deformation capacity this approach is safe, but not economical. On the other hand, using shear connectors which do not fulfil the requirements of Eurocode 4 may be a crucial safety issue. In awareness of these considerations, extensive experimental, analytical and numerical investi-gations have been performed in the last two decades. Some publications are listed below.The fundamental theory of composite beams with elastic shear connection was developed - among many others - by Stüssi [59], [60], [61] and Hoischen [41]. Möhler [51] gives a simpli-fied design model for composite beams with flexible shear connection. Natterer and Hoeft [52], Mischler and Kenel [50] use these models for the elastic analysis of timber-concrete composite beams.

δu ∞=

P x( ) t x( ) ∆c⋅ N n⁄= =

Composite Beams

14

Oehlers et al. ([53], [54] and [55]) have performed comprehensive investigations on composite beams with partial shear connection with shear connectors of limited slip capacity.Frangi and Fontana [33] present an elasto-plastic design model for timber-concrete composite beams in which the shear interface may exhibit nonlinear behaviour.

2.4.2 Numerical models

Kim et al. [45] list some of the well known finite element models for composite beams. Among others, the numerical models of Aribert (cf. [3], [4], [5] and [6]) and Bode/Schanzenbach (cf. [15] and [16]) are used for the determination of the minimum degree of partial shear connection in composite beams. Aribert [4] presents numerical investigations on composite beams with Hilti HVB shear connectors. Aribert [6] presents numerical, experimental and analytical investigations on the minimum de-gree of partial shear connection for steel grades between S235 and S460. In the numerical in-vestigations a set of 225 composite beams with spans from 5 to 20 m is analysed. For the char-acteristic slip of the shear connection values of 2, 6 and 8 mm are treated. The degree of partial shear connection amounted to 0.2, 0.4, 0.6, 0.8 and 1.0.Johnson and Molenstra [43] report on a parametric study on composite beams performed using their finite element programme EPPIB. The investigations cover 15 two-span beams and more than 100 simulations on simply-supported beams. In [15] and [16] Becker reports numerical investigations on a set of 96 composite beams with the aim of defining the minimum degree of partial shear connection for different values of the shear connector’s deformation capacity. The load-slip behaviour of the shear connectors is elas-tic-perfectly plastic. Composite beams of different types with degrees of partial shear connec-tion from 0.1 to 1.0 are loaded to 97.5 % of their theoretical bending resistance. The required deformation capacity of the shear connectors is then determined as the maximum slip occurring. The minimum degree of partial shear connection is then derived by evaluating the results for different values of deformation capacity. This evaluation is carried out for ultimate slip values of the shear connectors of 6, 8 and 15 mm.Hegger et al. [35] reports on finite element modelling of a shear connection with a headed stud in concrete.

2.4.3 Experimental investigations

In [31] and [32] (referring to [8], [9], [10], [14], [19], [20], [46], [48], [49], [56] and [58]) the results from the experimental investigations performed at ETH with powder-actuated fasteners are shown. Push test and beam tests with the Hilti X-HVB shear connectors are reported in [38]and [39] respectively. This work is assessed by Badoux in [7]. Thomann et al. [62] report on push tests with an adherence shear connection used for bridges. Gündel [34] and Lehner [47]report on a total of 21 push tests with X-HVB shear connectors with different types of powder-actuated fasteners. Gündel [34] further reports numerical modelling of the local anchorage of powder-actuated fasteners in a steel flange while Lehner [47] developed analytical models for the same problem. In [36] Hegger and Döinghaus report on shear connection with high strength materials.

Overview

15

3 Analytical Models

3.1 Overview

Today‘s well known analytical models for composite beams can be divided into two groups:a) Models describing the elastic load-deformation behaviour of a composite beam are mainly

used for the serviceability limit state analysis of composite beams or for the elastic design of composite beams. The differential equation for elastic composite action is reported for ex-ample in [33], [50], [51] or [52]. Möhler [51] developed an elastic design method called the γ-method which is described be-low in section 3.5.2 on page 23, and Aribert [6] published a design method for composite beams with elastic chords and rigid-ideal plastic behaviour of the shear interface which is de-scribed in section 3.5.3 on page 24.

b) Rigid-ideal plastic design models are used to determine a composite beam‘s bending resist-ance. These models neglect the fact that the rotational capacity of the composite beam may be limited due to limited rotational capacity of the steel beam or of the concrete slab or due to a limited deformation capacity of the shear connection. Therefore, the application range of the rigid-ideal plastic model is limited in the different codes. For example, in Eurocode 4 [26] the range of application of the rigid-ideal plastic model is limited to composite beams of Class 1 or Class 2 without pre-stressing with tendons. Further, the degree of partial shear connection must be sufficient, and the shear connectors must be „ductile“. In Eurocode 4 [26] „ductile“ shear connectors are defined as shear connectors with a characteristic defor-mation capacity of at least δuk = 6 mm at PRk = 0.9 Pu.

Apart from these specific models for composite beams, elementary beam theory can also be used for composite beams with full, rigid shear connection and for composite beams without composite action.Composite beams can be formulated as a pair of beams showing the same vertical displacement at every longitudinal position. Each of the two members carries a normal force Ni(x) and a bend-ing moment Mi(x) (cf. section 3.2 on page 16). In order to determine the axial force Ni(x) for each beam, a differential equation for linear elastic composite action (cf. section 3.3 on page 18) or for nonlinear composite action (cf. section 3.4 on page 20) is used. However, application of these differential equations is rather complex. Nevertheless, the differ-ential equation for linear elastic composite action is solved for two cases and used for the elastic analysis of a composite beam in the present work. The differential equation for nonlinear composite action is given as an indication for further re-search only and is not used in the present work. A simple design model for bilinear component behaviour is shown in section 6.4 on page 69. This design model is based on the kinematic behaviour of a composite beam.

Analytical Models

16

3.2 Fundamental equations for composite beams in pure bending

3.2.1 Loads, forces and moments

Loads are applied in the vertical axis of symmetry as concentrated loads or distributed loads. Local effects and shear deformations are not considered. The relations between external actions, internal forces and internal moments are shown in equation (3.1).

(3.1)

In composite beams the global forces and moments are redistributed between the single compo-nents. For example, the total bending moment of a composite beam without external axial force is established as shown in equation (3.2).

(3.2)

For each chord (of number i) of the composite beam (typically the concrete slab and the steel beam) equations (3.3) and (3.4) apply.

or (3.3)

or (3.4)

The above equations are valid also for normal non-composite beams. Given are the functions M(x), N(x). The functions z0(x) and w(x) (see equation (3.6)) are sought. z0(x) is the vertical position where the strain is zero in each chord (cf. Figure 3.1) at the corresponding longitudinal position x, while w(x) stands for the vertical displacement of the composite beam at the corre-sponding longitudinal location. In composite beams the axial force at every position in each member is given by equation (3.5), whereby the slip at the shear interface is determined using equation (3.10) with the shear flow in the interface t(x).

, with (cf. e.g. equation (3.35)) (3.5)

3.2.2 Kinematic relations

Assuming that for each chord (of number i) all sections remain plane and kinematic consistency is given and deformations are small in comparison to the system geometry, the strain is deter-mined from the deformation of the member as shown in equation (3.6).

(3.6)

From equation (3.6) w(x) and z0,i(x) can be determined for any beam in bending with or without axial force using equations (3.3) and (3.4).

q x( )xd

d Q x( )x2

2

∂

∂ M x( )= =

M x( ) Mi x( ) N x( ) e x( )⋅+i∑ Ma x( ) Mc x( ) N x( ) e x( )⋅+ += =

Ni x( ) σz∫ x z,( ) b x z,( )⋅ zd⋅=

zdd Ni x( ) σ x z,( ) b x z,( )⋅=

Mi x( ) σz∫ x z,( ) b x z,( ) z⋅ ⋅ zd⋅=

zdd Mi x( ) σ x z,( ) b x z,( ) z⋅ ⋅=

Ni x( ) t x( )x∫± xd= t x( ) f d x( )( )=

εi x z,( )x2

2

∂

∂ w x( ) z0 i, x( ) z–( )⋅=

Fundamental equations for composite beams in pure bending

17

Concrete in tension exhibits cracks which represent discontinuities in longitudinal strain. As a simplification, averaged strains (resulting from longitudinally smearing the cracks) are used in this report and also for equation (3.6).In composite beams slip is a function of w(x) and the vertical distance between z0,a(x) in the steel beam and z0,c(x) in the concrete slab as shown in Figure 3.1. Slip (cf. equations (3.9) and (3.10)) is the integral of the difference in longitudinal strain between the underside of the con-crete slab (or at the underside of a gap of height ∆, if present) as given in equation (3.7) and on top of the steel beam as shown in equation (3.8).

(3.7)

(3.8)

(3.9)

(3.10)

The relations shown above for composite beams can be used for the formulation of a differential equation. For linear elastic behaviour of all materials and the shear interface the differential equation is shown below in section 3.3 on page 18. For nonlinear behaviour of the material or the shear interface the differential equation becomes much more complicated as shown in sec-tion 3.4 on page 20 and can only be solved in general using numerical methods. For the appli-cation of numerical methods it is very convenient if all influences are formulated as a continu-ous function. Therefore, for the load distribution a continuous formulation is given in equation (3.18) with concentrated loads and in (3.20) with uniform load (cf. Vogel [64]). Further, contin-uous functions are given for section geometries (3.36), material properties (equations (3.32), (3.33) and (3.34)) and the behaviour of the shear interface (3.35).

Figure 3.1 Composite section, strain planes for rigid, flexible and no shear connection. For cracked concrete averaged strains are shown. For a given curvature w‘‘(x) the dif-ference in slip at the shear interface ∆ε(x) depends on the vertical distance z0,c - z0,a between the neutral axes of the two chords.

εc,inf x( )x2

2

∂

∂ w x( ) z0 c, x( ) ha x( ) ∆ x( )––{ }⋅x2

2

∂

∂ w x( ) z0 c, x( ) z0 a, x( )–{ }⋅–=

εa,sup x( )x2

2

∂

∂ w x( ) z0 c, x( ) z0 a, x( )–{ }⋅x2

2

∂

∂ w x( ) ha x( ) z0 a, x( )–{ }⋅–=

xdd d x( ) εc,inf x( ) εa,sup x( )

x2

2

∂

∂=– w x( ) z0 c, x( ) z0 a, x( )–{ }⋅=

d x( )x2

2

d

d

x∫ w x( ) z0 c, x( ) z0 a, x( )–{ }dx=

Analytical Models

18

3.3 Differential equation for linear elastic composite action

For the linear material behaviour of all components and neglecting geometrical nonlinearities, the differential equation for elastic composite action (cf. [33], [50], [51] or [52]) is written as:

or alternatively as (3.11)

(3.12)

with , and . (3.13)

Equation (3.13) has the general solution

. (3.14)

The shear force in the interface is expressed by equation (3.15), slip by equation (3.16).

(3.15)

(3.16)

Equation (3.12) can be solved for special cases. For any number of concentrated loads and for uniform load solutions are shown below. In order to find a consistent formulation for both uni-form load and concentrated loads, concentrated loads are expressed using the Dirac function δas shown in equation (3.18). For example, is a concentrated load Qi at the lon-gitudinal position xi. The differential equation then has the general solution shown in equation (3.19) where i is the number of the respective concentrated load and H(x) is the Heaviside step function defined as shown in equation (3.17).

(Heaviside step function) (3.17)

(3.18)

(3.19)

For uniform load as shown in equation (3.20) the differential equation is simplified to the equa-tion given in equation (3.21) and has the general solution shown in equation (3.22).

x6

6

∂

∂ w x( ) β2

x4

4

∂

∂ w x( )⋅– 1ϕ---

x2

2

∂

∂ q x( ) ω2 q x( )⋅–⎝ ⎠⎜ ⎟⎛ ⎞

=

x2

2

∂

∂ N x( ) β2 N x( )⋅–k eel M x( )⋅ ⋅

ϕ--------------------------------–=

βϕ ω⋅ 2 eel

2 k⋅+ϕ

--------------------------------------= ϕ EaIa EcIc+= ω k 1EaAa------------- 1

EcAc-------------+⎝ ⎠

⎛ ⎞=

N x( ) C1 e βx–⋅ C2 e⋅ βx+k eel e⋅ ⋅ βx–

2 ϕ β⋅ ⋅----------------------------- eβx M x( ) xd⋅ ⋅∫ e β– x M x( ) x e2βx⋅d⋅ ⋅∫–

⎩ ⎭⎨ ⎬⎧ ⎫

+=

t x( )xd

d N x( ) β C– 1e βx– C+ 2eβx

( )⋅=k eel⋅2 ϕ⋅

--------------– e β– x eβxM x( ) xd∫⋅ eβx e β– xM x( ) xd∫⋅+⎩ ⎭⎨ ⎬⎧ ⎫

=

d x( ) t x( )k

--------- βk--- C– 1e βx– C+ 2e

βx( )⋅=

eel2 ϕ⋅-----------– e β– x eβxM x( ) xd∫⋅ eβx e β– xM x( ) xd∫⋅+

⎩ ⎭⎨ ⎬⎧ ⎫

=

q x( ) Qiδ x xi–( )=

H x( ) δ x( ) xd∞–

x∫=

q x( ) Qi∑ i

δ x xi–( )=

w x( )C1e βx– C2eβx+

β4--------------------------------------- C3

x3

6----- C4

x2

2----- C5x C6

H x xi–( )Qiϕ-----⎝ ⎠⎛ ⎞ β ω2

β------–⎝ ⎠

⎛ ⎞ β x xi–( )( )sinh

β4-------------------------------------–

x xi–

β3-------------

x xi–( )3

6β---------------------+ +

⎝ ⎠⎜ ⎟⎛ ⎞ x xi–( )3

3---------------------+

i∑

+ + + + +=

Differential equation for linear elastic composite action

19

(3.20)

(3.21)

(3.22)

If the origin of the x-axis is set to the centre of symmetry of the beam, a symmetrical formulation is found for the first term in equation (3.22) by using C1 = C2 as shown in equation (3.23).

(3.23)

In all cases, the constants C1..6 are determined with the boundary conditions. From each support or free end of a beam located at x = xsi, three boundary conditions are given. For three cases boundary conditions are given in equations (3.24), (3.25) and (3.26).

Fixed end support: , , (3.24)

Cantilever support: , , (3.25)

Free end of a beam: , , (3.26)

From equation (3.23) it can be seen that the deformation of a composite beam with elastic shear connection exposed to uniform load is the sum of a fourth order polynomial and a catenary curve. The fourth order polynomial is well-known for usual beams in bending. The catenary curve is typical for the deformation of a member with no bending stiffness under uniform trans-verse load. If a beam is loaded with concentrated loads (cf. equation (3.18)) the deformation is the sum of a third order polynomial and a catenary curve, whereby the influence of each con-centrated load is shown as a distortion term.For a simply-supported beam, equation (3.12) results in equations (3.27) and (3.28) for uniform load q(x) = q0 = const

(3.27)

(3.28)

and in equations (3.29) and (3.30) for two concentrated loads at x1 = a and x2 = (L-a): q(x) = Q0*(δ(a) + δ(L-a)).

q x( ) q0=

x6

6

∂

∂ w x( ) β2

x4

4

∂

∂ w x( )– ω2

ϕ------– q x( )=

w x( )C1eβx C2e βx–+

b4--------------------------------------- x4 ω2q0

24β2ϕ---------------- x3C3

6------ x2C4

2------ xC5 C6+ + + + +=

w x( )2C1 βx( )cosh

β4--------------------------------- x4 ω2q0

24β2ϕ---------------- x3C3

6------ x2C4

2------ xC5 C6+ + + + +=

x2

2

∂

∂ w xsi( ) 0= w xsi( ) 0= N xsi( ) 0=

xdd w xsi( ) 0= w xsi( ) 0= d xsi( ) 0

xdd N xsi( ) 0=⇒=

x2

2

∂

∂ w xsi( ) 0= N xsi( ) 0= Q xsi( ) Qi xi x= si( )=

N x( )q0 k eel⋅ ⋅

ϕ β4⋅------------------------ βx( ) β x L–( )( )sinh–sinh

βL( )sinh----------------------------------------------------------------- β2x L x–( ) 2–⋅( )

2---------------------------------------------+

⎩ ⎭⎨ ⎬⎧ ⎫

=

t x( )q0 k eel⋅ ⋅

ϕ β3⋅------------------------ h βx( ) β x L–( )( )cosh+cos–

βL( )sinh----------------------------------------------------------------------- β x L

2---–⎝ ⎠

⎛ ⎞⋅+⎩ ⎭⎨ ⎬⎧ ⎫

=

Analytical Models

20

(3.29)

The two last lines containing the Dirac terms in equation (3.30) can be neglected in general, as they amount to zero if . However, they are shown to indicate the discontinuity in shear flow at the locations where the beam is subjected to concentrated loads.

(3.30)

3.4 Differential equation for nonlinear composite action

The differential equation for elastic composite action shown in section 3.3 on page 18 is limited to composite beams in which all components exhibit linear elastic behaviour. If at least one component exhibits nonlinear behaviour, the differential equation becomes much more compli-cated. In the following a nonlinear approach for the differential equation is considered. Howev-er, in general the differential equation for nonlinear composite action can only be solved by means of numerical methods.

3.4.1 Material properties

All materials and the shear interface must exhibit a behaviour which is described by a continu-ous mathematical function. Below a few examples are given. • Elastic material

(3.31)

• Steel, bilinear formulation (see Figure 3.2, left)

(3.32)

N x( )Q0keel

ϕ β3⋅----------------- H a x–( ) β x a– L–( )( ) β x a– L+( )( )cosh–cosh

2 h βL( )sin⋅--------------------------------------------------------------------------------------------------- β x a–( )+⎝ ⎠⎛ ⎞

H L a– x–( ) β x a 2L–+( )( ) β x a+( )( )cosh–cosh2 h βL( )sin⋅

--------------------------------------------------------------------------------------------- β x a L–+( )+⎝ ⎠⎛ ⎞

β x a L–+( )( ) h β x a 2L–+( )( ) β x a–( )( ) β x a– L–( )( )cosh–cosh+cos–cosh2 h βL( )sin⋅

------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- β L x–( )+⎝ ⎠⎛ ⎞+

⋅+

⋅⋅=

x a L a–;( )≠

t x( )Q0keel

ϕ β2⋅----------------- H a x–( ) h β x a– L–( )( ) h β x a– L+( )( )sin–sin

2 h βL( )sin⋅-------------------------------------------------------------------------------------------------- 1+⎝ ⎠⎛ ⎞

H L a– x–( ) h β x a 2L–+( )( ) h β x a+( )( )sin–sin2 h βL( )sin⋅

-------------------------------------------------------------------------------------------- 1+⎝ ⎠⎛ ⎞

h β x a L–+( )( ) h β x a 2L–+( )( ) h β x a–( )( ) h β x a– L–( )( )sin–sin+sin–sin2 h βL( )sin⋅

---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1–⎝ ⎠⎛ ⎞

δ– x a–( ) β x a– L–( )( ) β x a– L+( )( )cosh–cosh2 h βL( )sin⋅

--------------------------------------------------------------------------------------------------- β x a–( )+⎝ ⎠⎛ ⎞⎝ ⎠⎛ ⎞

δ– x L a+–( ) β x a 2L–+( )( ) β x a+( )( )cosh–cosh2 h βL( )sin⋅

--------------------------------------------------------------------------------------------- β x a L–+( )+⎝ ⎠⎛ ⎞⎝ ⎠⎛ ⎞⋅

⋅

+

⋅+

⋅⋅=

σ x z, )( E ε x z,( )⋅=

σ x z, )( E ε x z,( )⋅

Et E–( ) H ε x z,( )fyE----–⎝ ⎠

⎛ ⎞ ε x z,( )fyE----–⎝ ⎠

⎛ ⎞ H ε– x z,( )fyE----–⎝ ⎠

⎛ ⎞ ε x z,( )fyE----+⎝ ⎠

⎛ ⎞⋅+⋅⋅+

=

Differential equation for nonlinear composite action

21

• Steel, hyperbolic formulation (see Figure 3.2, right)

(3.33)

• Concrete (see Figure 3.3, left)

(3.34)

• Shear interface (see Figure 3.3, right)

(3.35)

Figure 3.2 Material law formulations for steel: bilinear (left, cf. equation (3.32)) and hyper-bolic (right, cf. equation (3.33))

Figure 3.3 Bilinear formulation for concrete material law (left, cf. equation (3.34)) Exponential load-slip law for shear interface (right, cf. equation (3.35))

σ x z, )(fy E Et–( ) ε x z,( )⋅( )tanh⋅ Et+ ε x z,( )⋅

a1 ε x z,( )⋅( )ea2 ε x z,( )2⋅( )

cosh

--------------------------------------------------------------------------------------------------=

σ x z, )( H ε x z,( ) εcu+( ) HfctE------ ε– x z,( )⎝ ⎠⎛ ⎞Eε x z,( ) H ε x z,( )–

fcE----–⎝ ⎠

⎛ ⎞ ε x z,( )fcE----+⎝ ⎠

⎛ ⎞ Et E–( )+=

P x )( PuC1d x( )( )tanhC2d x( )( )cosh

----------------------------------- Pue

C1d x( )e–

C– 1d x( )

eC1d x( )

e+C– 1d x( )

------------------------------------------- 2

eC2d x( )

eC– 2d x( )

–-------------------------------------------⋅

⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

= =

Strain [-]

Stre

ss [

N/m

m²]

fy

-fy

-εy

1

E 1Et

εy

-400-300-200-100

0100200300400

-0.4 -0.2 0 0.2 0.4Strain [-]

Stre

ss [N

/mm

²]

V(u)

fy*tanh(E*eps)

fy*tanh((E-Et)*eps)+Et*eps

fy/cosh(a1*eps)^exp(a2*eps²)

Strain [-]

Stre

ss [

N/m

m²]

fy

fct

εct

1

E 1Et

fc

εcy εcu

-150

-100

-50

0

50

100

150

-30 -20 -10 0 10 20 30Slip u(x) [mm]

Shea

r For

ce P

(u) [

kN]

Analytical Models

22

3.4.2 Section geometry

The geometry of the section is symmetrical in the vertical direction (with respect to the z-axis). Torsional effects and questions of stability are not considered. The section is defined by a con-tinuous mathematical function. Equation (3.36) is an example for a typical I-shaped steel sec-tion.

with (3.36)

3.5 Well known design models

3.5.1 Rigid-ideal plastic design model in Eurocode 4

Eurocode 4 [26] gives a simple design method for composite beams with headed studs both for ultimate load capacity and deformations. For composite beams with ductile shear connectors (shear connectors exhibiting a characteristic deformation capacity of at least δuk = 6 mm) the rigid-ideal plastic design method can be used if the degree of partial shear connection η is great-er than the minimum degree of partial shear connection ηmin. The minimum degree of partial shear connection ηmin is determined e.g. for steel sections with equal flanges in beams with Le(cf. [26]) less than or equal to 20 m. For unequal flanges a similar formula applies.

(3.37)

Under the conditions given in [26] (section 7.3.1, (4)) the deformations of a composite beam may be determined assuming rigid shear connection. In earlier versions of Eurocode 4 a simple formula (cf. equation (3.38)) was given to calculate the deflections of composite beams with low degrees of partial shear connection, whereas a0 is 0.5 for propped and 0.3 for unpropped construction.

(3.38)

The deflection obtained with the stiffness of the composite section assuming rigid shear connec-tion is called wrigid, while wa stands for the deflection of the composite beam considering the stiffness of the steel beam only.

b x z,( ) H z( )H ha x( ) z–( ) b x( ) Web– LowerFillet UpperFillet+ +{ }=

Web H z tf x( )–[ ] H ha x( ) tf x( )– z–[ ] b x( ) tw x( )–[ ]⋅ ⋅=

LowerFillet 2 H z tf x( )–[ ] H tf x( ) r x( ) z–+[ ] r x( ) r x( )2 r x( ) z tf x( )–{ }–( )2––[ ]⋅ ⋅ ⋅=

UpperFillet 2 H z ha x( ) tf x( )– r x( )–{ }–[ ] H ha x( ) tf x( )–{ } z–[ ]

r x( ) r x( )2 z ha x( ) tf x( )– r x( )–{ }–( )2––[ ]

⋅ ⋅

⋅

=

ηminnmin

nf----------- 1=

fy355---------⎝ ⎠⎛ ⎞ 0.75 0.03Le–( ) 0.4≥–=

weff wrigid 1 a0 1 η–( )wa

wrigid--------------- 1–⎝ ⎠⎛ ⎞⋅ ⋅+

⎩ ⎭⎨ ⎬⎧ ⎫

⋅=

Well known design models

23

3.5.2 γ-method

The γ-method developed by Möhler [51] is a simplified design procedure which is frequently used in timber construction. For linear material behaviour of all components of the steel beam the stiffness of a composite member is determined as shown in equation (3.39). In addition, the elastic bending resistance of the composite beam may be determined by means of this method.

(effective bending stiffness from γ-method) (3.39)

(3.40)

The variables ϕ and ω are defined in equation (3.13). The value γ may be regarded as the degree of partial interaction (cf. section 2.1.2.14 on page 9). γ = 0 describes a pair of beams with no interaction, while γ = 1 stands for a composite beam with rigid shear connection and thus full interaction.

with (3.41)

By means of the γ-method also the shear flow (the transferred longitudinal shear force per length) and the slip in the shear interface may be determined as shown in equations (3.42) and (3.43).

(shear flow from γ-method) (3.42)

(slip from γ-method) (3.43)

The γ-method gives exact results only for sinusoidal loading of the beam. For uniform or con-centrated loads the γ-method represents an approximation. However, the accuracy of the γ-method may be refined by defining the loads as Fourier series. With Fourier series of infinite length the exact result of the differential equation for linear elastic composite action (cf. section 3.3 on page 18) is achieved. However, in many cases the γ-method provides a sufficient approx-imation even if the load is defined without Fourier series. The solutions obtained with the γ-method are located in between the solutions of the differential equation for a uniform load and two concentrated loads at L/3 and 2 L/3 as may be seen from Figure 3.4.

EIeff ϕ γ S⋅+=

S k

ω2------ eel

2⋅=

γ 1

1 π2 S⋅

k L⋅ 2 eel⋅2

---------------------------+

-------------------------------------= kSi∆c------=

t x( ) Q x( ) γ S⋅eel EIeff⋅-----------------------⋅=

d x( ) Q x( ) γ S⋅eel EIeff⋅----------------------- 1

k---⋅ ⋅ t x( )

k---------= =

Analytical Models

24

3.5.3 Design model for linear elastic chords and rigid-plastic connectors

Aribert [6] proposes a simple design model for composite beams with low degrees of shear con-nection, assuming linear elastic behaviour of both the steel beam and the concrete slab, while the shear connection shows rigid-ideal plastic load-slip behaviour with limited deformation ca-pacity δu. The interaction between slip and shear load in the shear interface which is accounted for in the differential equation (cf. section 3.3 on page 18) is avoided by assuming a rigid-plastic load-slip behaviour for the shear interface. This also allows one to assume that all shear connec-tors are at the ultimate load irrespective of the actual slip. The bending resistance of a composite beam is determined using equation (3.44).

with (3.44)

For a uniform load q(x) = q0 and symmetrical conditions, slip in the shear interface is deter-mined as given in equation (3.45).

with (3.45)

, and

Maximum slip may occur in two places, either at the support or some distance away from the support in the field (cf. equations (3.46)).

or whereas (3.46)

This design model developed by Aribert in [6] permits the easy design of composite beams with very low degrees of partial shear connection where shear connectors are in the plastic state while the chords are still in the elastic state.

Figure 3.4 Left: Comparison between end slip from γ-method and differential equation. Slip determined using the γ-method is in between the results from the differential equa-tion for linear composite action for uniform load and for concentrated loads at one and two thirds of the span. Right: γ as function of k for the composite beam referred to in Figure 2.2.

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1E-061E-03

1E+001E+03

1E+06

k [N/mm²]

d end

(k) [

mm

]

gamma-methoddifferential equation, concentrated loadsdifferential equation, uniform load 1E-09

1E-06

0.001

1

1.E-061.E-03

1.E+001.E+03

1.E+06

k [N/mm²]

γ [-]

M x( )x2

2

∂

∂ w x( ) ϕ N x( ) eel⋅+⋅= N x( ) n PRk2 x⋅

L----------⋅ ⋅=

d x( ) A1x3

3----- A1 L A2–⋅( ) x2

2----- A2

L2

8------ A1

L2

12------⋅–⋅+⋅+⋅–=

A1 q0eel

2 ϕ⋅-----------⋅= A2

2 1 A3+( ) eel2 n PRk⋅ ⋅ ⋅ ⋅

ϕ L⋅----------------------------------------------------------------= A3

ω2

k------ ϕ

eel2

---------⋅=

x1 0= x2 LA2A1------–= 0 x2

L2---≤ ≤

Overview

25

4 Experimental Investigations

4.1 Overview

A large number of shear tests and push tests as well as three beam tests on composite beams in partial shear connection with novel shear connectors were conducted at the Institute of Struc-tural Engineering (IBK). These tests are reported in [31] and summarised in section 4.2 on page 25 below. However, none of the push tests reported in [31] was performed with a steel beam similar to the one used in the beam tests [31]. Therefore, push tests series S3 (reported in section 4.3 on page 27) was performed on specimens of the same geometry and material properties as found in the beam tests A-01 and A-02 [31].

4.2 Experimental studies reported in [31]

Experimental investigations were performed at ETH to develop new types of shear connectors fixed by means of powder-actuated fasteners. The shear connectors to be developed had to meet the following requirements:• The shear capacity per fastener has to exceed the shear resistance per fastener of the Hilti X-

HVB shear connector• The deformation capacity must meet the Eurocode 4-criteria (cf. [26])• The connector is made by cold forming out of a thin steel sheet• The geometry of the connector must allow the use of automatic fastening toolsThe connectors were developed to be used for composite slabs with re-entrant trough (Holorib HR 51) and open trough (Vikam TR 60/235) profiled steel sheeting and for solid concrete slabs.

45 basic steel-to-steel shear tests were performed at ETH. The aim of these tests was to study the behaviour of the connection between a thin steel sheet and a thick baseplate by means of powder-actuated fasteners. The effect of group fixings in a row (in comparison with a single fas-tener), the effect of sheet thickness, the effect of the double washer and of nailhead standoff and the effect of an intermediate sheet (simulating a profiled sheet) were investigated.Additionally, 21 push tests according to Eurocode 4 [26] were performed (9 push tests with Rib-Con connectors (series R), 12 push tests with StripCon connectors (series S)). The parameters investigated were:• steel sheet thickness of the connector• steel strength of the connector• geometry of the connector• type of the profiled steel sheeting• concrete strengthFigure 4.1 contains an extract of the results of these push tests.

Experimental Investigations

26

Finally, three simply-supported composite beams with a span of 7.2 m were tested. Two beam tests were conducted on composite beams with StripCon connectors in different degrees of par-tial shear connection, and one beam test was performed with RibCon connectors. The results of these beam tests are summarised in Figure 4.2. In Figure 4.3 composite beams A-01 and B-01 are shown after test.

Figure 4.1 Load-slip diagrams of some push tests with RibCon shear connectors (left) and StripCon shear connectors (right)

Figure 4.2 Results of beam tests A-01, A-02 and B-01: load-deflection diagram (left) and load-slip diagram (right)

Figure 4.3 Beam tests A-01 (left) and B-01 (right) after failure. In beam tests A-01 and A-02 the loads were introduced from above to the upperside of the concrete slab. In beam test B-01 the loads were introduced from underneath to the lower flange of the steel beam.

0100200300400500600700

0 5 10 15 20 25Slip [mm]

Load

[kN

]R2.1 uMeanR2.2 uMeanR2.3 uMean

0100200300400500600700

0 5 10 15 20 25Slip [mm]

Load

[kN

]

S2.3 uMeanS2.4 uMeanS2.5 uMeanS2.6 uMean

0

20

40

60

80

100

120

0 50 100 150 200 250Deflection [mm]

Load

[kN

]

A-01

A-02

B-01

0

20

40

60

80

100

120

0 10 20 30 40Slip [mm]

Load

[kN

]

A-01 u1

A-02 u7

B-01 u7

Push tests with StripCon on IPE 270 steel profile

27

4.3 Push tests with StripCon on IPE 270 steel profile

A series of 3 push tests was performed to determine the load-slip behaviour of the StripCon shear connector under the conditions of beam tests A-01 and A-02. Therefore the push tests were conducted with a steel section IPE 270 taken from the undamaged part of specimen A-02.The whole test setup (including preparation of tests, naming and allocation of measurements and test procedure) followed the procedure shown in [31]. The fasteners are numbered as shown in Figure 4.4 (left).

One test (S3.1) was performed with identical shear connectors as in the beam tests (from the same production lot as used in push test S2.6 and beam tests A-01 and A-02). Two more tests (S3.2 and S3.3) were performed with strip connectors made of steel S280GD instead of DX51D (from the same production lot as used in push test S2.5). Also, the fasteners and the profiled steel sheeting were taken from the same lot as in beam tests A-01 and A-02.Specimen S3.1 and S3.2 were pre-loaded to 130 kN (40 % of the predicted ultimate load) in 25 load cycles. Specimen S3.3 was pre-loaded to 261 kN (80 % of the predicted ultimate load) in 18 load cycles. After 18 load cycles it was loaded to failure under displacement control. How-ever, the maximum load reached after the load cycles amounted to 239 kN and was thus lower than the upper boundary load during the load cycles.In specimen S3.2 and 3.3 the profiled steel sheeting was placed only on the inner half of the slab‘s width due to a lack of material. In the outer parts of the ribs in the concrete slabs, poly-styrene blocks were placed in the formwork before casting to ensure the concrete slabs to exhibit the same geometry. However, the influence of this procedure is judged to be negligible. In the middle of the slab where the powder-actuated fasteners are placed the profiled steel sheet was present as usual.

Test Test date StripCon

materialProfiled

steel sheetingCylinder com-

pressive strengthUltimateload Pu

Slip at ulti-mate load

Slip at 90 % Pu

S3.1 16.04.2004 DX51D VIKAM TR60/235 Slab 1: 29.4 N/mm²

Slab 2: 41.3 N/mm²

334.4 kN 3.9 mm 4.6 mm

S3.2 15.04.2004S280GD VIKAM TR60/235

over 1/2 width of slab293.4 kN 9.1 mm 11.9 mm

S3.3 14.04.2004 261.0 kN (239.0) 1.2 mm 7.4 mm

Table 4.1 Overview on specimens tested

Experimental Investigations

28

Figure 4.4 (right) gives an overview on the mean load-slip curves of all three tests. Tests S3.1 and S3.2 show a similar deformation capacity, but an exact evaluation of the „ductility“ as the slip at a residual load of 90 % Pu gives a value below 6 mm for push test S3.1, while for push test S3.2 a value greater than 6 mm results, because S3.1 reached a higher ultimate load than S3.2. The load-slip curves in Figure 4.4 show that failure of a shear connector takes place stepwise, whereby each step is identified as failure of one or several single nails and occurs at a charac-teristical amount slip. In S3.1 no fasteners have failed until ultimate load is reached at a slip of d = 3.9 mm. Then three fasteners on side 1 (#1, #2, #5) were sheared off at the same time, while fastener #6 was starting to be pulled off. Shortly after that (with d = 4.6 mm) on side 2 fasteners #1 and #2 were sheared off while fasteners #5 and #6 were starting to be pulled off. The slow rising of the load-slip curve shows the stretching of the ridge between the two fastener rows of each rib. At a slip of d = 11.4 mm (side 1) and d = 12.4 mm (side 2) the fasteners on the compression side of the rib failed as well.In S3.2 the fasteners on side 1 near to the tension legs of each rib were not sheared off but failed continuously due to washer pullover. However, the fasteners near to the tension leg on side 2 were sheared off, which took place at the same slip (d = 4.6 mm) and at the same load as in test S3.1. The difference in behaviour between the two sides is explained by the difference in the concrete strength between the two sides. Push test S3.2 reached ultimate load with several fas-teners already failed. With d = 9.8 mm the load-slip curve from test S3.2 shows another step. At a slip of d = 11.4 mm both tests showed the same stepwise load decrease again. Shortly after that, S3.1 failed completely, while in S3.2 failure occurred stepwise at further deformation.

Figure 4.4 Left: Geometry of specimens S3.1, S3.2 and S3.3 Right: Overview: mean slip in all three tests

050

100150200250300350

0 5 10 15 20 25Slip [mm]

Load

[kN

]

S3.1 Slip meanS3.2 Slip meanS3.3 Slip mean

Push tests with StripCon on IPE 270 steel profile

29

A comparison between push tests S3.2 and 3.3 illustrates the effect of pre-damage due to cyclic loading. As seen in numerical simulation of the beam tests A-01, A-02 and B-01 (cf. section 5.3.2 on page 48 and [31]) the shear forces and the local stresses in a shear connector can be relatively high even at service load level. This shows that for shear connectors with large, con-centrated force transmission the effects of repeated loading e.g. due to changes in live load or seismic actions may have a considerable influence on the residual connector resistance. In Fig-ure 4.7 (right) the typical liftoff-slip behaviour found in the shear interface is shown. This be-haviour is explained by the swaying movement of the fasteners coming progressively loose from the steel flange.

Figure 4.5 Test S3.1

Figure 4.6 Test S3.2

Figure 4.7 Test S3.3

050

100150200250300350

0 5 10 15 20 25Slip [mm]

Load

[kN

]uMean

aoMean

auMean

0

2

4

6

8

10

0 5 10 15 20 25Slip [mm]

Lifto

ff [m

m]

1vo2vo1ho2ho

050

100150200250300350

0 5 10 15 20 25Slip [mm]

Load

[kN

]

uMean

aoMean

auMean

0

2

4

6

8

10

0 5 10 15 20 25Slip [mm]

Lifto

ff [m

m]

1vo2vo1ho2ho

050

100150200250300350

0 5 10 15 20 25Slip [mm]

Load

[kN

]

uMean

aoMean

auMean

0

2

4

6

8

10

0 5 10 15 20 25Slip [mm]

Lifto

ff [m

m]

1vo2vo1ho2ho

Experimental Investigations

30

Figure 4.8 Shear interface of specimen S3.1 after test (side 1 left, side 2 right)

Figure 4.9 Shear interface of specimen S3.2 after test (side 1 left, side 2 right)

Two-dimensional beam models

31

5 Numerical Investigations

Several numerical models were developed in order to describe the load-bearing behaviour of composite beams. As the effects of nonlinear connector and material behaviour and the behav-iour of the beam after failure of some connectors were found to be of crucial importance, the models were required to consider these effects. Further, the models had to be able to process the behaviour of a composite beam after ultimate limit state not only of some connectors, but of the whole composite beam. By means of this information the „ductility“ of a composite beam should to be assessed. The groups of numerical models developed are listed below:• Two-dimensional beam models were created and verified against data from beam tests A-01

and A-02 (described in [31]) as well as against analytical models for beams with full, rigid shear connection and without shear connection (cf. section 5.1 on page 31).

• Two-dimensional plane models were considered as a good means of overcoming some of the problems encountered with the beam models. However, due to practical restrictions, no de-tailed analysis was performed on two-dimensional plane models (cf. section 5.2 on page 36).

• Therefore, a three-dimensional volume model based on the ANSYS finite element pro-gramme [2] was established for beams with profiled steel sheeting perpendicular to the beam and verified against the data from beam tests A-01 and A-02 (cf. [31]) as well as from the beam tests described in [39]. Another three-dimensional volume model was established for beams with profiled steel sheeting parallel to the beam and verified against the data from beam test B-01 (cf. [31]) (cf. section 5.3 on page 37).

• Finally, a comprehensive parametric numerical 3D-volume model based on the ANSYS fi-nite element program [2] was developed and used for the simulation of a large number of composite beams in order to investigate the effect of the shear interface behaviour on the load-bearing behaviour and further parameters of simply-supported composite beams.

5.1 Two-dimensional beam models

As a first step in the numerical simulation, two-dimensional beam models were developed using the software programme STATIK-N [1]. The beam models were only developed for beam tests A-01 and A-02 performed at ETH [31].

5.1.1 Model geometry

For efficiency reasons the simulations were performed with half-span models. Figure 5.1 shows the four truss and two frame models tested for the simulation. The concrete slab was considered as the upper chord of the trusses and frames, while the steel beam is modelled by means of the lower chords. Additionally, calculations on a simple beam model were performed as a reference.

Numerical Investigations

32

5.1.2 Section geometries

For all simulations the cross-sections of the concrete slab, the steel beam and the verticals were kept constant over the length of the beam. The truss members simulating the shear connection between steel beam and concrete slab were varied between rigid and quasi-non-existent.• The section of the steel beam (lower chord) was assumed to have the theoretical dimensions

of an IPE 270 profile.• The section of the concrete slab (upper chord) was chosen as 1200 mm in width and 130 mm

in depth (which equals to the mean depth of the concrete slab when considering the profiled steel sheeting) with a gap of 20 mm in height between the concrete slab and the steel beam.

• The rigid truss and frame members were considered with an axial stiffness of 210*109 N*mm²/mm² and a flexural stiffness of 17.5*1015 N*mm² (steel section of 1 m²). Axial stiffness was reduced to 52.5*109 N*mm²/mm² and flexural stiffness to 1.1*1015 N*mm² (steel section of 0.25 m²) in the nonlinear analysis of the I-shaped frame (all cases) and for the K-shaped truss (case with full, rigid connection only) to avoid numer-ical instabilities caused by a badly conditioned stiffness matrix. However, the elastic results of the system with reduced stiffness did not differ from the results obtained for the system with original stiffness in the first three digits.

• The truss members simulating the shear connection between concrete slab and steel beam were assumed to have the same section and stiffness as the rigid truss and frame members for the simulation of a rigid shear connection. For the linear analysis of non-rigid, elastic cas-es the sections (and thereby the stiffness) of the truss members simulating the shear connec-tion were gradually adapted. In places where the shear connection was omitted the shear connection truss members were inactivated. For the nonlinear analysis of the systems with partial shear connection (according to beam tests A-01 and A-02 [31]) the non-existent shear connection bars were deleted.

Figure 5.1 Geometry of two-dimensional truss and frame models investigated with STATIK-N

Two-dimensional beam models

33

5.1.3 Materials

The materials properties were chosen according to the materials used in the beam tests.

5.1.4 Nonlinear component behaviour

Nonlinear behaviour was assumed for the steel beam, for the concrete slab and for the shear con-nection truss members. For the upper and the lower chords an M-N interaction diagram was es-tablished using the Software FAGUS-3 (cf. Figure 5.2). For the shear connection truss members a nonlinear load-strain relation to fit the results from the push tests performed at CTU, Prague (cf. [31]) was used (cf. Figure 5.3).

MemberModulus of elasticity E

Shear mod-ulus G

Yield strength fy

Ultimate strength fu

Cylinder compressive strength fc

Tensile strength fc

[N/mm²] [N/mm²] [N/mm²] [N/mm²] [N/mm²] [N/mm²]

Steel section 210000 81000 317 436 -

Concrete slab 28705 9000 - 34.35 3.5

Verticals210000 81000

-

Shear connection derived from push tests (cf. Figure 5.3)

Table 5.1 Material properties for STATIK-N simulations

Figure 5.2 M-N-Interaction diagrams for steel section (left) and concrete slab (right)

Figure 5.3 Nonlinear load-strain relation for shear connectors in STATIK-N Models

136; 290153.6; 0

136; -290-136; -290-153.6; 0

-136; 290

0; 1455

0; -1455-1500

-1000

-500

0

500

1000

1500

-200 -100 0 100 200M [kNm]

N [k

N]

-10; 250

-75; -3500

-87; -2800

-90; -2000

-80; -1300

-10; 250

7; -5360

86; -3600

94; -290090; -2400

80; -1800

-5500

-4500

-3500

-2500

-1500

-500

500-100 -50 0 50 100

M [kNm]

N [k

N]

0

50

100

150

200

250

0.0 0.1 0.2 0.3STATIK-N Strain [-]

Load

[kN

]

SLASH-FW (Base = 322 mm)X_FW (Base = 322 mm)K-FW (Lower diags, Base = 271 mm)Z-FW (Base = 235 mm)H-FRAME (Base = 235 mm)I-FRAME (Base = 100 mm)

0.0; 8.40.0; 16.9

0.1; 33.9

0.1; 50.8

0.1; 67.7

0.2; 84.7

0.4; 101.6

0.6; 118.5

1.3; 135.5

2.6; 152.4

6.4; 169.3

10.6; 152.4

0255075

100125150175

0 5 10 15Displacement [mm]

Load

[kN

]

Mean curve of Series T1(DX51, 25.7MPa)

Numerical Investigations

34

5.1.5 Simulation arrangements

In a first stage, the specific deformation and the specific internal forces and moments caused by a load of 1 kN applied to the system with rigid shear connection 2.4 m from the support were calculated. The results from these calculations were used to decide whether a specific model was well-suited to reflect the actual physical processes of the composite beam.In a second stage, the load-deformation behaviour of the system with consideration of the non-linear behaviour of the chords and the shear connection was determined. The results of these simulations were compared to the results of the composite beam tests.

5.1.6 Results

The results of the two-dimensional numerical simulations shown in this section are helpful to discuss the properties of the different truss models. Simulations have been performed and results are shown for composite beams with no shear con-nection (cf. section 5.1.6.1 on page 34), rigid shear connection (cf. section 5.1.6.2 on page 34) and for the two beam tests A-01 and A-02 (cf. section 5.1.6.3 on page 35). Further, some more detailed results for rigid shear connection are shown in the appendix.

5.1.6.1 No shear connection

For composite beams without shear connection all truss models deliver results which agree well with analytical considerations. The I-frame model is a little stiffer than the other models and thus a little closer to the analytical model. All other truss systems show the same stiffness and ultimate load. This is due to the fact that with the I-frame model the inclinations of the upper and the lower chord are forced to be the same at each pair of nodes at the top and at the bottom of a vertical.

5.1.6.2 Full, rigid shear connection

For composite beams with rigid shear connection the cross-shaped truss model is too stiff and too strong. All other models agree rather well with the analytical model. The additional strength

Model (all with consideration of shear defor-mation and reinforcement) Units

Beam Truss shape Frame shape

\ X K Z H I

Project Directory Names for linear and nonlinear analysis (*.$S3)

Man

ual

Cal

cula

tion

SLA

SH_F

0 SL

ASH

_N0

X-F

W0

K-F

W0

Z-FW

0

H-F

RA

ME0

I-FR

AM

E0

I-FR

A0N

L

Deflection per kN at no shear connection [mm/kN] 0.717 0.720 0.718

First crackLoad [kN]

-1.2 1.2

Deflection [mm] 0.9 0.9

YieldLoad [kN] 56.7

-Deflection [mm] 40.6

Ultimate limit stateLoad [kN] 64.6 64.3 64.3

Deflection [mm] - 69.9 63.1

Table 5.2 Results of 2D beam models for composite beam without shear connection

Two-dimensional beam models

35

and stiffness of the cross-shaped truss is because the rigid diagonals act not only as a shear con-nection between the chords, but also as bending stiffeners. However, this effect is only of sub-stantial influence if the shear connection is very stiff. For a typical stiffness of the shear connec-tion (cf. tests A-01 and A-02, see below) this effect is of minor importance.The ultimate loads predicted by the numerical models are too low compared to the results from analytical models (except for the cross-shaped model which gives far too large results). The nu-merical models show only a limited ability for plastic behaviour of a system. Therefore, the full plastic resistance taking into account all plastic redistribution is not reached.The I-frame model shows the best agreement with the analytical model. Apart from the resist-ance values also the bending moments and axial forces in the chords are as predicted by the an-alytical model (cf. appendix). In all other truss models the axial forces in the chords are trans-mitted longitudinally through the diagonals, resulting in incorrect axial forces in the chords. In the H-frame model the axial forces in the chords are transmitted in the longitudinal direction and the verticals introduce a moment due to eccentricity into the chords at each connector.

5.1.6.3 Beam tests A-01 and A-02

All investigated truss and frame models are able to predict the ultimate load of beam test A-01 well. For beam test A-02 the numerical models show higher ultimate loads than observed in test. In the test, beam A-02 showed premature failure. The numerical model is not able to simulate the descending branch of the shear connector‘s load-slip behaviour in an appropriate way. All models show a similar stiffness since the stiffness of the shear connectors is lower than with the rigid shear connection. The stiffness of the models is similar to the initial stiffness of the tested beams.

Model

Uni

ts

Beam Truss shape Frame shape

rigid shear connection \ X K Z H I

Shear deformat. considered no yes yes yes

Reinforcement considered no yes yes yes

Project Directory Names for linear and nonlinear analysis (*.$S3)

Man

ual

calc

ulat

ion

BEA

M_M

BEA

M_M

_V

SLA

SH_F

W

SLA

SH_N

F

X-F

W

X-F

WFN

L

K-F

W

K-F

WFN

L

Z-FW

Z-

FWFN

L

H-F

RA

ME

H-F

RA

FNL

I-FR

AM

E I-

FRA

FNL

Deflection per kN at full, rigid shear connection

[mm

/kN

]

0.233 0.233 0.231 0.232 0.225 0.160 0.219 0.232 0.232 0.235

First crackLoad [kN] 15.2

-

130.6 67.2 37.0 17.1 16.9 13.4

Deflection [mm] 3.5 31.6 10.8 8.1 4.0 3.9 3.1

YieldLoad [kN] 113.2 125.3 67.2 99.5 123.0 123.4 35.6

Deflection [mm] 26.4 28.4 10.8 21.8 28.6 28.6 8.4

Ultimate limit state

Load [kN] 160.2 - 162.0 162.0 133.5 287.4 132.6 123.5 123.9 130.9

Deflection [mm] - - 37.4 37.5 33.7 49.0 32.0 28.7 28.7 34.2

Table 5.3 Results of 2D beam models for full, rigid shear connection

Numerical Investigations

36

The displacements at ultimate load obtained from the numerical models are much smaller than the displacements observed in the test, because the numerical models assume linear elastic be-haviour until the resistance of a member in bending and axial force is reached.

5.2 Two-dimensional plane models

Composite beams can be easily simulated as a set of at least three two-dimensional planes (low-er member, shear connector layer and upper member). This is considered to be an economic way to achieve good results. However, after preliminary investigations this type of analysis was abandoned because the concrete-specific material properties (brittle cracking in tension and ductile behaviour in compression) were not possible for a two-dimensional model with the available version of the software programme used [2], but only for three-dimensional solid models.

Model (all with consideration of shear deforma-tion and reinforcement) Units Beam

TestTruss shape Frame shape

\ X K Z H I

Project Directory Names for both linear and nonlinear analysis (*.$S3)

SLA

SH_N

L

X-F

W_N

L

K-F

W_N

L

Z-FW

_NL

H-F

RA

_NL

I-FR

A_N

L

Deflection per kN at partial shear connection [mm/kN] ~0.23 0.230 0.219 0.225 0.244 0.235 0.236

First crackLoad [kN] - 5.7 7.0 5.4 5.6 5.6 6.7

Deflection [mm] - 1.5 1.9 1.4 1.5 1.4 1.7

YieldLoad [kN] - 105.0 105.3 106.0 106.6 106.8 106.2

Deflection [mm] - 43.2 43.7 42.7 42.8 43.0 41.4

Ultimate limit stateLoad [kN] 106.9 105.5 105.4 106.4 106.8 107.1 106.4

Deflection [mm] 148.9 43.5 43.8 42.9 42.9 43.1 41.5

Table 5.4 Results of 2D beam models for beam test A-01

Model (all with consideration of shear deforma-tion and reinforcement) Units Beam

TestTruss shape Frame shape

\ X K Z H I

Project Directory Names for both linear and nonlinear analysis (*.$S3)

SLA

SH2N

L

X-F

W2N

L

K-F

W2N

L

Z-FW

2NL

H-F

RA

2NL

I-FR

A2N

L

Deflection per kN at partial shear connection [mm/kN] ~0.25 0.232 0.222 0.227 0.247 0.236 0.237

First crackLoad [kN] - 5.2 4.6 4.9 5.2 5.1 6.2

Deflection [mm] - 1.4 1.3 1.3 1.4 1.4 1.6

YieldLoad [kN] - 98.1 98.4 92.4 98.3 88.6 98.2

Deflection [mm] - 50.6 51.3 45.0 50.0 41.7 47.4

Ultimate limit stateLoad [kN] 89.4 98.5 98.4 92.4 98.3 88.6 98.5

Deflection [mm] 115.5 50.9 51.3 45.0 50.0 41.7 47.6

Table 5.5 Results of 2D beam models for beam test A-02

Three-dimensional volume models

37

5.3 Three-dimensional volume models

A comprehensive parametric numerical 3D-volume model software applet for the ANSYS APDL [2] API platform was developed to account for a wide variety of geometry, materials and loading conditions (see also [11]). It is able to process composite beams without profiled steel sheeting and with profiled steel sheeting spanning perpendicular or parallel to the beam.The model was verified against data from the beam tests A-01, A-02 and B-01 (cf. [31]) as well as from the beam tests described in [39]. In order to create a reliable numerical model which accurately describes all physical nonlinear-ities in composite beams, extensive investigations were performed on the various components of the model, such as the concrete elements and material options, the formulation and the non-linear behaviour of the shear interface and the nonlinear behaviour of the steel material options.The parametericised model was used for the simulation of a large number of composite beams in order to investigate the effect of the shear interface behaviour and further parameters on the load-bearing behaviour of composite beams.

5.3.1 Description of the model

5.3.1.1 General information

In order to save computation time, symmetry conditions are used and only half of the span and half of the width of the composite beam is modelled.All simulations are performed as a static analysis. Transient effects are not considered. For all simulations geometric nonlinearities are excluded. This is due to the following facts:a) The SOLID 65 element is used for the concrete section. This element does not support geo-

metric nonlinearities and would produce a major error if used in a simulation with geometric nonlinearities included.

b) The block model used to simulate the shear connection would establish tension in the vertical (Y) direction with geometric nonlinearities included (cf. Figure 5.8, right) and thus wrongly create additional stiffness and bearing capacity.

The finite element models consist of 8-noded volume elements and one 1-noded mass element at the point where the load is applied to the system (cf. section 5.3.1.7 on page 46).At the support local buckling due to transverse compression of the steel section is avoided by coupling all nodes of the steel section in the vertical direction.For the concrete slab the element SOLID 65 is applied while for all other volumes the SOLID 185 element is used. The SOLID 185 element exhibits an excellent capacity for large strain, as occurs mainly in the shear connector elements. The element options used are shown in Table 5.6.

Elem. type Used in K1 K2 K5 K6 K7

SOLID 65 Concreteslab

1 (suppress extra displace-ment shapes)

-0 (concrete linearsolution output

at centroid only)

0 (concrete nonlinear solution output

at centroid only)

1 (include stressrelaxation after

cracking)

SOLID 185 Shear interface - 0 (full integra-

tion)

- 1 (mixed U/P) -

SOLID 185 Other model parts - - 0 (pure displacement) -

Table 5.6 Element types, key options and formulations used in the 3D-finite element model

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38

5.3.1.2 Geometry

The geometry of the finite element model is parameterised and thus depends on the actual spec-imen simulated. Below a few examples are shown (Figure 5.4, Figure 5.5 and Figure 5.6).

Figure 5.4 Geometry of finite element model for simulation of beam tests A-01 and A-02 [31]

Figure 5.5 Geometry of the finite element model for the simulation of beam test B-01 [31]

Three-dimensional volume models

39

5.3.1.3 Modelling the steel beam

The steel beam is modelled according to the theoretical dimensions of the actual profiled steel section. The fillets between the flanges and the web are replaced by a wedge at both the upper and the lower end of the web. The size of these wedges is chosen in such a way that the cross-sectional area of the steel beam with the wedges equals the theoretical cross-sectional area of the actual steel beam used. Figure 5.4, detail A, shows an example of the wedges.The material properties of the steel sections are chosen as bilinear isotropic with isotropic hard-ening plasticity (BISO) in general, where the linear-elastic material properties are assumed as Ea = 210000 N/mm² and ν = 0.3. For the simulations based on the beam tests performed at ETH [31] the material properties for the steel beam are derived from the tensile tests shown in [31]and formulated using the multilinear isotropic material option with isotropic hardening plastic-ity (MISO). However, the tensile tests were modelled with a finite element volume model for verification. For these simulations, material properties following the engineering and the true stress-strain curve (cf. Figure 5.7) were used with geometrical nonlinearities either included or excluded. As the composite beam models had to work with geometrical nonlinearities excluded, this option was given special attention.For the conversion from engineering stress and strain to true stress and true strain an approxi-mation which assumes incompressible material shown in equation (5.1) was used [2].

and (5.1)

Figure 5.6 Geometry of finite element model for the simulation of the Hilti beam tests [39]

εtrue 1 ε+( )ln= σtrue σ 1 ε+( )⋅=

Numerical Investigations

40

The load-deformation curves obtained from the finite element model with the different options were compared to those of the tensile tests. This comparison is shown in Figure 5.8 (right). With

geometrical nonlinearities excluded the engineering stress-strain curve shows good agreement with the behaviour of the tensile tests. The numerical problem of this combination has no effect on the composite beam models, because the steel strain remains below 5 % in all investigated cases which is far below the amount of strain at which the problem occurred.

5.3.1.4 Modelling the concrete slab

The concrete slab is simulated with its plan-based geometry (cf. Figure 5.4, Figure 5.5 and Fig-ure 5.6). The profiled steel sheet is considered as openings in the concrete slab, but is not mod-elled itself as e.g. a steel shell. This simplification is justified for thin-walled steel sheeting with open trough profiles. For re-entrant trough profiled steel sheeting and profiled steel sheeting of greater thickness, where the steel sheeting plays an important role, this simplification may pro-duce inaccurate results.Above the upper flange of the steel beam a block of a certain height (5 mm in general) is cut out of the concrete slab to provide space for the shear connection. Additionally, wedges are cut out of the concrete slab on the side of the shear connectors to ensure that all forces between the steel beam and the concrete slab are directed through the connectors (cf. Figure 5.4, Detail A).

Figure 5.7 Stress-strain diagrams for material used for the steel beam in beam tests [31]

Figure 5.8 Left: Geometry of tensile model. Symmetry conditions and loading point. Right: Load-displacement diagrams of tensile tests and of simulations with differ-ent simulation options.

0

100

200

300

400

500

600

0 10 20 30 40True Strain [%]

True

Str

ess

[N/m

m2 ]

Tensile Test A-01-2Tensile Test A-02-2Tensile Test B-01-2ANSYS Material Model: MultilinearANSYS Material Model: Bilinear

0

100

200

300

400

500

600

0 10 20 30 40Strain [%]

Stre

ss [N

/mm

2 ]

Tensile Test A-01-2Tensile Test A-02-2Tensile Test B-01-2ANSYS Material Model: MultilinearANSYS Material Model: Bilinear

020406080

100120140160

0 20 40 60Displacement [mm]

Load

[kN

]

Tensile Test A-01-2Tensile Test A-02-2Tensile Test B-01-2FE geom. nonlin., log. stress-strainFE geom. lin., log. stress-strainFE geom. lin., eng. stress-strain

Three-dimensional volume models

41

The reinforcement is simulated as smeared reinforcement with elastic isotropic material behav-iour (Es = 200000 N/mm² and ν = 0.3) and bilinear isotropic hardening plasticity (BISO). In general, a yield stress of fys = 550 N/mm² and Est = 1 N/mm² is used.

The amount of reinforcement is defined as real constants containing the area ratio of the rein-forcing steel to the area of the concrete slab in each of the three directions X (longitudinal), Y (vertical) and Z (transversal).The slab is divided in two main parts. In the upper part (generally this region measures 50 mm from the top of the slab) the actual amount of reinforcement is simulated in the X- (longitudinal) and Z- (transversal) directions. In the Y-direction (vertical) a very small amount of reinforce-ment (typically an area ratio of 10-6) is prescribed to prevent numerical instabilities caused by the very large displacement of single nodes after cracking. In the lower part of the concrete slab a reinforcement ratio of 10-6 in all three directions is used for the same reason. Above the steel beam, shear connectors have the same effect as vertical reinforcement in real composite beams. Therefore, a reinforcement ratio in the Y-direction (vertical) derived from the steel area of the shear connectors is defined for the parts of the concrete slab which are located on top of the steel beam. Table 5.7 gives an overview of the default reinforcement configura-tions.

Realconstant

set number

Reinforcement area ratio [-]Comment

longitudinal X vertical Y transversal Z

1 0.008482 10-6 0.01047 Values determined from beam tests A-01, A-02 and B-01 [31] and used as default values in all simulations except the simulations of the beam tests at Hilti [39]. Figure 5.9 shows which real constant set was used in which part of the concrete slab.

2 0.008482 0.005 0.01047

3 10-6 10-6 10-6

4 10-6 0.005 10-6

5 0 0 0 reference (used in material tests only): 0 % reinforcement

6 0.01 0.01 0.01 reference (used in material tests only): 1 % reinforcement

1 (Hilti) 0.00377 0.00377 0.001 reinforced part of the concrete slab in beam tests [39]

2 (Hilti) 10-6 10-6 0.0001 unreinforced part of the concrete slab in beam tests [39]

Table 5.7 Real constant sets used for concrete material tests and beam simulations.

Figure 5.9 Section of a composite beam. Normal case (left) and special cases for simulations of Hilti beam tests [39]. Real constant sets #1 and #2 are used for the upper (rein-forced) part of the concrete slab, #3 and #4 for the lower (unreinforced) part.

Numerical Investigations

42

The linear elastic material properties for concrete are generally set to Ec = 35000 N/mm² and ν = 0.18. Modelling the nonlinear material properties of concrete is rather complex.Concrete in tension exhibits linear elastic behaviour and then brittle cracking failure on reaching its tensile strength. In compression concrete exhibits nonlinear behaviour and crushing failure on reaching its ultimate strain. To simulate these complex material properties of the concrete, a combination of several material models is used.• The concrete material option available in [2] provides linear elastic behaviour, brittle crack-

ing failure in tension when reaching the tensile strength and brittle crushing when reaching the compressive strength of the concrete. After cracking or crushing, only the reinforcement is active. Further, the ANSYS concrete material option offers the possibility of defining shear (friction) coefficients for open and closed cracks. While the cracking option is used to simulate the behaviour of concrete in tension, the crush-ing option is deactivated in the combined models by setting the compressive strength value to „-1“ (cf. [2]). In general the friction coefficient is set to 0.8 for closed cracks and to 0.5 for open cracks. Other options are not used.

• Plastic material behaviour is established using two different plasticity models. For compari-son with the analytical models and for most of the parametrical investigations the bilinear isotropic hardening plasticity option (BISO) is applied, while for the simulation of the beam tests [31] a multilinear stress-strain relation (MISO) is used.

Either the bilinear or the multilinear plasticity option is combined with the concrete material properties. As a major difference to the real behaviour of concrete, the material is assumed to be ideal plastic at strains greater than the ultimate compressive strain and the typical descent of concrete stress after passing the compressive strength is not modelled. This gives an increased numerical stability of the model, while the impact on the load-deformation of the beam is neg-ligible in all investigated cases. However, in cases where concrete compressive failure governs failure, the MISO option may be used to define a re-descending curve.

5.3.1.5 Material tests for the concrete slab

Since the use of the combined material options (CONC + BISO, CONC + MISO) is not com-mon, multiple simulation runs were performed to investigate the load-bearing behaviour of these combinations of material options and the influence of the reinforcement in each direction. The test models consisted of a single element of cubic shape. To these models geometrical loads of 1 % strain were applied. This was done for the 6 sets of real constants shown in Table 5.7with material data from [31] as shown in Table 5.11.The results of the simulations are listed in Table 5.8. For the two fundamental cases without any reinforcement and with 1 % of reinforcement in all three directions the stress-strain behaviour is shown in Figure 5.10.

Three-dimensional volume models

43

In compression, with the concrete model, brittle crushing failure occurs while the combined models both exhibit ductile behaviour. With the combined MISO material law the behaviour only differs in the fillet shortly before the ultimate stress is reached (cf. Figure 5.10, left).In tension, all material models show identical behaviour in which brittle cracking of the concrete is dominant (cf. Figure 5.10, right).In shear, the ANSYS concrete material option exhibits brittle behaviour. In spite of identical boundary conditions, the shear strengths in the YZ- and ZX-directions are lower than in the XY-direction. However, this inconsistency in the ANSYS concrete material option has no conse-quences for the presented model, since the pure concrete material was not used in the composite beam models but only used as a reference in the material tests. The material combinations with the BISO and the MISO model both exhibit ductile behaviour. Similar to the behaviour in com-pression, the combined MISO material law only differs from the combined BISO material law in the fillet shortly before the ultimate stress is reached (cf. Figure 5.10, lower left).

Load type Compression Tension ShearMaterial option

Direction X Y Z X Y Z XY YZ ZX

Real constant set #1

36.50 35.63 36.97 4.67 3.39 5.77 13.94 9.42 9.37 Concrete only

40.89 38.91 41.56 4.67 3.39 5.77 13.02 15.21 13.55 Concrete + BISO

40.88 38.90 41.56 4.67 3.39 5.77 13.02 15.21 11.35 Concrete + MISO

Real constant set #2

37.26 36.71 37.44 4.67 3.50 5.77 14.19 9.37 9.33 Concrete only

42.36 41.49 42.97 4.67 3.50 5.77 14.11 13.54 13.42 Concrete + BISO

42.35 41.48 42.96 4.67 3.50 5.77 14.11 15.15 14.66 Concrete + MISO

Real constant set #3

34.16 34.16 34.16 3.44 3.44 3.44 14.21 9.25 9.25 Concrete only

34.36 34.36 34.36 3.44 3.44 3.44 13.00 13.00 13.00 Concrete + BISO

34.35 34.35 34.35 3.44 3.44 3.44 12.95 12.95 12.95 Concrete + MISO

Real constant set #4

34.60 35.18 34.60 3.43 3.55 3.43 14.46 9.20 9.16 Concrete only

35.48 36.94 35.48 3.43 3.55 3.43 14.03 12.94 13.07 Concrete + BISO

35.47 36.93 35.47 3.43 3.55 3.43 14.03 12.89 13.07 Concrete + MISO

Real constant set #5

34.16 34.16 34.16 3.44 3.44 3.44 14.21 9.20 9.20 Concrete only

34.36 34.36 34.36 3.44 3.44 3.44 12.98 12.98 12.98 Concrete + BISO

34.35 34.35 34.35 3.44 3.44 3.44 12.98 12.98 12.98 Concrete + MISO

Real constant set #6

38.06 38.06 38.06 5.51 5.51 5.51 14.39 9.30 9.30 Concrete only

44.34 44.34 44.34 5.51 5.51 5.51 13.42 13.42 13.42 Concrete + BISO

44.33 44.33 44.33 5.51 5.51 5.51 11.21 11.21 11.21 Concrete + MISO

Table 5.8 Results of test simulations for concrete material models and combinations. With real constant sets # 5 and #6 the results are identical in all three directions because both material options and reinforcement are identical in all three directions. With real constant sets #1 to #4 the results for the three directions show slight differenc-es because the amount of reinforcement is not the same in all three directions. With the combined material options accurate results are obtained, while with the pure CONC options some inconsistencies are observed.

Numerical Investigations

44

5.3.1.6 Modelling the shear interface

The shear interface is modelled as a layer of brick elements which is placed on top of the steel beam. The height of the brick elements defaults to 5 mm. For the first simulations of the beam tests it was chosen as 20 mm in order to reduce shear strain in the connector elements, which was later found to be unnecessary because the SOLID 185 elements exhibit excellent behaviour even at large strains. For the parameterised model the height of the shear interface defaults to 5 mm. The width of the bricks is equal to that of the upper flange of the steel beam. With solid slabs and with profiled steel sheeting spanning parallel to the beam, the interface layer is placed over the whole length of the concrete slab (cf. Figure 5.5). The shear interface is then divided into a number of shear connectors. With a profiled steel sheet spanning perpendicular to the slab, the interface consists of one connector brick in each trough of the profiled steel sheet. The actual length of the brick is calculated to fit the length of the concrete rib on top of the connector brick. Examples are given in Figure 5.4 and Figure 5.6 (detail B). For each connector the load-slip behaviour may be chosen separately. In places where no connectors are placed the stiffness of the connector material is modelled as extremely low (cf. Table 5.9).Vertical forces are transmitted across the shear interface by coupling all nodes at the same lon-gitudinal (X-) position in the vertical (Y-) direction. Further, the nodes at the same longitudinal position on top of the shear interface are coupled in the longitudinal direction.

Figure 5.10 Load-bearing behaviour in compression (upper left), tension (upper right) and shear (lower left) of a cubic element using the ANSYS concrete material option (CONC), a combination of the concrete material option and bilinear isotropic hardening plasticity (CONC + BISO) and a combination of the concrete material option and multilinear isotropic hardening plasticity (CONC + MISO), all without reinforcement and with 1 % of reinforcement in each direction (X, Y, Z)

0

10

20

30

40

50

0 0.2 0.4 0.6 0.8 1Strain [%]

Stre

ss [N

/mm

²]

CONC option, 1 % of reinforcementCONC option,

no reinforcement

CONC+MISO, no reinforcement

CONC+BISO, no reinforcement

CONC+BISO, 1 % of reinforcement

CONC+MISO, 1 % of reinforcement

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1Strain [%]

Stre

ss [N

/mm

²]

All material options, 1 % of reinforcement in each direction

All material options, no reinforcement

0

4

8

12

16

0 0.2 0.4 0.6 0.8 1Strain [%]

Stre

ss [N

/mm

²]

CONC option, no reinforcementCONC option, 1 % of reinforcement

CONC+MISO

CONC+BISO

Three-dimensional volume models

45

The materials to simulate the shear connection are orthotropic materials with Hill‘s plasticity law ([2], [37]). For the simulations of the beam tests the shear characteristics in the X-Y direc-tion were derived from the results of push tests (cf. [31] and [7] and the push tests shown in chapter 4 on page 25) as a verification of the finite element model against test data. In the pa-rameterised model the load-slip characteristics of the shear connectors can be chosen very free-ly. In all directions other than X-Y shear the material strength is increased by a large factor (typ-ically 103) in comparison. The linear orthotropic material option is used for the non-bearing el-ements and for the simulation of a rigid shear connection with stiffness values as shown in Table 5.9. In cases without profiled steel sheeting or with profiled steel sheeting parallel to the beam, the shear connectors form a compact layer along the whole of the beam. In this configuration, the shear connector layer might carry longitudinal normal forces and thus produce inaccurate results. These errors would appear only if the elastic stiffness of the shear interface is rather high. In order to make the model work, also in cases with a very stiff shear connection, several measures are taken. As orthotropic material properties are used for the shear connectors, the ma-terial stiffness in the Y- and Z-directions (but not in the X-direction) may be chosen independ-ently of the X-Y shear stiffness. Therefore, as a solution for the elastic domain, the shear con-nector elements are rotated by 90° about the transverse axis (Z) of the composite beam. Addi-tionally, the height of the shear interface may be reduced and the shear interface may be divided into many connectors, whereby every second connector is non-bearing and thus does not carry longitudinal forces. However, in all investigated cases the effect of the longitudinal stiffness of the shear connectors was found to be negligible.

The SOLID 185 elements simulating the shear interface must be able to deal with shear strains in the X-Y direction of much more than 100 % in some of the models treated. Therefore, the behaviour of these elements with different sets of key options at very large strain was investi-gated using a model which consisted of a single element (cf. Figure 5.11). To this single element the same boundary conditions as expected in the beam model were applied. On the lower side of the element for all nodes all degrees of freedom were blocked, while on the upper side of the element the displacement of all nodes was blocked in the Y- and in Z-directions. In the X-direc-tion a displacement of 100 mm was applied as a geometrical shear load.The results obtained with different sets of key options are shown in Figure 5.11.For key option K2 (element technology) the „full integration“, the „reduced integration“ and the „enhanced strain“ settings were tested. The „enhanced strain“ setting would produce erroneous results by reducing strain above a certain value while with the „full integration“ setting the load-displacement data fits well with the target data obtained from push test S2.6 [31]. The results obtained with the „reduced integration“ setting could not be distinguished from the results pro-duced with the „full integration“ setting under the boundary conditions given for this analysis. For further use in the composite beam models the „full integration“ option is used.

EX EY EZ νXY νYZ νXZ GXY GYZ GXZ

[N/mm²] [N/mm²] [N/mm²] [-] [-] [-] [N/mm²] [N/mm²] [N/mm²]

rigid shear connector 0.0202 0.000202 0.000202 0.0001 0.0001 0.0001 210000 0.01 0.01

no shear connection 0.0202 0.0202 0.0202 0.01 0.01 0.01 0.01 0.01 0.01

soft layer 500 10000 500 0.01 0.01 0.01 500 500 500

Table 5.9 Orthotropic material properties for the shear interface and for the soft layers at the load distribution blocks

Numerical Investigations

46

For key option K6 (element formulation) the „pure displacement“ and the „mixed U/P formula-tion“ settings were tested. However, the results obtained with pure displacement formulation could not be distinguished from those obtained with mixed U/P formulation. In the end, the mixed U/P formulation is used, because - according to [2] - for materials in which plastic defor-mation is dominant, mixed formulation of the SOLID 185 element is more robust and has better performance.As shown in Figure 5.11, a simulation with consideration of geometrical nonlinearities („gNL...“ graphs) gives completely different results. This is because tension is established in the Y-direction (vertical) with geometrical nonlinearities included.

5.3.1.7 Applying the loads

The model is able to simulate one or more symmetrically-placed concentrated loads and a uni-form load. Distributed loads are applied to the top of the concrete slab. Concentrated loads may be applied either from above to the concrete slab or from below to the steel beam. Applying con-centrated loads to a finite element model may cause singularities and local failures within the model if the concentrated load values are too large. Therefore, concentrated loads are usually replaced by distributing pressure over a certain area. However, since not only force-controlled loading, but also displacement-controlled loading is applied to the model, this is not possible. In finite element simulations, often coupling equations are then used. In laboratory tests, load distribution blocks are used in locations where concentrated loads are applied to the specimen (e.g. at the support or under hydraulic jacks). In the finite element models explained in this section, load distribution blocks are used. This allows for better visual control of the model in the graphical user interface while the increase of computational effort is negligible. The geometry of all load distribution blocks is chosen by de-fault as usual in laboratory tests. As a major difference to laboratory tests, the load distribution blocks are simulated as glued to the composite beam structure, while in laboratory tests the dis-tribution blocks transfer loads via contact and friction only. All blocks exhibit a width equal to that of the upper flange of the structural steel section. The dimensions of the blocks are user-definable. The height is in general 30 or 50 mm. The length of the load distribution blocks is adjusted by the ANSYS APDL macro applet to fit the rib grid of the composite beam.The material option for the load distribution blocks is intended to simulate steel of infinite strength with E = 210000 N/mm² and ν = 0.3. In order to prevent the load introduction blocks from stiffening and strengthening the composite beam, a soft layer which corresponds to a

Figure 5.11 Left: Single element model to test shear load-deformation behaviour Right: Load-slip behaviour of a shear interface element with different key options

0

20

40

60

80

100

0 10 20 30Slip [mm]

Load

[kN

]

Target: Push Test S2.6Full Integration, Mixed U/PEnhanced Strain, Mixed U/PgNL, Full Integr., Mixed U/P

Three-dimensional volume models

47

wooden block or elastomer layer as used in laboratory tests is placed between the load distribu-tion blocks and the composite beam. For this soft layer a linear orthotropic material option with material values similar to timber is used (cf. Table 5.9).In some laboratory tests (cf. [31]) loads are applied by hydraulic jacks fixed directly to the spec-imen. In some other tests (cf. [39]), a load-distributing steel structure is placed on top of the specimen, and the hydraulic jack is attached to the steel structure. The steel structure may have a considerable weight which is carried by the specimen in addition to the self-weight and the load introduced by the jacks.In order to treat all these cases systematically in one model, all loads are applied to one single node. This load introduction node is meshed with a mass element (MASS 21) which offers a simple way of considering the mass of the load distribution structure in the model. From the load distribution blocks attached to the composite beam, one node each (the node above the longitu-dinal axis of the composite beam at the exact longitudinal position where the load is introduced into the specimen) is coupled to the load introduction node by means of a constraint equation. This constraint equation assures that the mean vertical displacement of all load introduction nodes on the composite beam equals the vertical displacement of the load introduction node with the MASS 21 element. Applying distributed loads is accomplished without load introduc-tion blocks because the loads per node are relatively small. The constraint equation is estab-lished between the load introduction node and all nodes on top of the concrete slab on the lon-gitudinal axis of the beam. At the support the actual discrete boundary conditions are established for all nodes at the position X = 0 on the lower surface of the load distribution block.

5.3.1.8 Simulation arrangements

The models are subjected to the following load steps:

1) Self-weight (vertical acceleration of 9810 mm/s²). The self-weight is gradually applied in load step 1 and held constant through all other load steps.

2) Force-controlled loading to service load level (optional). In some laboratory tests the beams were pre-loaded. Load step 2 is used to simulate this pre-loading. The force is ap-plied at the load introduction node.

3) Force-controlled unloading (optional). If pre-loading was simulated in load step 2, the pre-load is gradually reduced to zero again.

4) Deformation-controlled loading. The deformation is applied at the load introduction node. Usually, the deformation at the load introduction node is gradually increased from zero to L/50. Thus, the beams are loaded beyond the ultimate limit state, and the descending branch of the beam‘s load-deflection curve can also be simulated.

5.3.1.9 Model optimization

In order to save computation time, mesh density optimisations were performed. Special atten-tion was given to the aspect ratio of the elements in the web of the steel beam, because the size of these elements is crucial for the performance of the model. The web elements have an aspect ratio of up to 20. As this may produce inaccurate results, a simulation with a maximum aspect ratio of 10 was performed for beam test A-01. The refined model consisted of 2761 8-noded volume elements (compared to 1725 with normal mesh) and 4889 nodes (compared to 3317 for

Numerical Investigations

48

the unrefined model). However, the results obtained from the fine-meshed model could not be distinguished from the results obtained from the final model (cf. Figure 5.12, right).

5.3.2 Verification and first applications

The finite element model was verified against data from the beam tests reported in [31] and [39]. An overview on the beam tests used for verification is given in Table 5.10.

For the beam tests performed at ETH [31] detailed material data was available and used in the simulations. For the beam tests performed at Hilti [39] bilinear material behaviour was assumed according to the test data shown in Table 5.11. The geometry of the finite element models is shown in Figure 5.4, Figure 5.5 and Figure 5.6.

Figure 5.12 Refined model (left: finite element model, right: load-deflection diagram)

Beam Span [m] Steel profile

Concrete slab b x h [mm]

Profiled steelsheeting

Shear connectors used (besides rigid & no shear connection)

A-01

7.2 m IPE 270 1200 x 150VIKAM TR60/235

perpendicular S2.5, S2.6, S3.1, HVB1, SR1 to SR6A-02

B-01 VIKAM TR60/235 parallel RT3

Träger 1

6.0 m IPE 220 2000 x 120 HI-BOND 55 perpendicular

HVB2

Träger 2 HVB1

Träger 3 HVB1

Table 5.10 Overview on beam tests used for verification of the numerical model

BeamSteel Concrete

Yield strength [N/mm²]

Compressive strength [N/mm²]

Tensile strength [N/mm²]

Modulus of elasticity[N/mm²]

Materialmodel

A-01317 (MISO)

(cf. Figure 5.7) 34.35

3.5

28705 CONC+MISOA-02

B-01

Träger 1 460 (BISO) 32.1

35000 CONC+BISOTräger 2 456 (BISO)31.2

Träger 3 467 (BISO)

Table 5.11 Materials used for verification of finite element model

0

20

40

60

80

100

120

0 100 200 300Deflection [mm]

Load

per

jack

[kN

]

refined mesh A300_A01coarse mesh A200_A01

Three-dimensional volume models

49

Table 5.12 gives an overview of the shear connectors used in the finite element simulations. Connector characteristics S2.5, S2.6, S3.1 and S3.2 are taken from the corresponding push test reported in [31] and in chapter 4 on page 25, respectively. Load-slip characteristics SR1 to SR6 were adapted from S3.2 to match the connector behaviour in beam tests A-01 and A-02.

The beam tests performed at Hilti [39] were simulated using two connector characteristics named „HVB1“ and „HVB2“. Each of these characteristics was determined graphically as the

Connector type Source Shown in Remarks

S2.5 [31], test S2.5 Figure 5.13 StripCon made of S280GD on HEB 260

S2.6 [31], test S2.6 Figure 5.13 StripCon made of DX51D on HEB 260

S3.1 chapter 4 on page 25, test S3.1 Figure 5.13 StripCon made of DX51D on IPE 270

S3.2 chapter 4 on page 25, test S3.2 Figure 5.13 StripCon made of S280GD on IPE 270

SR1 to SR6 generic Figure 5.13 Adapted to match beam tests A-01, A-02

HVB1 [38], series 5 (averaged) Figure 5.14, left HVB100 on IPE 220

HVB2 [38], series 7 (averaged) Figure 5.14, right HVB100 on IPE 220

RT3 [49], series T3 (averaged) Figure 5.15, left RibCon made of DX51D on HEB 260

Param

generic

Figure 5.15, right Idealised headed stud for parametric studies

el Figure 5.16, left elastic-brittle behaviour, varying Si

ep Figure 5.16, right elastic-ideal plastic behaviour, varying δi

Table 5.12 Shear connectors used with finite element model

Figure 5.13 Load-slip curve for shear connectors from push tests (left) and generic ones (right)

Figure 5.14 Load-slip curve for shear connectors from push tests [38] used in beam tests [39]

0

20

40

60

80

100

0 10 20 30Slip [mm]

Load

per

rib

[kN

/rib]

Material Model S2.5Material Model S2.6Material Model S3.1Material Model S3.2

StripCon from push tests

0

20

40

60

80

100

0 10 20 30Slip [mm]

Load

per

rib

[kN

/rib]

Material Model SR1Material Model SR2Material Model SR3Material Model SR4Material Model SR5Material Model SR6

StripCon, adapted from push tests

0

10

20

30

40

50

0 2 4 6 8 10 12 14 16 18Slip [mm]

Load

per

rib

[kN

]

Material Model HVB1

HVB 100 from push tests series 5 with two connectors per rib

0

10

20

30

40

50

0 2 4 6 8 10 12 14 16 18Slip [mm]

Load

per

rib

[kN

]

Material Model HVB2

HVB 100 from push tests series 7 with one connector per rib

Numerical Investigations

50

mean curve of the three push tests reported in [38] for each series. Connector behaviour HVB1 is determined from test series 5 [38] with two shear connectors per rib, while HVB2 was taken from series 7 with one HVB100 shear connector per rib.The load-slip behaviour for the RibCon shear connector was determined graphically as the mean curve of the three push tests of series T3 reported in [49].The parametric load-slip characteristics („Param“, „el“ and „ep“) are used in the parametric studies and are explained in section 5.3.4 on page 58 and section 5.3.5 on page 58.

Figure 5.15 Load-slip curve for shear connectors RT3 (left) and parametric studies (right)

Figure 5.16 Load-slip curve for parametric studies groups elX (left) and epX (right)

Pt #series C2/3/4

(cf. Figure 5.15, right)groups el1/2/3/4/5

(cf. Figure 5.16, left)groups ep1/2/3/4/5

(cf. Figure 5.16, right)

P d P d P d

1 0.8 Pu P1/Si 1.0 Pu P1/Si 1.0 Pu0.001/0.01/0.1/1/6 mm

2 0.9 Pu d1+0.2 [d3-d1] 0.1 Pu 1.1 d1 1.0 Pu 6.0 mm

3 1.0 Pu 0.75 * δu 0.01 Pu 3.0 d1 0.1 Pu 7.0 mm

4 0.9 Pu 1.0 * δu Si = 10000 kN/mm in general Si = (10/100/1000/10000/ 100000) kN/mm in groups elX, Si according to d1 in groups epX.

0.01 Pu 10.0 mm

5 0.3 Pu (C2), 0.1 Pu (else) 1.5 * δu

6 0.01 Pu 3.0 * δu

Table 5.13 Parametric load-slip behaviours of shear interface

050

100150200250300350

0 2 4 6 8 10 12 14 16 18Slip [mm]

Load

per

con

nect

or [k

N] Material Model RT3

Test T3S1Test T3S2Test T3S3

RibCon from push tests

0.75; 1

d1; 0.8d2; 0.9

3; 0.011.5; 0.1

1; 0.9

1.5; 0.3

0

0.2

0.4

0.6

0.8

1

0 1 2 3Slip [d/δu]

Shea

r for

ce [P

/Pu]

Si

1

used in series C3/4/5

used in series C2

δu=6mm;1δi; 1

10; 0.017; 0.1

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10Slip [mm]

Shea

r for

ce [P

/Pu]

Si

1

used for epX-simulations(elasto-plastic behaviour)

d1; 1

3d1; 0.011.1d1; 0.1

0

0.2

0.4

0.6

0.8

1

Slip [mm]

Shea

r for

ce [P

/Pu]

Si

1

used for elX-simulations(linear elastic behaviour

with brittle failure)

Three-dimensional volume models

51

Figure 5.17 shows the results of the finite element simulations of beam test A-01 with the test-based shear connector characteristics S2.5, S2.6, S3.1 and S3.2, compared to the behaviour of the specimen in the test. Figure 5.18 shows the same data for beam test A-02.

The results of the simulations on beam tests A-01 and A-02 differ from the results of the beam tests. The differences between the test results and the data obtained from the numerical simula-tions with shear connectors characteristics S2.5 and S2.6 can be explained by the fact that push tests S2.5 and S2.6 were performed with a different steel section (HEB 260 instead of IPE 270) of higher strength than used in the beam tests. However, characteristics S3.1 and S3.2 were obtained from push tests performed with a steel profile taken from specimen A-02 and with concrete properties very close to those used in the beam tests.A detailed analysis clearly shows that the behaviour of the shear connectors in the beam tests considerably differs from the load-slip characteristics found in the push tests both with a steel section HEB 260 (S2.5, S2.6) and IPE 270 (S3.1, S3.2). Therefore, several simulations were performed with adapted connector characteristics (SR1 to SR6, cf. Figure 5.19). For beam test A-01, a simulation with generic behaviour SR6 shows the best agreement with the test data. For beam test A-02 all generic characteristics show similar results. Ultimate loads and deformation capacity are both larger in the simulations than in the beam test.This effect is also observed when comparing push tests S3.2 and S3.3 (cf. chapter 4 on page 25 ff). Specimens S3.2 and S3.3 are identical, but S3.3 was loaded to 80 % of the expected ul-timate load during load cycles while S3.2 was loaded to 40 % only. In the ultimate load test per-formed afterwards, S3.3 fails at much lower loads and at lower deformations than S3.2. As a system-inherent effect of most shear connectors, large local stress concentrations and thus plas-

Figure 5.17 Results of simulations on beam test A-01 with connector behaviour from push tests

Figure 5.18 Results of simulations on beam test A-02 with connector behaviour from push tests

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A01_S25 A01_S26 A01_S31A01_S32 no con full conTest A01

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n [k

N]

A01_S25 A01_S26 A01_S31

A01_S32 no con Test A01

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n [k

N]

A02_S25 A02_S26 A02_S31A02_S32 no con full conTest A02

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f spa

n [k

N]

A02_S25 A02_S26 A02_S31

A02_S32 no con Test A02

Numerical Investigations

52

tic local bearing effects occur. However, the numerical model does not consider such pre-dam-age effects.

In the beam tests reported in [31] the pre-loading procedure differs between the three tests.• With specimen A-01 25 + 1 load cycles to the assumed service load level were performed

prior to the ultimate load test. At the ultimate load state the beam was unloaded. On re-load-ing the beam, the ultimate load could not be reached again. This may be explained by con-sidering the observations made in push test S3.3: When unloading the beam the ENPH2-fasteners became loose in their nail-holes. Without unloading both deformation capacity and ultimate load of the beam would presumably have been larger than observed with unloading.

• Specimen A-02 was first subjected to a stepwise increasing long-term load over 17 h in total and then to 3000 load cycles (instead of the 25 of A-01). It is very probable that during the long-term loading, but even more during the load cycles, pre-damage of the shear connection occurred. According to the numerical simulations, the maximum shear force per rib at serv-ice load level was up to 65 kN per rib or 75 % of the ultimate shear resistance of 88 kN per rib assumed with shear connector S2.6.

• Specimen B-01 was subjected to the same procedure as A-02, whereby the long-term load was increased over a period of 70 h instead of 17 h. Further, a total of 29 dynamic tests (not reported in [31]) were performed to determine eigenfrequencies and damping of the beam. Again, pre-damaging of the shear interface is very likely to have occurred during this proce-dure.

The load-slip data for the simulation of beam test B-01 was taken from push tests [49] which had been performed with a HEB 260 steel section. Beam test B-01 was performed with a steel section IPE 270 with a yield strength of fy = 317 N/mm². From a comparison of shear charac-teristics S2.5/S2.6 to S3.1/S3.2 it may be seen that the thickness of the steel flange and the ma-terial strength of the steel section have a crucial influence on the load-slip behaviour of a nailed connection with powder-actuated fasteners or similar products (and thus the shear connector which uses these fasteners). Further, the pre-damage which has occurred during pre-loading in the test has an important influence as mentioned above. These considerations explain the differ-ences found in Figure 5.20 between test data and simulation results. However, as the data base is rather poor for composite beams with shear connectors of type RibCon RT3 with only one beam test and three push tests, no further simulations and evaluations were performed.

Figure 5.19 Results of simulations with adapted connector behaviour (A-01 left, A-02 right)

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A02_SR1 A02_SR2 A02_SR3A02_SR4 A02_SR5 A02_SR6no con full con Test A02

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n [k

N]

A01_SR1 A01_SR2 A01_SR3A01_SR4 A01_SR5 A01_SR6no con full con Test A01

Three-dimensional volume models

53

For the beam tests performed at Hilti [39] the results from the simulations, the analytical model and the tests agree well. This is due to the fact that the push tests [38] were performed under almost the same conditions as the beam tests [39]. Furthermore, no pre-loading is reported in [39]. Thus, these beam tests meet very closely the conditions assumed for the numerical model.

5.3.3 Parametric investigation series A200

Series A200 of the parametric investigations was performed to study the behaviour of simply-supported composite beams with the StripCon shear connector of type S2.6.

Figure 5.20 Results of simulations on beam test B-01

Figure 5.21 Results of simulation on beam tests performed at Hilti

Figure 5.22 Geometry of beams simulated in series A200. The sectional geometry is identical to that in beam tests A-01 and A-02 (cf. Figure 5.4). For each span the beam over-hangs the support by 150 mm.

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Load

per

jack

[kN

]

Simu: Rigid connection, bilinear materialsSimu: Rigid connection, multilinear materialsSimu: No shear connection, bilinear materialsSimu: No shear connection, multilinear materials

Beam test B-01

Numerical simulation of beam test B-01 with shear connectors of type RT3 and multilinear material options

Analytical: full, rigid shear connection

Analytical: steel beam

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[kN

] Test B-01 (Slip u7)

Simu: no shear connection,multilinear materialSimu: B-01 with RT3,multilinear materialAnalytical: No shear connection

0255075

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0 40 80 120 160 200Deflection [mm]

Load

per

hal

f spa

n [k

N]

Analytical: Rigid shear connectionAnalytical: No shear connectionSimu: Rigid shear connectionSimu: No shear connectionSimu: Träger 1 (1 con per 2 ribs)Simu: Träger 2 (1 con per 1 rib)Simu: Träger 3 (2 cons per 1 rib)

All tests shown in grey

0255075

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Load

per

hal

f spa

n [k

N]

Analytical: No shear connectionSimu: No shear connectionSimu: Träger 1 (1 con per 2 ribs)Simu: Träger 2 (1 con per 1 rib)Simu: Träger 3 (2 cons per 1 rib)

All tests shown in grey

Numerical Investigations

54

The beams investigated had the same geometry and the same material properties as beam tests A-01 and A-02. The variable parameters investigated are the span (varying from 2.4 m to 14.4 m in steps of 1.2 m) and the number of shear connectors used (one connector in each rib, every second, third, fourth, fifth or sixth rib). Additionally, beam tests A-01 and A-02 are treated in series A200. In series A200 a total of 68 simulations were performed.The simulation jobs are named A200_<Span in dm>_<spacing between connectors>. For ex-ample, a simulation of a beam with 8.4 m span and a shear connector placed every 5th rib is named „A200_084m_05“. The first connector from the support was always placed in the first complete rib. From Figure 5.22 it can be seen e.g. that rib #6 is not complete if the span is 2.4 m. Therefore, the first connector is then placed in rib #5. Table 5.14 gives an overview on the sim-ulations performed. The load-deflection curves from all simulations of series A200 are shown in the appendix.

Job name

Span

n η MR

,ana

lytic

al

MR

(FEM

)

MR

(FEM

)/MR

,ana

lytic

alScheme: Allocation of shear connectors in ribs (right boundary of the table = centre of the beam).„1“ in the cells below means: connector placed in this rib.„0“ in the cells below means: connector placed in this rib, but not considered in analytical model, because it is not located be-tween the support and the loading point.An empty cell stands for a rib without a shear connector.

[m] [-] [-]

[kN

m]

[kN

m]

32

31

30

29

28

27

26

25

24

23

22

21

20

19

18

17

16

15

14

13

12

11

10 9 8 7 6 5 4 3 2 1

A200_024m_01 2.4 3 0.17 217 246 1.14 1 1 1 0 0

A200_024m_02 2.4 2 0.12 198 210 1.06 1 1 0

A200_024m_03 2.4 1 0.06 176 198 1.12 1 0

A200_024m_04 2.4 1 0.06 176 187 1.06 1 0

A200_024m_05 2.4 1 0.06 176 181 1.03 1

A200_024m_06 2.4 1 0.06 176 181 1.03 1

A200_036m_01 3.6 5 0.29 250 282 1.13 1 1 1 1 1 0 0 0

A200_036m_02 3.6 3 0.17 217 232 1.07 1 1 1 0

A200_036m_03 3.6 2 0.12 198 211 1.07 1 1 0

A200_036m_04 3.6 2 0.12 198 202 1.02 1 1

A200_036m_05 3.6 1 0.06 177 199 1.13 1 0

A200_036m_06 3.6 1 0.06 177 188 1.06 1 0

A200_048m_01 4.8 8 0.47 289 313 1.08 1 1 1 1 1 1 1 1 0 0 0

A200_048m_02 4.8 4 0.23 235 252 1.07 1 1 1 1 0 0

A200_048m_03 4.8 3 0.17 217 228 1.05 1 1 1 0

A200_048m_04 4.8 2 0.12 198 212 1.07 1 1 0

A200_048m_05 4.8 2 0.12 198 203 1.03 1 1 0

A200_048m_06 4.8 2 0.12 198 202 1.02 1 1

Table 5.14 Parameters and results of the simulations performed in series A200

Three-dimensional volume models

55

A200_060m_01 6.0 9 0.52 301 328 1.09 1 1 1 1 1 1 1 1 1 0 0 0 0

A200_060m_02 6.0 5 0.29 250 263 1.05 1 1 1 1 1 0 0

A200_060m_03 6.0 3 0.17 217 226 1.04 1 1 1 0 0

A200_060m_04 6.0 3 0.17 217 224 1.03 1 1 1 0

A200_060m_05 6.0 2 0.12 198 209 1.06 1 1 0

A200_060m_06 6.0 2 0.12 198 205 1.04 1 1 0

A200_072m_01 7.2 11 0.64 324 358 1.10 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0

A200_072m_02 7.2 6 0.35 264 281 1.06 1 1 1 1 1 1 0 0

A200_072m_03 7.2 4 0.23 235 238 1.01 1 1 1 1 0 0

A200_072m_04 7.2 3 0.17 217 232 1.07 1 1 1 0

A200_072m_05 7.2 3 0.17 217 226 1.04 1 1 1 0

A200_072m_06 7.2 2 0.12 198 107 0.54 1 1 0

A200_072m_A01 7.2 8 0.47 289 107 0.37 1 1 1 1 1 1 1 1 0 0

A200_072m_A02 7.2 6 0.35 264 107 0.40 1 1 1 1 1 1 0 0

A200_084m_01 8.4 12 0.70 335 373 1.11 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0

A200_084m_02 8.4 6 0.35 264 290 1.10 1 1 1 1 1 1 0 0 0

A200_084m_03 8.4 4 0.23 235 258 1.10 1 1 1 1 0 0

A200_084m_04 8.4 3 0.17 217 240 1.10 1 1 1 0 0

A200_084m_05 8.4 3 0.17 217 231 1.06 1 1 1 0

A200_084m_06 8.4 2 0.12 198 220 1.11 1 1 0

A200_096m_01 9.6 14 0.82 357 391 1.10 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0

A200_096m_02 9.6 7 0.41 277 309 1.12 1 1 1 1 1 1 1 0 0 0 0

A200_096m_03 9.6 5 0.29 250 274 1.09 1 1 1 1 1 0 0

A200_096m_04 9.6 4 0.23 235 254 1.08 1 1 1 1 0 0

A200_096m_05 9.6 3 0.17 217 232 1.07 1 1 1 0 0

A200_096m_06 9.6 3 0.17 217 109 0.50 1 1 1 0

Job name

Span

n η MR

,ana

lytic

al

MR

(FEM

)

MR

(FEM

)/MR

,ana

lytic

al

Scheme: Allocation of shear connectors in ribs (right boundary of the table = centre of the beam).„1“ in the cells below means: connector placed in this rib.„0“ in the cells below means: connector placed in this rib, but not considered in analytical model, because it is not located be-tween the support and the loading point.An empty cell stands for a rib without a shear connector.

[m] [-] [-]

[kN

m]

[kN

m]

32

31

30

29

28

27

26

25

24

23

22

21

20

19

18

17

16

15

14

13

12

11

10 9 8 7 6 5 4 3 2 1

Table 5.14 Parameters and results of the simulations performed in series A200

Numerical Investigations

56

When comparing the simulations A200_072m_02 and A200_072m_A02 (the simulation of beam test A-02 with connector behaviour S2.6) the only difference between the two beams is that in simulation A200_072m_02 a connector is placed in every second rib while in simulation A200_072m_A02 the ribs are provided with connectors in pairs, followed each time by a pair of ribs without a connector. However, the two simulations show no noticeable difference in their behaviour (cf. Figure 5.23).

A200_108m_01 10.8 16 0.93 378 396 1.05 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0

A200_108m_02 10.8 8 0.47 289 322 1.11 1 1 1 1 1 1 1 1 0 0 0 0

A200_108m_03 10.8 6 0.35 264 284 1.07 1 1 1 1 1 1 0 0

A200_108m_04 10.8 4 0.23 235 222 0.95 1 1 1 1 0 0

A200_108m_05 10.8 4 0.23 235 174 0.74 1 1 1 1 0

A200_108m_06 10.8 3 0.17 217 0 0.00 1 1 1 0

A200_120m_01 12.0 17 0.99 388 401 1.04 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0

A200_120m_02 12.0 9 0.52 301 401 1.33 1 1 1 1 1 1 1 1 1 0 0 0 0

A200_120m_03 12.0 6 0.35 264 299 1.13 1 1 1 1 1 1 0 0 0

A200_120m_04 12.0 5 0.29 250 276 1.10 1 1 1 1 1 0 0

A200_120m_05 12.0 4 0.23 235 259 1.10 1 1 1 1 0 0

A200_120m_06 12.0 3 0.17 217 228 1.05 1 1 1 0 0

A200_132m_01 13.2 19 1.11 389 407 1.04 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0

A200_132m_02 13.2 10 0.58 313 355 1.13 1 1 1 1 1 1 1 1 1 1 0 0 0 0

A200_132m_03 13.2 6 0.35 264 310 1.17 1 1 1 1 1 1 0 0 0

A200_132m_04 13.2 5 0.29 250 284 1.14 1 1 1 1 1 0 0

A200_132m_05 13.2 4 0.23 235 243 1.03 1 1 1 1 0 0

A200_132m_06 13.2 4 0.23 235 217 0.92 1 1 1 1 0

A200_144m_01 14.4 21 1.22 389 412 1.06 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

A200_144m_02 14.4 11 0.64 324 374 1.15 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0

A200_144m_03 14.4 7 0.41 277 326 1.18 1 1 1 1 1 1 1 0 0 0

A200_144m_04 14.4 6 0.35 264 276 1.05 1 1 1 1 1 1 0 0

A200_144m_05 14.4 5 0.29 250 159 0.63 1 1 1 1 1 0 0

A200_144m_06 14.4 4 0.23 235 159 0.68 1 1 1 1 0 0

Job name

Span

n η MR

,ana

lytic

al

MR

(FEM

)

MR

(FEM

)/MR

,ana

lytic

al

Scheme: Allocation of shear connectors in ribs (right boundary of the table = centre of the beam).„1“ in the cells below means: connector placed in this rib.„0“ in the cells below means: connector placed in this rib, but not considered in analytical model, because it is not located be-tween the support and the loading point.An empty cell stands for a rib without a shear connector.

[m] [-] [-]

[kN

m]

[kN

m]

32

31

30

29

28

27

26

25

24

23

22

21

20

19

18

17

16

15

14

13

12

11

10 9 8 7 6 5 4 3 2 1

Table 5.14 Parameters and results of the simulations performed in series A200

Three-dimensional volume models

57

The ultimate loads obtained from the simulations (cf. Figure 5.24) show good agreement with the analytical predictions with degrees of partial shear connection η greater than about 1.0. The analytically predicted ultimate loads are lower by a few percent than the ultimate loads obtained from the numerical investigations, because the analytical model does not consider either the bending strength of the concrete slab or strain hardening. Also, no reinforcement is considered in the analytical calculations. At lower degrees of partial shear connection the ultimate loads reached during the finite element calculations are higher than the predicted values. In the simulations shear connectors were placed equidistant over the whole length of the beam. However, the analytical model only con-siders the connectors located between the support and the position where the load is brought up at a third of the span.In some cases (A200_060m_03, A200_072m_03, A200_096m_05/06, A200_108m_04/05/06, A200_120m_06, A200_132m_05/06 and A200_144m_04/05/06) the numerical solution was aborted before ultimate load was reached. In these cases the ultimate loads shown in Figure 5.24are lower than the predicted value.The results of simulation series A200 show that shear connectors with load-slip behaviour of type S2.6 can be used with the plastic design model according to Eurocode 4 [26] as long as the degree of partial shear connection is sufficient, whereby the value for the minimum degree of partial shear connection as determined using Eurocode 4 [26] is rather conservative for the in-vestigated cases.

Figure 5.23 Series A200: comparison between A200_072m_02 and A200_072m_A02

Figure 5.24 Results of series A200: comparison with results from analytical calculations

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f spa

n [k

N]

072m_02 072m_A02 full con no con0

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Load

per

hal

f spa

n [k

N]

072m_02 072m_A02 no con

0.00.20.40.60.81.01.21.4

2.4m

3.6m

4.8m

6.0m

7.2m

8.4m

9.6m

10.8

m

12.0

m

13.2

m

14.4

m

Span [m]

MR

(FEM

)/MR

(Ana

lytic

al) [

-]

_01: Connector in each rib_02: Connector in every 2nd rib_03: Connector in every 3rd rib_04: Connector in every 4th rib_05: Connector in every 5th rib_06: Connector in every 6th rib

0.8

0.9

1.0

1.1

1.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

η [-]

MR

(FEM

)/MR

(Ana

lytic

al) [

-]2.4m 3.6m 4.8m6.0m 7.2m 8.4m9.6m 10.8m 12.0m13.2m 14.4m

Numerical Investigations

58

5.3.4 Parametric investigations series C2

Series C2 of the parametric investigations was performed on simply-supported composite beams of identical geometry (chosen according to beam test B-01, but without profiled steel sheeting). The material properties chosen were based on the values for the beam tests shown in [31], but rounded for practical reasons. The shape of the load-slip behaviour for the shear con-nectors was chosen as shown in Figure 5.15 (right) as an approximation of a headed stud shear connector with the two curve parameters δu (deformation capacity) and Pu (shear resistance).

Series C2 was performed as a pilot series to series C3 which is shown below.

5.3.5 Parametric investigation series C3

Parametric investigation series C3 was conducted with simply-supported beams with IPE steel sections and solid concrete slabs. Series C3 was used to determine load-deformation behaviour of composite beams and the minimum degree of partial shear connection depending on the load-slip behaviour of the shear connection. Table 5.16 shows the investigated parameters.The simulations are grouped as shown in Table 5.17. In each group the beams were simulated with two concentrated loads at one third and two thirds of the span as well as with uniform load. For each parameter set the characteristic slip of the shear interface was varied between 1 mm and 15 mm in steps of 1 mm and the degree of partial shear connection was varied between 0.1 and 1.5 in steps of 0.1. (In simulation groups el1/2/3/4/5 and ep1/2/3/4/5 the degree of partial shear connection was varied to values up to η = 3.0). Additionally, for each beam a simulation without any shear connection was performed.

Span 7.2 m

Steel section IPE 270

Concrete slab width: bc = 1200 mm, height: hc = 150 mm, solid slab (no profiled steel sheeting)

Concrete properties fc = 35 N/mm², fct = 3.5 N/mm² Ec = 35000 N/mm², CONC+BISO, Ect = 1 N/mm²

Steel properties fy = 300 N/mm², Ea = 210000 N/mm², BISO, Eat = 1 N/mm²

Shear connectors load-slip curve cf. Figure 5.15, right. δu = 3/4/5/6/9 mm, η = 0.1 to 1.5 in steps of 0.1

Table 5.15 Overview of parameters for series C2

Span 7.2 m, 10.0 m, 15.0 m, 20.0 m

Steel section IPE 270, IPE 550

Concrete slab width: bc = 1200 mm, bc = 2400 mm,height: hc = 150 mm, hc = 200 mm, hc = 300 mm, solid slab (no profiled steel sheeting)

Concrete properties fc = 35 N/mm², fct = 3.5 N/mm² Ec = 35000 N/mm², CONC+BISO

Steel properties fy = 235 N/mm², fy = 300 N/mm², fy = 460 N/mm², Ea = 210000 N/mm², Eat = 1 N/mm², BISO

Shear connectors load-slip curve cf. Figure 5.15, right. δu = 1 to 15 mm in steps of 1 mm, η = 0.1 to 1.5 in steps of 0.1 (0.1 to 3.0 in simulation groups el1/2/3/4/5 and ep1/2/3/4/5).

Load type uniform load or 2 concentrated loads at one and two thirds of the span

Connector allocation evenly spaced, either over the whole length of the beam or in the outer thirds of the beam only

Table 5.16 Overview of parameters for series C3

Three-dimensional volume models

59

The behaviour of the shear interface is given in Figure 5.15 and Table 5.13. The shear resistance Pu of the shear interface was calculated according to the given degree of shear connection. The initial stiffness of the shear interface was usually to Si = 10000 kN/mm. In simulation groups el1, el2, el3, el4 and el5 the initial stiffness was varied between 10 kN/mm (group el1) and 100000 kN/mm (group el5).The simulation jobs are named e.g. „C3P_N_IPE270_D_72_6_70_S460“ using the scheme:<SER><PAR>_<PSS>_<PROF>_<LT>_<SPAN>_<DSP>_<DGR>_<SPC>_<GROUP>. An explanation and the used values for each variable are shown in Table 5.18.

Group Steel fy[N/mm²]

Concrete fc [N/mm²]

Concrete slab [mm²]

Steel section Span [m] Remarks

S235 235

351200 x 150

IPE

270,

IPE

550

7.2/15 m Steel yield strength 235 N/mm²

S300 300 7.2/10/15/20 m Steel yield strength 300 N/mm²

S460 460

7.2/15 m

Steel yield strength 460 N/mm²

outer3

300

Connectors placed in outer thirds of the span only

bc2400 2400 x 150 Concrete slab 2400 mm wide

fc50 50 1200 x 150 Concrete compressive strength 50 N/mm²

hc200

35

1200 x 200 Concrete slab 200 mm deep

hc300 1200 x 300 Concrete slab 300 mm deep

el1

1200 x 150

IPE2

70

7.2 m

Brittle connector behaviour, Si = 10 kN/mm

el2 Brittle connector behaviour, Si = 100 kN/mm

el3 Brittle connector behaviour, Si = 1000 kN/mm

el4 Brittle connector behaviour, Si = 10000 kN/mm

el5 Brittle connector behaviour, Si = 100000 kN/mm

ep1 Elasto-plastic connector δi = 6 mm, δu = 6 mm

ep2 Elasto-plastic connector δi = 1 mm, δu = 6 mm

ep3 Elasto-plastic connector δi = 0.1 mm, δu = 6 mm

ep4 Elasto-plastic connector δi = 0.01 mm, δu = 6 mm

ep5 Elasto-plastic connector δi = 0.001 mm, δu = 6 mm

Table 5.17 Overview of simulation groups for series C3

Variable Possible values

SER C3, C4

PAR Simulation parameter set. E = Simulation of beam tests performed at ETH [31], H = Simulation of beam tests performed at Hilti [39], P = Simulation using parametric data sets

PSS Allocation of a profiled steel sheeting. N = No profiled steel sheeting, Q = Profiled steel sheeting per-pendicular to the beam axis, L = Profiled steel sheeting parallel to the beam axis

PROF Type of profiled steel section IPE270, IPE550

LT Type of load distribution. 2 = 2 concentrated loads at L/3 and 2L/3, D = uniform load

Table 5.18 Explanation of nomenclature for simulation series C3 and C4

Numerical Investigations

60

The results of these simulations form part of the key findings and are shown in chapter 6 on page 63 and in the appendix. Some extracts of the results are also shown in [12].

5.3.6 Parametric investigation series C4

Parametric investigation series C4 is almost identical to series C3. As the only difference, series C4 simulates unpropped construction of the composite beam while series C3 simulates contin-uously propped construction. The scope of simulation series C4 is limited to group S300 and to spans of 7.2 m and 15 m.In order to simulate unpropped construction, initial stress is applied to the steel section of the composite beam. The initial stress data is determined as the stress present in the steel beam due to the loads due to self-weight of the concrete and the steel section. The results from these simulations form part of the key findings and are shown in chapter 6 on page 63 and in the appendix. Some extracts of the results are also shown in [12].

5.4 Evaluation: minimum degree of partial shear connection

From the results of the numerical simulation with the three-dimensional models series C3 and C4 the minimum degree of partial shear connection was determined for each parameter set.For this purpose, the ultimate loads obtained from the numerical simulations were compared to the ultimate loads calculated using the rigid-ideal plastic design method according to Eurocode 4 with all partial factors set to 1.0. The effects of the longitudinal reinforcement in compression and the concrete in tension were not considered in the analytical model. This com-

SPAN Span of beam in [dm]

DSP δu in [mm]

DGR η in [%]

SPC Every nth connector bears loads (usually SPC = 1 (each connector bearing))

GROUP Simulation group (no group = S300)

Figure 5.25 Comparison of bending resistances from numerical and analytical models for the cases of numerical simulations C3P_N_IPE270_2_72_x_xx_1_S300 (left) and C3P_N_IPE270_D_72_x_xx_1_S300 (right). Curve parameter is the deformation capacity of the shear connectors δu (varying from 1 mm to 15 mm).

Variable Possible values

Table 5.18 Explanation of nomenclature for simulation series C3 and C4

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50η [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50η [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_S300

Discussion of numerical models

61

pa r i son i s shown fo r s imu la t i ons C3P_N_IPE270_2_72_x_xx_1 and C3P_N_IPE270_D_72_x_xx_1 (cf. designations in section 5.3.5 on page 58) in Figure 5.25.In the next step, ηmin is determined as the minimum degree of partial shear connection at which the bending resistance obtained from the numerical model is equal to or greater than that deter-mined analytically (see vertical arrows in Figure 5.25).Alternatively, the determination of ηmin could also be performed on a level MR(FEM)/MR(Ana-

lytical) different from 1.0, e.g. on the level reached with η >> 1.0 (shown by a light grey, hori-zontal line in Figure 5.25) as was done in [12]. The rationale for this higher level of comparison is that composite beams with partial shear connection must exhibit the same ratio between the bending resistances from numerical and analytical calculations MR(FEM)/MR(Analytical) as com-posite beams with full shear connection. However, the ratio MR(FEM)/MR(Analytical) is typically small. Further, the numerical simulations typically show a slight increase of the bending resist-ance as the deformation is further increased (cf. e.g. Figure 5.20, left) due to hardening with a tangent modulus of elasticity of 1 N/mm² in the plastic domain of the bilinear steel material def-initions. This has an effect on bending resistance for composite beams with no and full shear connection only. For composite beams with partial shear connection the bending resistance de-creases to values lower than bending resistance before these hardening effects become consid-erable.Therefore, ηmin is determined at a comparison level of MR(FEM)/MR(Analytical) = 1.0.

5.5 Discussion of numerical models

5.5.1 Two-dimensional beam models

By means of two-dimensional beam models an appropriate simulation of the load-bearing be-haviour is difficult to obtain with the computer program used [1]. Initial stiffness and ultimate load are predicted accurately for some cases, but the deformations with nonlinear behaviour dif-fer considerably between the numerical models and the beam tests.The truss models exhibit coupling of the horizontal and vertical shear forces. Further, with truss models the chord forces are not directed vertically between the chords but along the diagonals from a position nearer to the middle of the beam to a node nearer to the support. Thus, the chord forces and the deformations exhibit a principal error. With the cross-shaped model this error is excluded. Further, the diagonals of the truss models may also stiffen the beam in bending. With the cross-shaped truss this effect is very strongly observed. The other models only show this ef-fect in the middle of the beam.The investigated frame models do not couple vertical and horizontal shear forces and do not stiffen the beam excessively in bending. On the other hand, the horizontal shear forces are in-troduced eccentrically into the chords and thus create local moments of eccentricity. Also, the H-shaped frame does not transfer the shear forces vertically and thus exhibits the same principal error as the truss models.The I-shaped frame directs the chord forces vertically between the chords. The verticals are stiff in bending over the whole depth of the frame which considerably reduces the local moments of eccentricity. The high bending stiffness of the verticals also forces the two chords to remain par-allel which stiffens the beam in bending.For the elastic cases the stiffness of a composite beam is well-reflected by many models.

Numerical Investigations

62

Due to restrictions in the computer program [1] the load-bearing behaviour with consideration of the material nonlinearities is only poorly approximated by the investigated beam models. This is due to the fact that the chords are assumed to exhibit linear elastic-ideal plastic behav-iour. However, the ultimate loads are approximated well for some cases.The cross-shaped truss is a good way to simulate the load-bearing behaviour of a composite beam, as long as the shear connection between the chords is not too stiff. The I-shaped frame exhibits excellent behaviour for all investigated cases.

5.5.2 Three-dimensional volume models

By means of three-dimensional solid models the load-bearing behaviour of composite beams with consideration of nonlinear material and connector behaviour can be described in an excel-lent manner. However, these models require a very large computational effort.Brittle material and connector behaviour often causes numerical convergence problems, which can be avoided to some extent by appropriately choosing the solver parameters and smoothing the material behaviour in the numerical model.Creation, solution, post-processing and evaluation of all jobs was performed using a custom-made macro programme suite (cf. [11]). This macro programme suite is recommended for de-tailed analysis of composite beams in both research and design.

5.5.3 Asymmetrical effects

All numerical investigations were performed with half-span models exploiting symmetry con-ditions. Thus, asymmetrical effects are not considered in any of the numerical models shown previously. However, the failure of a composite beam with partial shear connection in a building and in a laboratory test is generally asymmetric, as a composite beam is never perfectly sym-metrical in geometry, loading and material properties. Therefore, the interface will fail on one side only (the side on which its resistance is a little smaller than on the other side), while the other side does not fail. In the numerical simulations this effect is not considered and symmet-rical failure is assumed (cf. Figure 5.26). Thus, the deflections at mid-span after ultimate load obtained from the numerical models are greater than the deflections observed in laboratory tests. However, this has no effect on the behaviour of the composite beam before and at ultimate limit state.

Figure 5.26 Symmetrical and asymmetrical failure in tests and simulations. In buildings and in laboratory tests asymmetrical failure occurs while the finite element model as-sumes symmetrical failure.

Overview

63

6 Synthesis

6.1 Overview

This chapter comprises the synthesis of all findings of the research shown in the present report.

1) In composite beams with partial shear connection the critical cross-section is not always located at the place where the maximum bending moment occurs. Therefore, a simple method to determine the position of the critical cross-section is given in section 6.2 on page 63.

2) With evenly-spaced non-ductile shear connectors the degree of partial shear connection required for full shear connection is greater than η = 1.0. The actual required degree of partial shear connection is discussed in section 6.3 on page 67.

3) In section 6.4 on page 69 the application range for the rigid-ideal plastic design method according to Eurocode 4 [26] is re-defined and extended. Basically, two criteria are rec-ognised: the deformation capacity criterion and the ductility criterion. The ductility (or de-formation capacity, respectively) available with a shear connector at a chosen value of PRk is compared to the ductility (or deformation capacity, respectively) required by the com-posite beam in the case of no shear connection. The minimum degree of partial shear con-nection ηmin is then derived from the minimum of these two ratios.

4) In section 6.6 on page 78 the re-defined application range determined in section 6.4 on page 69 for simply-supported composite beams is applied and generalised to continuous composite beams.

5) In section 6.7 on page 80 the findings in this chapter are summarised on the basis of an application example.

6.2 Position of the critical cross-section in composite beams

Bending failure in a beam occurs at the first position where the bending moment caused by all actions becomes larger than the bending resistance at the respective place. Usually, beams in steel or concrete construction show a constant bending resistance over their whole length. Fail-ure will then occur at the place where the bending moment is maximum. For the typical case of symmetrical loading failure this occurs at the centre of the beam. In composite beams the bending resistance is not constant over the length of a member. While the bending resistance of the steel beam is constant over the length of the beam, the bending resistance from composite action varies from zero at the support to maximum at the mid-span of a simply-supported beam. Therefore, in some cases, failure occurs in a different place and at lower loads than predicted using basic beam theory with constant bending resistance over the length of the beam.The following parameters are recognised to be of key importance to the position of the critical cross-section:

Synthesis

64

• The ratio Ma/MR,mid between the bending resistance of the pure steel section and the bending resistance of the composite beam at mid-span. As the bending resistance at mid-span depends on the degree of partial shear connection η, also the location of the critical cross-section is influenced by η.

• The longitudinal distribution of shear forces in the interface. Below, the two most-commonly used distributions are used: the linear elastic distribution and the constant distribution as used in the rigid-plastic design method.

• The longitudinal distribution of loads. Below the cases of concentrated loads and of uniform load are discussed.

6.2.1 Position of the critical cross-section with concentrated loads

The longitudinal distribution of bending moment from concentrated loads exhibits a polygonal shape. If the concentrated loads are dominant compared to a uniform load (e.g. self-weight) the critical cross-section is located at one of the points where a concentrated load is applied.

Figure 6.1 Distribution of bending resistance between the support (x/L = 0) and mid-span of the beam (x/L = 0.5) assuming rigid-ideal plastic behaviour of the interface in nu-merical simulations series C3, group S235. Composite beams with steel sections IPE 270 (left) and IPE 550 (right) η between 0 and 1.5.

Figure 6.2 Position of the critical cross-section in the composite beams used for finite element simulations series C3 for 2 concentrated loads (left) and for uniform load (right). For composite beams subjected to concentrated loads the critical cross-section is located in the position where the load is introduced. For composite beams subject-ed to uniform load the location of the critical cross-section is described by the sim-plified model shown in equation (6.1).

00 0.1 0.2 0.3 0.4 0.5

x/L [-]

MR [k

Nm

]

η = 1.5, 2 concentrated loads @ L/3

η = 0

Ma

Mpl

00 0.1 0.2 0.3 0.4 0.5

x/L [-]

MR [k

Nm

]

η = 1.5, 2 concentrated loads @ L/3

η = 0

Ma

Mpl

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5xcrit/L [-]

Ma/M

R,m

id [-

]

Non-simplified evaluation of all beams used in numerical simulations series C3 with two concentrated loads at L/3.

With partial shear connection the critical cross-section is located at the position where the concentrated load is introduced.

Without shear connection and with full shear connection the critical cross-section is located at mid-span.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5xcrit/L [-]

Ma/M

R,m

id [-

]

Non-simplified evaluation for all beamsused in numerical simulations seriesC3 with uniform load

Simplified model for uniform load

Minimum ratio Ma/Mpl

according to Eurocode 4

Position of the critical cross-section in composite beams

65

6.2.2 Position of the critical cross-section with uniform load

The longitudinal distribution of bending moment from a uniform load is parabolic. The location of the critical cross-section depends on the material behaviour of the chords.

6.2.2.1 Elastic material and interface behaviour

With elastic behaviour of all beam components the longitudinal distribution of axial force in the chords shows a parabolic distribution. The height of concrete in compression increases from zero at the support towards the centre of the beam, also following a second order polynomial (parabola). Therefore, the vertical distance between the two resulting axial forces in the concrete slab and the steel beam decreases from the support towards the centre of the beam. Thus, the bending resistance of the composite beam increases less than quadratic and thus less than the bending moment from a uniform load from the support towards the centre of the beam. Therefore, the critical cross-section for elastic beams under uniform load may be assumed to be located in the centre of the composite beam.

6.2.2.2 Plastic material and interface behaviour

In the plastic design of composite beams the shear force in the interface is assumed to be con-stant from the support to the critical cross-section. Constant distribution of shear forces in the interface results in a linear increase of axial force in the chords from zero at the support to a maximum towards the centre of the beam. Thus, the height of concrete in compression also shows a linear increase from the support towards the centre of the beam. Therefore, the bending resistance from composite action with plastic material and interface behaviour increases a little less than linearly from the support towards the centre of the beam. However, the height of the concrete slab in compression is usually rather small compared to the vertical distance of the re-sulting axial forces in the concrete slab and the steel beam. Therefore, in order to develop a sim-plified model, the height of the concrete slab in compression is assumed to be constant over the whole length of the beam. Thus, the bending resistance from composite action shows a linear increase from zero at the support (cf. Figure 6.3, right).In the worst case the bending resistance of the steel beam Ma is zero and the critical cross-sec-tion is located at L/4. With Ma/Mmid = 1.0 the critical cross-section is located at L/2. A detailed analytical evaluation of the composite beams used for the numerical analysis series C3 (cf. sec-

Figure 6.3 Position of the critical cross-section in a composite beam with elastic (left) and rigid-plastic (right) interface behaviour.

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 0.25 0.5 0.75 1x/L [-]

M/M

mid

[-] MR,elastic ME(q)

MR,elastic-ME(q)Critical cross-section

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 0.25 0.5 0.75 1x/L [-]

M/M

mid

[-]

MR,plasticME(q)

Critical cross-sectionsMR,plastic-ME(q)

Synthesis

66

tion 5.3.5 on page 58 ff) shows that for ratios Ma/Mmid between 0 and 1.0 a linear interpolation as given in equation (6.1) is justified (cf. Figure 6.2, right).

(6.1)

In Eurocode 4 [26] the effect of an eccentric critical cross-section for a uniform load is neglect-ed. On the other hand, the ratio Ma/Mpl must be greater than 0.4 (cf. Figure 6.2). According to the simplified model in equation (6.1) the critical cross-section is located at xcrit = 0.35 Le for Ma/Mpl = 0.4. Thus, the shear resistance of the interface is up to 30 % less than predicted by Eurocode 4 with the critical cross-section assumed at mid-span.Therefore, it is recommended not to neglect the eccentricity of the critical cross-section in com-posite beams with partial shear connection subjected to uniform load when using the rigid-ideal plastic design method.

6.2.3 Position of the critical cross-section with large values of η

If the degree of partial shear connection in a composite beam is smaller than or equal to η = ηf, the maximum bending resistance is reached at the centre of the beam. From the centre of the beam towards the support, the bending resistance decreases. With η > ηf the axial forces in the chords are built up over a shorter length. Thus, the bending resistance of the composite beam in full shear connection is reached closer to the support. Be-tween the positions where full bending resistance is reached, the bending resistance is constant, limited by the axial resistance of the concrete slab or the steel beam (cf. Figure 6.4).

With elastic interface behaviour (found in particular with a non-ductile shear connection) this has no effect on the position of the critical cross-section. However, with rigid-ideal plastic in-terface behaviour (typically found with ductile shear connectors and thus ηf = 1.0 (cf. section 6.3 on page 67)) the critical cross-section is located either closer to the centre of the beam if Ma/Mmid is sufficiently large or closer to the support if Ma/Mmid is rather small. With degrees of partial shear connection η = 2.0 or greater the critical cross-section is always located at the centre of the beam because the initial slope of a parabola is twice the mean slope.However, with values of η greater than ηf the bending resistance of a composite beam in the critical cross-section is higher than with lower values of η. Therefore, as a safe approach, the

Figure 6.4 Position of the critical cross-section in a composite beam with η = 2.0 and rigid-ideal plastic interface behaviour.

xcritLe4

------ 1MaMpl---------+⎝ ⎠

⎛ ⎞⋅=

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 0.25 0.5 0.75 1x/L [-]

M/M

mid

[-]

MR,plastic

η = 2.0

ME(q)

Critical cross-sectionMR,plastic(η=2.0)-ME(q)

The degree of partial shear connection required for full shear connection

67

position of the critical cross-section can be determined as for a composite beam with full shear connection with η = ηf.

6.3 The degree of partial shear connection required for full shear connection

6.3.1 Overview

With elastic design of the shear interface the shear connectors are placed such that the interface is stronger in places where the shear flow t(x) is larger. This is usually done by varying the spac-ing between the shear connectors over the shear length of a composite beam.With plastic design of the shear interface, shear connectors are evenly spaced over a whole shear length, e.g. between the support and the critical cross-section. The required shear resistance of a shear connector is then determined according to the mean shear flow in the interface as

with as given in equation (6.2). is the maximum shear flow in the interface.

and are the maximum shear flow due to a uniform load q and to concentrated loads Q, re-spectively (cf. Figure 6.5).

(6.2)

If ductile shear connectors are used in a composite beam, full shear connection is reached with a degree of partial shear connection η = 1.0. With non-ductile shear connectors the degree of partial shear connection required for full shear connection is larger. With shear connectors whose ductility is much smaller than the ductility required by the com-posite beam, elastic design of a composite beam is in many cases more economical than plastic design. If a composite beam is designed using the rigid-ideal plastic method with evenly-spaced shear connectors in spite of the plastic method‘s inefficiency in this case, all shear connectors must show enough shear resistance to resist the maximum shear force (cf. Figure 6.5).

6.3.2 Rigid-brittle shear connection

With rigid-brittle shear connectors (deformation capacity δu = 0 and thus ductility δu - δi = 0, cf. Figure 2.4, left, type a)) no redistribution of shear forces is possible at all. In this case the shear forces in the interface are proportional to the longitudinal distribution of the vertical shear force Q(x) (cf. Figure 6.5).In composite beams subjected to just one or two concentrated loads the shear force is constant and no redistribution of shear forces is required. Full shear connection is thus established with a degree of partial shear connection of ηf = 1.0.In composite beams subjected to uniform load the shear forces exhibit a triangular distribution over the length of the beam. The maximum shear force is twice the mean shear force. Thus the degree of partial shear connection required for full shear connection is ηf = 2.0. However, if the critical cross-section is not located at the centre of the beam‘s field but somewhat eccentric, the required degree of partial shear connection is smaller as the difference in shear force between the most-loaded and the least-loaded connector becomes smaller (cf. equation (6.3)).For other load distributions analogous considerations apply.

t

PRk t≥ ∆c⋅ t t̂ t̂ q( )

t̂ Q( )

t t̂ Q( ) t̂ q( ) 1LsL-----–⎝ ⎠

⎛ ⎞⋅+=

t̂

Synthesis

68

For composite beams subjected to a combination of concentrated and distributed loads, full shear connection requires a degree of partial shear connection ηf between 1.0 and 2.0.

The degree of partial shear connection required for full shear connection depends on the shear resistance of the connector at which the ductility δu - δi is zero, and thus on the ultimate shear resistance Pu of a shear connector, not on the characteristic value PRk. Therefore, the term shown in equation (6.3) is multiplied by the factor PRk/Pu. This implies that the lower the ratio PRk/Pu, the lower is the required degree of partial shear connection for full shear connection.The degree of partial shear connection required for full shear connection with rigid-brittle shear connectors is thus determined as shown in equation (6.3).

with (6.3)

• Q(uniform): Vertical shear force due to uniform load q at the support of the composite beam: Q(uniform) = q*L/2.

• Q(concentrated): Vertical shear force due to concentrated loads Q at the support of the com-posite beam: Q(concentrated) = Q*(number of concentrated loads applied to the beam)/2.

If a composite beam yields, a plastic hinge develops at the critical cross-section. In the region of this critical cross-section large strains develop in the chords. Therefore, relatively large, more localised slip occurs at this location. If shear connectors with very small deformation capacity are used, the shear connectors in the vicinity of the critical cross-section may fail immediately after yield. Thus, the composite beam fails at the load at which the steel beam starts to yield, and the plastic bending resistance is not reached. A more detailed analysis of composite beams with shear connectors exhibiting very small de-formation capacity is beyond the scope of this work and is reported in [62] and [63].

6.3.3 Flexible brittle shear connection

With flexible brittle shear connectors (deformation capacity δu > 0 and ductility δu - δi = 0, cf. Figure 2.4, left, type b)) no plastic redistribution of the shear forces is possible, but only elastic redistribution.

Figure 6.5 Longitudinal distribution of shear forces with rigid-brittle shear connectors (left), Real and 3 examples of characteristic load-slip behaviour of a shear connector (cf. Figure 2.1) (right).

δPu

δ i,1δu,1

δu,2δi,2δu,3

δi,3

Slip [mm]

Shea

r for

ce [k

N]

Real behaviourCharacteristic behaviour 1Characteristic behaviour 2Characteristic behaviour 3

PRk,3

PRk,2

PRk,1

Pu

Si,1

1

ηf rigid,t̂t-

PRkPu

---------== 1Q uniform( )

LsL 2⁄----------⋅

Q uniform( ) Q concentrated( )+---------------------------------------------------------------------------------+

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

⋅

Application range for the rigid-ideal plastic design method

69

(Elastic redistribution of the shear forces is observed when comparing the two diagrams in Fig-ure 6.6, where the longitudinal distribution of slip (and thus of shear flow for linear elastic con-nector behaviour) is shown. In the left diagram the distribution of slip is shown for a rather stiff shear connection (γ = 0.98, k = 104 N/mm²), while in the left diagram the same data is shown for a rather soft shear connection (γ = 6.3*10-7, k = 10-4 N/mm²). The longitudinal distribution of slip and shear flow (and thus the shear forces) changes as the stiffness of the shear interface changes. This effect is termed elastic redistribution.)In this case (with flexible brittle shear connectors) not all shear connectors reach their ultimate load at the same time. Therefore, the bending resistance of the composite beam is lower than determined using the rigid-plastic design method, and a higher degree of partial shear connec-tion is required for full shear connection. With shear connectors of very low stiffness the longitudinal shear forces are distributed over the length of a beam as shown in Figure 6.8 for a composite beam without shear connection. As the stiffness of the shear interface increases, the distribution of the longitudinal shear forces be-comes more and more similar to that shown in Figure 6.5. A comparison for elastic behaviour of all components using the differential equation is shown in Figure 6.6.

Thus, the required ductility of a composite beam with flexible shear connectors is smaller than with rigid shear connection if the beam is subjected to uniform load. On the other hand, with concentrated loads the required ductility is greater with flexible shear connection than with rigid shear connection. The relative peak shear force is maximum for rigid shear connection with uniform loads. For concentrated loads the relative peak shear force increases as the stiffness of the interface decreases.

6.4 Application range for the rigid-ideal plastic design method

6.4.1 Overview

In the following the minimum degree of partial shear connection for the application of the rigid-ideal plastic design method according to Eurocode 4 [26] is re-defined based on the comparison between the slip occurring in the composite beam to be designed or assessed and the deforma-tion capacity and ductility of the shear connector used.

Figure 6.6 Longitudinal distribution of slip in the beam mentioned in Figure 2.2 for k = 104 N/mm² (left) and k = 10-4 N/mm² (right) determined using the differential equation for linear elastic composite action

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.00 0.25 0.50 0.75 1.00x/L [-]

d(x)

[mm

]

d(x),uniform

d(x),concentrated

-2.0-1.5-1.0-0.50.00.51.01.52.0

0.00 0.25 0.50 0.75 1.00x/L [-]

d(x)

[mm

]

d(x),uniform

d(x),concentrated

t̂ t⁄

t̂ t⁄

Synthesis

70

The description of the shear connector‘s behaviour is given in section 2.1.1 on page 5. The de-termination of slip in a composite beam with partial shear connection under consideration of nonlinear material and connector behaviour is rather complicated. Therefore, slip is determined analytically for composite beams without shear connection (cf. section 6.4.3.1 on page 71). Fur-ther, some indications on slip are given for composite beams with full shear connection (cf. sec-tion 6.4.3.2 on page 74). In composite beams with partial shear connection, slip is expected to be greater than with full shear connection and smaller than without shear connection. The results of the finite element simulations justify the assumption that slip in a composite beam exhibits a linear decrease as the degree of shear connection increases (cf. section 6.4.3.3 on page 74).

6.4.2 Design criteria

6.4.2.1 Application range in Eurocode 4 [26] for the rigid-plastic design method [26]

In Eurocode 4 a shear connector is termed „ductile“ if its characteristic deformation capacity is at least δuk = 6 mm. With „ductile“ shear connectors, the rigid-ideal plastic design method can be used if the degree of partial shear connection is greater than ηmin determined according to equation (3.36) for composite beams with steel beams having equal flanges, or with a similar formula defined in [26] with unequal flanges.

6.4.2.2 Re-defined application range for the rigid-plastic design method [26]

The design criteria in the work reported here is re-defined compared to Eurocode 4. The defini-tion of a shear connector‘s ductility is adapted as described in section 2.1.1.5 on page 6 and in section 2.1.2.10 on page 8.A composite beam can be designed using the rigid-plastic design method [26] if both of the fol-lowing criteria are fulfilled:

1) The deformation capacity δu(PRk) of the shear connector is greater than or equal to the required deformation capacity for a shear connector in the beam at the chosen degree of partial shear connection (see below, equation (6.19)).

2) The ductility δu(PRk) - δi(PRk) of the shear connector is greater than or equal to the re-quired ductility for a shear connector in the beam at the chosen degree of partial shear con-nection (see below, equation (6.18)).

6.4.3 Deformation behaviour of a composite beam

The deformation of a composite beam is modelled as the sum of a linear-elastic and a rigid-plas-tic part. The elastic part of the deformation is described by the differential equation for elastic composite action (cf. section 3.3 on page 18) or by approximation methods such as the γ-method (cf. section 3.5.2 on page 23). The plastic behaviour of a composite beam is characterised by a plastic hinge at the critical cross-section, while the other parts of the beam remain in an elastic state (cf. Figure 6.7). The plastic hinge is assumed to be discrete (of zero length). In reality, the plastic hinge has a certain length that depends on the material behaviour of the steel beam. How-ever, not the longitudinal strain at each position, but only the integral of the longitudinal strain over the length is of interest for the model developed in this chapter (cf. equations (3.10) and (6.4)).

Application range for the rigid-ideal plastic design method

71

6.4.3.1 Slip in a composite beam without shear connection

In a composite beam without shear connection the elastic part of deformation leads to unevenly-distributed slip over the length of the beam. This difference in slip between the maximum slip and the slip in the critical cross-section governs the minimum ductility (δu-δi) of a shear con-nector.The plastic part of the deformation causes a constant slip from the plastic hinge to the support. The total slip at the support from elastic and plastic deformations governs the deformation ca-pacity δu of the shear connector.Slip is determined using equation (3.10). For symmetrical load distributions equation (3.10) can be re-written as shown in equation (6.4).

(6.4)

In composite beams without shear connection the neutral axes are at a constant vertical position over the length of the beam. The neutral axis of the concrete slab is assumed to be located on the upperside of the concrete slab (cf. equation (6.5)), while the neutral axis of the steel beam is located in its centroid. For steel sections with equal flanges the neutral axis of the steel beam is given in equation (6.6).

(6.5)

(6.6)

For linear elastic material behaviour strain distribution is similar to the distribution of bending moment. For this case, equation (6.4) simplifies to equation (6.7) with α defined as shown in equation (6.8).

(6.7)

Figure 6.7 Elastic and plastic behaviour of a composite beam. The plastic behaviour is char-acterised by plastic hinges (assumed as discrete) at the critical cross-sections, while the other parts remain in an elastic state. Slip due to the plastic part of de-formation is zero between the critical cross-sections at mid-span and constant be-tween the critical cross-sections and the supports.

d x( )x2

2

∂

∂ w x( ) z0 c, x( ) z0 a, x( )–{ }⋅ xdx

L2---

∫=

z0 c, x( ) h=

z0 a, x( )ha2-----=

dend α L2--- εc,inf

L2---⎠⎞ εa,sup

L2---⎝ ⎠⎛ ⎞–⎝

⎛⎝ ⎠⎛ ⎞⋅ ⋅=

Synthesis

72

(6.8)

Values α are given in Table 6.1 for special cases of load distribution.

When the steel beam yields the strain on the upperside of the steel beam is equal to the value given in equation (6.9) (left). Strain at the underside of the concrete slab is then determined as shown in equation (6.9) (right).

and (6.9)

If a beam is erected in unpropped construction the stress in the steel beam at the moment when the concrete is cast is not zero. The shear interface must therefore not resist the slip due to the whole yield strength fy,a, but only due to the stress value fy,a - σa,c. If stress during casting has the same sign as the stress at the yield state, slip at yield is reduced. On the other hand, if stress during casting is of opposite sign to that at the yield state, slip will be larger than after continu-ously-propped construction.End slip when the steel beam yields is therefore determined as shown in equation (6.10). Slip at the critical cross-section is given in equation (6.11) for one or two symmetrically-placed con-centrated loads and in equation (6.12) for a uniform load.

(elastic end slip at yield with η = 0) (6.10)

(concentrated loads) (6.11)

(uniform load) (6.12)

After the steel beam yields, a plastic hinge develops at the critical cross-section, but further elas-tic deformations occur. The maximum elastic deformation occurs when ultimate load of the beam is reached. The elastic end slip at the ultimate limit state is therefore determined by mul-tiplying the slip at yield by Mpl,a/My,a as shown in equation (6.13). Slip at the critical cross-sec-tion (equations (6.11) and (6.12)) can be scaled in the same way, which results in equations (6.14) and (6.15).

Distribution of load

α α = 1/2 α = 1-a/L α = 2/3 α = 2/π

Distribution of bending moment and of longitudi-nal strain

Table 6.1 End slip determination from strain for special load distributions

αM x( ) xd

0

L∫L Mmax⋅

--------------------------=

εa sup y, ,f– y a,Ea

------------= εc inf y 0, , , ε– a sup y, ,hc ∆+ha 2⁄

---------------⋅=

dend el y 0, , , α L2---

fy a, σa c,–Ea

------------------------- 1hc ∆+ha 2⁄

---------------+⎝ ⎠⎛ ⎞⋅ ⋅ ⋅=

dcrit el y 0, , ,L2--- Ls–⎝ ⎠⎛ ⎞ fy a, σa c,–

Ea------------------------- 1

hc ∆+ha 2⁄

---------------+⎝ ⎠⎛ ⎞⋅ ⋅=

dcrit el y 0, , ,23--- 1

3---

4 Ls L Ls–( )⋅ ⋅

L2--------------------------------------⋅+

⎝ ⎠⎜ ⎟⎛ ⎞ L

2--- Ls–⎝ ⎠⎛ ⎞ fy a, σa c,–

Ea------------------------- 1

hc ∆+ha 2⁄

---------------+⎝ ⎠⎛ ⎞⋅ ⋅ ⋅=

Application range for the rigid-ideal plastic design method

73

(elast. end slip at ultimate load, η = 0)(6.13)

(concentrated loads) (6.14)

(uniform) (6.15)

Slip due to plastic deformation is constant between the critical cross-section and the support. At mid-span it is zero under symmetrical conditions. In order to reach the full plastic bending re-sistance in the plastic hinge, further rotation is necessary. However, both the required rotation and the length over which the plastic hinge develops are difficult to determine, as they depend mainly on the hardening behaviour of the steel. Therefore, the end slip at the ultimate load was determined from the results of the numerical simulations. In the numerical simulations of the beam tests shown in [31] and [39] (cf. section 5.3.2 on page 48) it was found that in composite beams without shear connection the full plastic bending resistance is reached at a slip which equals 1.5 times the slip at yield (cf. Figure 5.20, right and Figure 5.21, right). In these figures slip for η = 0 is shown as determined from the analytical assumption and from numerical sim-ulation. The first corner in the analytical curves stands for the yield point. The second corner (when reaching the horizontal plateau in analytical load-slip curves stands for the slip when the beam reaches the ultimate limit state (ULS).End slip when the steel beam reaches its full plastic bending resistance is therefore assumed as shown in equation (6.16).

(end slip at ultimate load, η = 0) (6.16)

For each connector to reach its full characteristic shear resistance, the total slip at the outer side of the critical cross-section must be at least as much as the initial slip of the shear connectors used in the composite beam. Therefore, if shear connectors with a large initial slip δi are used, the amount of slip due to plastic deformation and thus the maximum total end slip at the end of the beam must be further increased to dend,pl,u,0 = δi + dend,el,u,0 - dcrit,el,y,0. Therefore, dend,pl,u,0 must be determined as given in equation (6.17).

(6.17)

Figure 6.8 gives an overview on the longitudinal distribution of slip in a composite beam with-out shear connection under symmetrical loads. The total slip is the sum of the elastic and the rigid-plastic part. With two concentrated loads slip shows a linear distribution at mid-span of the beam between the critical cross-sections (e.g. L/3 and 2 * L/3). Between the critical cross-section and the sup-port the longitudinal distribution of slip is parabolic. With uniform load the longitudinal distri-bution of slip follows a third order polynomial both between the two critical cross-sections and from the critical cross-section to the support.

dend el u 0, , , α L2---

Mpl a,My a,-------------- fy a,⋅ σa c,–

Ea-------------------------------------------- 1

hc ∆+ha 2⁄

---------------+⎝ ⎠⎛ ⎞⋅ ⋅ ⋅=

dcrit el u 0, , ,L2--- Ls–⎝ ⎠⎛ ⎞

Mpl a,My a,-------------- fy a,⋅ σa c,–

Ea-------------------------------------------- 1

hc ∆+ha 2⁄

---------------+⎝ ⎠⎛ ⎞⋅ ⋅=

dcrit el u 0, , ,23--- 1

3---

4 Ls L Ls–( )⋅ ⋅

L2--------------------------------------⋅+

⎝ ⎠⎜ ⎟⎛ ⎞ L

2--- Ls–⎝ ⎠⎛ ⎞

Mpl a,My a,-------------- fy a,⋅ σa c,–

Ea-------------------------------------------- 1

hc ∆+ha 2⁄

---------------+⎝ ⎠⎛ ⎞⋅ ⋅ ⋅=

dend pl u 0, , , α L2---

1.5 f⋅ y a, σa c,–Ea

------------------------------------- 1hc ∆+ha 2⁄

---------------+⎝ ⎠⎛ ⎞⋅ ⋅ ⋅=

dend pl u 0, , , max α L2---

1.5 f⋅ y a, σa c,–Ea

------------------------------------- 1hc ∆+ha 2⁄

---------------+⎝ ⎠⎛ ⎞⋅ ⋅ ⋅ δi d+ end el u 0, , ,; dcrit el y 0, , ,–⎝ ⎠

⎛ ⎞=

Synthesis

74

At the critical cross-section the part of slip due to plastic deformation develops. In nature, the plastic hinge located at the critical cross-section exhibits a certain length, and plastic slip is built up from zero at the inner side of the plastic hinge to its maximum value at the outer side of the plastic hinge. However, in this simplified model the length of the plastic hinge is assumed to be zero, and slip therefore undergoes a sudden change at the critical cross-section (cf. Figure 6.8).

6.4.3.2 Slip in a composite beam with full shear connection

In a composite beam with full shear connection some slip may still occur, even though it is much smaller than without shear connection. Slip in the composite beam can still be determined using equation (6.4), but the vertical positions of the neutral axes are different from a composite beam without shear connection and are not constant over the length of a beam. At the support, the neutral axes are located at the same positions as for a composite beam with-out shear connection. At the critical cross-section the vertical positions of the neutral axes change as the beam is gradually loaded. The exact positions of the neutral axes and their longi-tudinal distributions depend on the nonlinear behaviour of the chords and the shear interface and must be determined using the differential equation for nonlinear composite action (section 3.4 on page 20) or a finite element analysis. As these methods require a large computational effort, simple assumptions are necessary. For the elastic behaviour of all components, the exact positions can be determined e.g. using the γ-method (cf. section 3.5.2 on page 23). For a very soft shear connection the vertical distance between the two neutral axes is large (maximum in the case of no shear connection). For rigid shear connection the two neutral axes coincide.

6.4.3.3 Slip in a composite beam with partial shear connection

Slip in a composite beam with partial shear connection is difficult to determine by means of an-alytical methods if nonlinear component behaviour is considered. Nevertheless, it is quite plau-sible that slip in a composite beam with partial shear connection is smaller than in a composite beam without shear connection and larger than in a composite beam with full shear connection. As a greatly simplified model, slip at yield is assumed to exhibit a linear decrease as the degree of partial shear connection η increases.

Figure 6.8 Longitudinal distribution of bending moment and slip for concentrated load at L/3 (left) and for uniform load (right). Results from analytical and numerical models. The longitudinal distribution of slip at yield according to the analytical model agrees well with the results from the numerical simulations series C3.

00.20.40.60.8

11.21.41.6

0 0.1 0.2 0.3 0.4 0.5x/L [-]

[-]

Bending Moment M(x)Elastic part of slip d,el(x)Plastic part of slip d,pl(x)Total slip d(x)

Slip at yield in all numericalsimulations of series C3 with concentrated loads at L/3

d(x)=del(x)+dpl(x)

00.20.40.60.8

11.21.41.6

0 0.1 0.2 0.3 0.4 0.5x/L [-]

[-]

Bending moment M(x)Elastic part of slip d,el(x)Plastic part of slip d,pl(x)Total slip d(x)

Slip at yield in all numericalsimulations of series C3 with uniform load

d(x)=del(x)+dpl(x)

Application range for the rigid-ideal plastic design method

75

Slip at the support („end slip“) at the ultimate load is governed by the deformation capacity δuof the shear connectors and therefore remains constant at dpl,u = δu up to full shear connection ηf. For η > ηf slip at ultimate load exhibits a linear decrease.

Figure 6.9 End slip at the ultimate load in numerical simulation series C3, groups S235, for two concentrated loads at L/3 (left) and uniform load (right) with deformation ca-pacities δu varying between 1 mm and 15 mm. The curves show that slip at the ul-timate load is governed by the deformation capacity δu of the shear connector. For low degrees of partial shear connection some simulations were aborted pre-maturely. Therefore, end slip at the ultimate load is also too small. In the simulations of composite beams without shear connection, the bending mo-ment is increased further as the deformation is increased, because bilinear mate-rial behaviour was assumed for the concrete slab and the steel beam. Thus the ultimate load is reached at the end of the simulation and the slip at the ultimate load is the slip at the end of the simulation.

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_S235

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_S235

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_S235

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_S235

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_S235

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_S235

Synthesis

76

6.4.4 The minimum degree of partial shear connection

6.4.4.1 Design method

The rigid-ideal plastic design method according to Eurocode 4 can be used for the design of a composite beam if the degree of partial shear connection is greater than or equal to the minimum degree of partial shear connection, as explained in section 6.4.2 on page 70.The ductility criterion is considered in equation (6.18) and the deformation capacity criterion in equation (6.19).

(ductility criterion) (6.18)

(deformation capacity criterion) (6.19)

In order to combine both criteria in one formula, a new variable θ containing the governing cri-terion is introduced (cf. equation (6.20)).

(6.20)

The minimum degree of partial shear connection ηmin is then determined as shown in equation (6.21).

(minimum degree of partial shear connection) (6.21)

Figure 6.10 and Figure 6.11 show the design model and a comparison to the results from the numerical simulations.

Figure 6.10 Design diagram for the minimum degree of partial shear connection. Values from results of numerical simulations series C3 (left) and C4 (right). PRk/Pu = 0.9. All evaluations were performed at the critical cross-section according to equation (6.1). In the numerical simulations with two concentrated loads the shear connec-tors in the middle part of the composite beam (between the critical cross-sections) make a considerable contribution to the shear resistance of the interface. There-fore, the design model for ηmin in equation (6.21) is more conservative for these beams. In simulation groups the three outer shear connectors were placed in the outer thirds only. These beams agree well with the analytical model. For θ-values below approximately θ = 0.2 ηmin is greater than predicted with equation (6.21).

ηmin 1δu δi–

dend el u 0, , , dcrit el y 0, , ,–-----------------------------------------------------------–≥

ηmin 1δu

dend pl u 0, , ,---------------------------–≥

θ minδu

dend pl u 0, , ,---------------------------

δu δi–( )dend el u 0, , , dcrit el y 0, , ,–( )

----------------------------------------------------------------;⎝ ⎠⎛ ⎞=

ηmin 1 θ–=

020406080

100120140

0 0.5 1θ [-]

ηmin

[%]

Filled dots: simulations with two concentrated loadsHollow dots: simulations with uniform load

020406080

100120140

0 0.5 1θ [-]

ηmin

[%]

IPE270 2 7.2 S300IPE270 D 7.2 S300IPE270 2 15 S300IPE270 D 15 S300IPE550 2 7.2 S300IPE550 D 7.2 S300IPE550 2 15 S300IPE550 D 15 S300

Application range for the rigid-ideal plastic design method

77

In order to apply this method, the following values need to be determined first:• Total end slip at ultimate load: dend,pl,u,0 from equation (6.17)• Elastic part of end slip at ultimate load: dend,el,u,0 from equation (6.13) and slip at yield at the

critical cross-section: dcrit,el,y,0 from equation (6.11) or (6.12)• Choose PRk and the related δu and δi from the load-slip diagram of the shear connector.With these values, the θ-value is determined according to equation (6.20). Then the minimum degree of partial shear connection ηmin is determined using equation (6.21).

6.4.4.2 Restrictions for the application of equation (6.21)

The application range of the simple method described above is restricted:a) The method is applicable only to composite beams with an interface of sufficient stiffness and

thus a sufficient degree of partial interaction γ. Below a certain degree of partial interaction the required ductility in the composite beam may be larger than predicted by equations (6.13), (6.14) and (6.15). The actual recommended limit is given in equation (6.22).

(6.22)b) If the ductility of a shear connector is much smaller than required in the composite beam, ηf

may be larger than 100 %. In order to avoid these effects, the method explained above should not be used if the ductility of a shear connector is too small. Figure 6.10 (left) shows that ηminbecomes greater than predicted using the method explained above if the ductility of a shear connector is smaller than about 20 % of the ductility required in the composite beam without shear connection. Therefore, the limitation given in equation (6.23) must be respected.

(6.23)

c) The deformation capacity of a shear connector must be large enough to allow plastification at the plastic hinges (as explained in section 6.3.2 on page 67) and must therefore not be too small compared to the required deformation capacity of the composite beam. More detailed investigations on this subject are reported in [63]. In the numerical simulations performed with values for this effect has not been observed. Therefore, the limita-tion in (6.24) is recommended.

Figure 6.11 Design diagram for the minimum degree of partial shear connection. Values from the results of simulation groups C3 elX (left) and C3 epX (right). PRk/Pu = 0.9. All evaluations were performed at the critical cross-section according to equation (6.1). For small θ-values ηmin is greater than predicted with equation (6.21).

0

50

100

150

200

250

300

0 0.5 1θ [-]

ηmin

[%]

IPE270 2 7.2 el1IPE270 D 7.2 el1IPE270 2 7.2 el2IPE270 D 7.2 el2IPE270 2 7.2 el3IPE270 D 7.2 el3IPE270 2 7.2 el4IPE270 D 7.2 el4IPE270 2 7.2 el5IPE270 D 7.2 el5

0

50

100

150

200

250

300

0 0.5 1θ [-]

ηmin

[%]

IPE270 2 7.2 ep1IPE270 D 7.2 ep1IPE270 2 7.2 ep2IPE270 D 7.2 ep2IPE270 2 7.2 ep3IPE270 D 7.2 ep3IPE270 2 7.2 ep4IPE270 D 7.2 ep4IPE270 2 7.2 ep5IPE270 D 7.2 ep5

γ 0.9≥

δu δi–( ) 0.2 dend el u 0, , , dcrit el y 0, , ,–( )⋅≥

δu 0.03 dend pl u 0, , ,⋅≥

Synthesis

78

(6.24)

d) Apart from shear forces due to vertical loads, shear connectors are also subjected to shear forces due to horizontal actions such as wind or diaphragm effects. With the low degrees of partial shear connection allowed with the method described above, these actions may be of the same order of magnitude as the shear resistance of the connectors. Therefore, the effects of these actions must be considered in the design of the shear interface.

6.5 Advanced design method

The rigid-plastic design model used within the boundary conditions mentioned above in section 6.4 on page 69 is conservative since not the real, but the characteristic load-slip behaviour is considered (cf. Figure 2.1). As a safe advanced approach, the slip values may be determined as shown in section 6.4.3.1 on page 71 and reduced according to the curves in Figure 6.10 for the actual degree of partial shear connection. Thus, the slip for each connector in the composite beam can be determined.In a second step, the shear force according to the real load-slip behaviour (cf. Figure 6.12) can be determined for each shear connector. From the sum of the shear forces of all connectors a higher, less conservative degree of partial shear connection and thus a higher bending resistance for the composite beam is determined.

6.6 Generalisation to continuous composite beams

Continuous composite beams can be described in the same way as simply-supported beams, if the spans of the beam are chosen such that the inclination of the beam at the inner supports amounts to zero, whereas the following modifications need to be made (cf. [43]):

Figure 6.12 Comparison of the rigid-ideal plastic and the advanced design method. As an ex-ample, a composite beam with three shear connectors is compared. Connectors number 2 and 3 are considered for design, connector number 1 is placed in the middle part of the beam to prevent uplift of the concrete slab. According to equa-tion (6.4) the three connectors exhibit slips of d1, d2 and d3. With the rigid-ideal plastic design method the axial force in the chords amounts to N = 2 * PRk and thus with PRk/Pu = 0.9 to N = 1.8 * Pu. With the advanced design method, the ax-ial force in the chords amounts to N = (0.72 + 0.96 + 0.93) * Pu = 2.61 * Pu.

δu 0.05 dend pl u 0, , ,⋅≥

δPu

δiδu

0

0.2

0.4

0.6

0.8

1

Slip [mm]

Shea

r for

ce P

/Pu [

kN]

RealbehaviourCharacteristicbehaviour

d1

PRk

d2 d3

Generalisation to continuous composite beams

79

• The difference in axial force in the chords and thus the shear force to be transferred through the shear interface over a shear span is greater than with a simply-supported beam of the same span, because the axial force in the chords at the support is not zero (cf. equation (2.1)).

• The span used to determine the minimum degree of partial shear connection is the length of the beam exhibiting a positive bending moment. Provided the composite section exhibits ductile behaviour, also at the inner supports, the amount of plastic redistribution of the bend-ing moment and thus the length of the beam in positive bending can be chosen freely.

The longitudinal position where the bending moment is zero (between x = 0 and x = L/2) is shown in equation (6.25) for uniform load and in equation (6.26) for two concentrated loads at x1 = a and x2 = L - a. The equivalent span (the length of the beam exhibiting a positive bending moment) is then determined as given in equation (6.27). Figure 6.13 gives some examples of redistribution of bending moment.

(6.25)

(6.26)

(equivalent span) (6.27)

The inclination at the inner supports of a continuous composite beam can be held at zero if the end spans are made somewhat shorter. The end supports therefore must be placed at the position where the bending moment in the continuous composite beam would be zero. The position of the end supports thus depends on the amount of plastic redistribution of the bending moment as can be seen from Figure 6.13. The amount of plastic redistribution may be chosen by the de-signer if the rotation capacity of the chords is sufficient. It affects the equivalent span (6.27) and the locations at which the bending moment is zero ((6.25) and (6.26)).

Figure 6.13 Plastic redistribution of bending moment in a continuous composite beam. The amount of plastic redistribution may be chosen by the designer if the rotation ca-pacity of the chords is sufficient. The elastic distribution of bending moment shows bending moments at the support which are twice as much as in the field. In con-crete structures bending moments are often redistributed such that the bending mo-ments at the support and in the field are of the same magnitude. For composite structures it is often economical if the bending moments are further redistributed, such that the bending moment in the field is greater than the bending moment at the support. The amount of plastic redistribution affects the equivalent span (6.27)and the locations at which the bending moment is zero ((6.25) and (6.26)).

x0 LMR support, MR field, MR support, MR field,+( ) MR field,⋅–+

2 MR support, MR field,+( )⋅--------------------------------------------------------------------------------------------------------------------------------------------------------⋅=

x0 aMR support,

MR support, MR field,+-------------------------------------------------------⋅=

Le L 2 x0⋅–=

0 1 2 3x/L

M [-

]

ElasticEvenMore in Field

Synthesis

80

6.7 Example: Application of the deformation-based method

6.7.1 Overview and design for construction state

As an example of application, a composite beam in a parking deck will be designed. The live load is assumed to be uniformly distributed over the whole deck area as q = 2 kN/m². The ge-ometry of the building is shown in Figure 6.14. The structure is stabilised horizontally in both directions by bracings. The beam is simply-supported with a span of L = 16 m, and the beams are spaced 4.8 m.

Figure 6.14 Geometry for example of application

Table 6.2 Data sheet for the Ribdeck 80 profiled steel sheeting (cf. http://www.holorib.com)

Example: Application of the deformation-based method

81

With a spacing of 4.8 m, a Ribdeck 80 profiled steel sheet and a concrete depth of 140 mm, the construction can be erected without propping (cf. Table 6.2). The steel beams are assumed to be stabilised against torsional buckling during construction by the profiled steel sheeting.

The design bending load for the construction state is qE,d = 33.0 kN/m‘, while the correspond-ing bending moment is ME,d = 1056 kNm. Therefore, for the construction state a steel beam of type HEB 400 of steel S355 (Ma = 1150 kNm) is chosen according to ULS analysis. Servicea-bility analysis is not performed. Deflections in the construction state are pre-cambered.

6.7.2 Design for final state

6.7.2.1 Actions

The loads considered for the final state are listed in Table 6.4. The resulting design load is qE,d = 43.4 kN/m‘ and the bending moment ME,d = 1390 kNm.

Load Dimensions kN/m² g [kN/m‘]

Dead loads

Steel beam HEB 400 1.55

Profiled steel sheeting t = 1.2 mm 0.145 0.7

Construction loading 1.5 kN/m² 1.5 7.2

Total gcons = 9.5

Live load

Concrete slab hc,eff = 140 mm - 42 mm + 4800/250 = 117 mm, 117 mm * 24 kN/m3 2.8 13.5

Total qcons = 13.5

Table 6.3 Loads considered for construction state

Load Dimensions kN/m² g [kN/m‘]

Dead loads

Steel beam HEB 400 1.55

Concrete slab hc,eff = 140 mm - 42 mm + 4800/250 = 117 mm, 117 mm * 24 kN/m3 2.8 13.5

Profiled steel sheeting t = 1.2 mm 0.145 0.7

Pavement Layer thickness: 50 mm, 24 kN/m3 1.2 5.76

Total g = 21.5

Live load

Light traffic (< 3.5 t) 2.0 9.6

Total q = 9.6

Table 6.4 Loads

Synthesis

82

6.7.2.2 Bending resistance of the composite beam with full shear connection

The axial resistance of the steel beam is

.

As the effective concrete depth is limited to

due to the profiled steel sheeting, with a concrete C40/50 with a design compressive strength of

and

the axial resistance of the concrete slab is lower than that of the steel beam and amounts to

.

The vertical distance between the resulting axial forces is

.

The bending resistance of the composite beam with full shear connection thus amounts to

and

compared to a bending moment of

However, the analysis should not be performed at mid-span, but at the critical cross-section.With full shear connection, the critical cross-section is located according to equation (6.1) at

.

If the beam is designed for full shear connection at mid-span, it is with partial shear connection at the critical cross-section. With rigid-ideal plastic behaviour of the interface the degree of par-tial shear connection at the critical cross-section amounts to

.

Thus the bending resistance at the critical cross-section is

and

while the bending moment at the critical cross-section is

.

Na pl, 7030kN=

hc eff, 140mm 90mm– 50mm= =

fcd 26N mm2⁄= bc eff, 4.0m=

Nc pl, 4420kN=

e 400mm2

------------------- 140mm 50mm2

----------------–+ 315mm= =

Mpl 1897kNm= MR d,MplγM0--------- 1897kNm= =

ME d,qE d, L⋅ 2

8--------------------- 1390kNm= =

xcritL4--- 1

MaMpl---------+⎝ ⎠

⎛ ⎞⋅ 6.4m= =

ηxcritηmid

xcritL 2⁄----------⋅ 0.80= =

MR xcrit, 1705kNm= MR d xcrit, ,

MR xcrit,

γM0------------------- 1705kNm= =

ME d xcrit, ,qE d,

2---------- xcrit L xcrit–( )⋅ ⋅ 1333kNm= =

Example: Application of the deformation-based method

83

6.7.2.3 Bending resistance of the composite beam with partial shear connection

6.7.2.3.1 Critical cross-section for partial shear connection

For the composite beam with partial shear connection the bending resistance at mid-span is smaller. Thus, the critical cross-section is located closer to mid-span (cf. equation (6.1)). The actual location of the critical cross-section depends on the chosen degree of partial shear con-nection. The smaller η is, the closer is the ratio Ma/Mpl, and the closer xcrit is to mid-span. How-ever, as a safe approach, the critical cross-section is assumed to be in the same place as for full shear connection at xcrit = 6.4 m.

6.7.2.3.2 The structurally required degree of partial shear connection

The minimum required bending resistance of the composite beam is

at mid-span.

at the critical cross-section.

With the simplified interpolation method given in Eurocode 4 the degree of partial shear con-nection required to reach the bending resistance MR,min = 1390 kNm is η = 0.32. With plastic theory (equilibrium method) the degree of partial shear connection must be at least η = 0.159 (based on an assumed shear length of L/2) for the composite beam to provide sufficient bending resistance at the critical cross-section (cf. Figure 6.15). The degree of partial shear connection at the critical cross-section is then

.

The minimum degree of partial shear connection according to Eurocode 4 (cf. equation (3.37)) is ηmin,EC = 0.73.

6.7.2.3.3 The required ductility and deformation capacity without shear connection

According to section 6.4.3.1 on page 71 slip in the shear interface and thus the required defor-mation capacity without shear connection is

Figure 6.15 Partial interaction diagram for the composite beam to be designed

MR min, ME d, γM0⋅ 1390kNm= =

MR min xcrit, , ME d xcrit, , γM0⋅ 1333kNm= =

ηxcritηmid

xcritL 2⁄----------⋅ 0.159 6.4m

8m------------⋅ 0.127= = =

100011001200130014001500160017001800

0.0 0.2 0.4 0.6 0.8 1.0η [-]

MR

,xcr

it [k

Nm

]

Minimum bending resistance: Mpl,min,xcritN/Nf,min, structural, interpolation methodN/Nf,min, structural, equilibrium methodInterpolation method: MR,xcritEquilibrium method: MR,xcritN/Nf,min (EC4)

Synthesis

84

according to equation (6.16) with

.

The required ductility without shear connection is determined according to equations (6.13) and (6.15)

(xcrit = L/2 without shear connection)

and amounts to

.

Thus, for a shear connector exhibiting a deformation capacity of at least δu = 12.4 mm and a ductility of at least (δu - δi) = 6.6 mm the minimum degree of partial shear connection is ηmin = 0.0.

6.7.2.3.4 The required ductility and deformation capacity for η = 0.159

For a minimum degree of partial shear connection of ηmin = 0.159 the required deformation ca-pacity and ductility are determined using equation (6.21):

and thus .

Thus, the deformation capacity of the shear connectors must be at least

and the ductility at least

.

If the shear connector used fulfils these criteria, the composite beam may be built with a degree of partial shear connection of η = 0.159.

6.7.2.3.5 The minimum degree of partial shear connection for a given shear connector

For comparison, the minimum degree of partial shear connection for a given shear connector is determined as follows. The connector characteristics are chosen according to the requirements given in Eurocode 4 for a „ductile“ shear connector as δu = 6 mm and δi = 0.3 mm. Thus, the shear connector exhibits a ductility of (δu - δi) = 5.7 mm.θ is determined using equation (6.20):

.

dend pl u 0, , ,23--- 16000mm

2------------------------- 1.5 355N mm2⁄ 245N mm2⁄–⋅

210000N mm2⁄------------------------------------------------------------------------------ 1 140mm

200mm-------------------+⎝ ⎠

⎛ ⎞ 12.4mm=⋅ ⋅ ⋅=

σa c, gserxcrit L xcrit–( )⋅

2 Wel a,⋅---------------------------------------- 23.0kN m⁄ 6.4m 16m 6.4m–( )⋅

2 2.88 106mm3⋅ ⋅--------------------------------------------------- 245N mm2⁄=⋅=⋅=

dend el u 0, , ,23--- 16000mm

2------------------------- 1.12 355N mm2⁄⋅ 245N mm2⁄–

210000N mm2⁄--------------------------------------------------------------------------------- 1 140mm

200mm-------------------+⎝ ⎠

⎛ ⎞ 6.6mm=⋅ ⋅ ⋅=

dcrit el y 0, , , 0=

dend el u 0, , , dcrit el y 0, , ,– 6.6mm=

ηmin 1 θ–= θmin 1 ηmin 0.841=–=

δu θmin dend pl u 0, , ,⋅ 10.4mm= =

δu δi–( ) θmin dend el u 0, , , dcrit el y 0, , ,–( )⋅ 5.6mm= =

θ min 6mm12.4mm-------------------- 5.7mm

6.6mm-----------------;⎝ ⎠

⎛ ⎞ 0.484==

Example: Application of the deformation-based method

85

Then, ηmin is determined using equation (6.21):

.

Thus, if shear connectors considered to be „ductile“ according to Eurocode 4 are used in the de-scribed composite beam, the minimum degree of partial shear connection is ηmin = 0.516 with the deformation-based model described in section 6.4 on page 69, compared to ηmin = 0.73 with the existing Eurocode 4 formula.

6.7.2.4 The minimum degree of partial shear connection with propped construction

This section determines the minimum degree of partial shear connection for the case that the beam was cast with the steel beam continuously propped.If the beam is cast continuously propped, σa,c = 0 N/mm². The required deformation capacity is then

compared to 12.4 mm for unpropped construction. The required ductility without shear connec-tion is determined according to equations (6.13) and (6.15)

(xcrit = L/2 without shear connection)

and is

compared to 5.6 mm with unpropped construction. For ηmin = 0.159, θmin is θmin = 0.841. Thus, the deformation capacity of the shear connectors must be at least

and the ductility at least

.

For a „ductile“ shear connector according to Eurocode 4 θ is given by

.

Then, ηmin is determined using equation (6.21):

.

Thus, if shear connectors considered to be „ductile“ according to Eurocode 4 are used in the de-scribed composite beam with continuously-propped construction, the minimum degree of par-tial shear connection is ηmin = 0.74 with the deformation-based model described in section 6.4 on page 69, compared to ηmin = 0.73 with the existing Eurocode 4 formula.

ηmin 1 0.484– 0.516= =

dend pl u 0, , ,23--- 16000mm

2------------------------- 1.5 355N mm2⁄ 0N mm2⁄–⋅

210000N mm2⁄------------------------------------------------------------------------ 1 140mm

200mm-------------------+⎝ ⎠

⎛ ⎞ 23mm=⋅ ⋅ ⋅=

dend el u 0, , ,23--- 16000mm

2------------------------- 1.12 355N mm2⁄⋅ 0N mm2⁄–

210000N mm2⁄--------------------------------------------------------------------------- 1 140mm

200mm-------------------+⎝ ⎠

⎛ ⎞ 17.2mm=⋅ ⋅ ⋅=

dcrit el y 0, , , 0=

dend el u 0, , , dcrit el y 0, , ,– 17.2mm=

δu θmin dend pl u 0, , ,⋅ 19.3mm= =

δu δi–( ) θmin dend el u 0, , , dcrit el y 0, , ,–( )⋅ 14.5mm= =

θ min 6mm23.0mm-------------------- 6mm

17.2mm--------------------;⎝ ⎠

⎛ ⎞ 0.26==

ηmin 1 0.26– 0.74= =

Synthesis

86

6.7.2.5 The required number of shear connectors

The shear connector to be used in the composite beam designed in this section is assumed to exhibit the load-slip behaviour shown in Table 6.5 when used in combination with a Ribdeck 80 profiled steel sheeting spanning perpendicular to the beam‘s longitudinal axis.

The required axial force for full shear connection is

.

With shear connector characteristics #1 (PRk/Pu = 0.9) the minimum degree of partial shear con-nection is η = 0.516. Thus, the required axial force is

.

Thus the required number of shear connectors with shear connector characteristics #1 is

.

Therefore, 32 shear connectors must be placed in each half span. The stiffness of the interface is

and thus .

This means that the deformation-based method may be used with this type of shear connector.With (PRk/Pu = 0.8) the ductility and the deformation capacity of the shear connector are greater than required by the beam. Therefore, the minimum degree of partial shear connection is ηmin = 0.159. The required axial force is then

.

Connector data sheet, Pu = 100 kN

Load P/Pu [-] Slip [mm]

0 0

0.5 0.02

0.8 0.05

0.9 0.3

1.0 3

0.9 6

0.8 12

PRk,1 = 90 kN => δi,1 = 0.3 mm, δu,1 = 6 mm, Si,1 = 300 kN/mm 0.7 20

PRk,2 = 80 kN => δi,2 = 0.05 mm, δu,2 = 12 mm, Si,2 = 1600 kN/mm 0.0 28

Table 6.5 Load-slip behaviour of the shear connector

δPu

δ i,1δu,1

δu,2δi,2

0 10 20 30Slip [mm]

Shea

r for

ce [k

N]

Real behaviourCharacteristic behaviour 1Characteristic behaviour 2

PRk,2

PRk,1

Pu

Nf N= c pl, 4420kN=

N ηN= c pl, 2281kN=

nminN γV⋅PRk 1,--------------- 2281kN 1.25⋅

90kN----------------------------------- 32= = =

k132 Si 1,⋅

xcrit-------------------- 1.5kN m⁄ m2= = γ 0.933=

N ηN= c pl, 703kN=

Example: Application of the deformation-based method

87

Thus the required number of shear connectors with shear connector characteristics #2 is

.

Therefore, 26 shear connectors must be placed in each half span.The stiffness of the interface amounts to

and thus .

This means that the stiffness of the shear interface is sufficient so that the deformation-based method to determine the minimum degree of partial shear connection may be used with this type of shear connector.

6.7.3 Comparison: deformation-based method vs. Eurocode 4 formulas

With the existing method given in Eurocode 4 the minimum degree of partial shear connection is ηmin,EC = 0.73 and the related axial force N = 3227 kN. The required number of shear con-nectors with the type given in Table 6.5 is then nEC = 45 shear connectors per half span. The use of the deformation-based method allows a reduction to n = 11 shear connectors per shear length. If the composite beam is erected in propped construction and shear connectors which are judged „ductile“ according to Eurocode 4 are used, the minimum degree of partial shear connection de-termined using the deformation-based method is ηmin = 0.74, which is almost the same as de-termined using the Eurocode 4 formula.The deformation-based method to determine the minimum degree of partial shear connection produces similar results to the Eurocode 4 formulas for a composite beam in propped construc-tion subjected to a uniform load and shear connectors with a deformation capacity of δu = 6 mm and an initial slip of δi = 0 mm. However, the Eurocode 4 formulas do not consider some pa-rameters such as the construction process or the actual load-slip behaviour of the shear connec-tors. The deformation-based method considers these parameters and is able to exploit the re-serves provided by the Eurocode 4 formulas under favourable conditions.However, the application of the deformation-based method requires greater design effort com-pared to the existing Eurocode 4 formulas.

nminN γV⋅PRk 2,--------------- 703kN 1.25⋅

80kN-------------------------------- 11= = =

k111 Si 2,⋅

xcrit-------------------- 2.75kN m⁄ m2= = γ 0.962 0.9>=

Synthesis

88

Critical cross-section

89

7 Conclusions

7.1 Critical cross-section

The design of a composite beam must not be performed on the basis of an analysis at the centre of a beam, but at the critical cross-section defined in section 6.2 on page 63. On the other hand, the rigid-ideal plastic design method according to Eurocode 4 may also be used for composite beams in which the bending resistance of the steel beam is less than 40 % the bending resistance of the composite beam Ma/Mpl < 0.4.

7.2 Minimum degree of partial shear connection

The existing formulas given in Eurocode 4 to determine the minimum degree of partial shear connection are a robust, but coarse approximation to the actual physical behaviour. The defor-mation-based method explained in section 6.4 on page 69 is much more detailed. This method allows the efficient use of a wide variety of shear connectors. With shear connectors exhibiting large ductility and deformation capacity very low minimum degrees of partial shear connection are possible. The use of shear connectors exhibiting poor deformation capacity and ductility is also possible with a considerably increased minimum degree of partial shear connection.

7.3 Shear connector database

In order to apply the method shown in section 6.4 on page 69, not only the shear resistance, but more detailed characteristics of the shear connector are required. In order to determine these characteristics, the complete load-slip curve of the shear connector including the descending branch must be known.The additional data needs to be determined for all shear connectors, e.g. based on the evaluation of push tests previously performed or on new push tests.However, many of the push tests performed in the past to determine a connector’s behaviour were only carried out to 80 % residual load. In order to produce a relevant database future push tests should be performed until the residual shear resistance has decreased to not more than 10 % of the connector’s shear resistance or until the slip reaches at least 5 times the slip at 90 % residual resistance.In the meantime, all shear connectors fulfilling the requirements of a „ductile“ shear connector according to Eurocode 4 may be used with the new method with PRk = 0.9 * Pu, δu = 6.0 mm and δi = 0 mm.

7.4 Serviceability limit state

For the determination of the deflections of a composite beam at the serviceability limit state (SLS) in many cases a rigid shear connection can be assumed. However, in general the use of the γ-method (cf. section 3.5.2 on page 23) is recommended.

Conclusions

90

7.5 Research-oriented analysis

For nonlinear high-accuracy analysis of composite beams, which typically requires a large com-putational effort, an analytical and a numerical model are presented in section 3.4 on page 20and in chapter 5 on page 31, respectively. The analytical model is suited to use with a computer algebra program while the numerical model requires a finite element program.By means of these tools the physically-nonlinear behaviour of composite beams can be analysed in detail.

7.6 Application range of the deformation-based method

The deformation-based method to determine the minimum degree of partial shear connection is well-suited to a rather stiff interface ( ) with shear connectors which exhibit a certain duc-tility ( ) and deformation capacity ( ) as as are mostly used in practice today. It does not cover design with a rather soft shear interface ( ) o r w i th shea r connec to r s wh ich exh ib i t ve ry poo r duc t i l i t y ( ) or deformation capacity ( ).

γ 0.9≥δu δi–( ) 0.2 dend el u 0, , , dcrit el y 0, , ,–( )⋅≥ δu 0.05 dend pl u 0, , ,⋅≥

γ 0.9<δu δi–( ) 0.2 dend el u 0, , , dcrit el y 0, , ,–( )⋅< δu 0.05 dend pl u 0, , ,⋅<

Application range of the deformation-based method

91

8 Outlook

8.1 Application range of the deformation-based method

The deformation-based method to determine the minimum degree of partial shear connection does not cover design with a rather soft shear interface ( ) or with shear connectors which exhibit very poor ductility ( ) or deformation capacity ( ). Further research is required for design with soft shear connectors. For de-sign with shear connectors of poor deformation capacity the publication of new findings is ex-pected soon [63]. In the meantime, composite beams with shear connectors of these types can be designed using the numerical model presented in chapter 5 on page 31.

8.2 Stability questions and local effects

In the present work a limitation of the rotational capacity of the chords due to stability questions or to local effects is not considered, though such effects are of importance in many cases. There-fore, further research on this field is recommended. For this research the numerical model pre-sented in chapter 5 on page 31 can be further adapted.

8.3 Connector behaviour and lift-off

In all analytical and numerical models mentioned previously, the characteristic load-slip behav-iour of the shear connectors is assumed as given. Usually, the load-slip behaviour determined in push tests is used in the models. However, an adequate parametric description of a shear con-nector‘s load-slip behaviour (including the effects of lift-off, concrete in tension and the influ-ence of profiled steel sheeting) is missing. Further research as reported e.g. for headed studs in [35] is also recommended for other types of shear connectors.

γ 0.9<δu δi–( ) 0.2 dend el u 0, , , dcrit el y 0, , ,–( )⋅<

δu 0.05 dend pl u 0, , ,⋅<

Outlook

92

93

Index of Figures

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Figure 1.1 Left: RibCon (left) and StripCon (right) shear connectors.

Right: Tecnaria (left) and Hilti X-HVB (right) shear connectors. . . . . . . . . . . . . . . . . . . . 1

2 Composite Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Figure 2.1 Real and 3 examples of characteristic load-slip behaviour of a shear connector. Characteris-

tic behaviour #1 shows the greatest characteristic shear resistance PRk, but the lowest initial stiffness Si, deformation capacity δu and ductility δu - δi. Characteristic behaviours #2 and #3 show lower shear resistances, but higher initial stiffnesses Si, deformation capacities δu and ductilities δu - δi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Figure 2.2 Load-deformation behaviour of a simply-supported composite beam with partial shear con-nection. η = 0.3, L = 7.2 m, Steel section IPE 270, fy = 300 N/mm². Shear connectors with δu (0.9 * Pu) = 2 to 6 mm evenly-distributed over the whole length of the beam.. . . . . . 10

Figure 2.3 Composite section, strain planes for rigid, flexible and no shear connection. For cracked concrete averaged strains are shown. For a given curvature w‘‘(x) the difference in slip at the shear interface ∆ε(x) depends on the vertical distance z0,c - z0,a between the neutral axes of the two chords.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Figure 2.4 Typical connector characteristics (left) and their effect on the bending resistance of the same composite beam mentioned in Figure 2.2 (right). For composite beams with ductile shear connectors of sufficient deformation capacity the rigid-ideal plastic design method accord-ing to Eurocode 4 [26] may be used for degrees of partial shear connection . With brittle shear connectors the plastic bending resistance of a composite beam is only reached with η >> 1.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Analytical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 3.1 Composite section, strain planes for rigid, flexible and no shear connection. For cracked

concrete averaged strains are shown. For a given curvature w‘‘(x) the difference in slip at the shear interface ∆ε(x) depends on the vertical distance z0,c - z0,a between the neutral axes of the two chords.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Figure 3.2 Material law formulations for steel: bilinear (left, cf. equation (3.32)) and hyperbolic (right, cf. equation (3.33)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Figure 3.3 Bilinear formulation for concrete material law (left, cf. equation (3.34)) Exponential load-slip law for shear interface (right, cf. equation (3.35)) . . . . . . . . . . . . 21

Figure 3.4 Left: Comparison between end slip from γ-method and differential equation. Slip deter-mined using the γ-method is in between the results from the differential equation for linear composite action for uniform load and for concentrated loads at one and two thirds of the span. Right: γ as function of k for the composite beam referred to in Figure 2.2. . . . . . . . . . . 24

4 Experimental Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Figure 4.1 Results of beam tests A-01, A-02 and B-01: load-deflection diagram (left) and load-slip di-

agram (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 4.2 Beam tests A-01 (left) and B-01 (right) after failure. In beam tests A-01 and A-02 the loads

were introduced from above to the upperside of the concrete slab. In beam test B-01 the loads were introduced from underneath to the lower flange of the steel beam. . . . . . . . . 26

Figure 4.3 Load-slip diagrams of some push tests with RibCon shear connectors (left) and StripCon shear connectors (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Figure 4.4 Left: Geometry of specimens S3.1, S3.2 and S3.3 Right: Overview: mean slip in all three tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Figure 4.5 Test S3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

94

Figure 4.6 Test S3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 4.7 Test S3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 4.8 Shear interface of specimen S3.1 after test (side 1 left, side 2 right) . . . . . . . . . . . . . . . . 30Figure 4.9 Shear interface of specimen S3.2 after test (side 1 left, side 2 right) . . . . . . . . . . . . . . . . 30

5 Numerical Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Figure 5.1 Geometry of two-dimensional truss and frame models investigated with STATIK-N . . 32Figure 5.2 M-N-Interaction diagrams for steel section (left) and concrete slab (right). . . . . . . . . . . 33Figure 5.3 Nonlinear load-strain relation for shear connectors in STATIK-N Models . . . . . . . . . . . 33Figure 5.4 Geometry of finite element model for simulation of beam tests A-01 and A-02 [31] . . . 38Figure 5.5 Geometry of the finite element model for the simulation of beam test B-01 [31] . . . . . . 38Figure 5.6 Geometry of finite element model for the simulation of the Hilti beam tests [39] . . . . . 39Figure 5.7 Left: Geometry of tensile model. Symmetry conditions and loading point.

Right: Load-displacement diagrams of tensile tests and of simulations with different simu-lation options.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Figure 5.8 Stress-strain diagrams for material used for the steel beam in beam tests [31] . . . . . . . . 40Figure 5.9 Section of a composite beam. Normal case (left) and special cases for simulations of Hilti

beam tests [39]. Real constant sets #1 and #2 are used for the upper (reinforced) part of the concrete slab, #3 and #4 for the lower (unreinforced) part. . . . . . . . . . . . . . . . . . . . . . . . 41

Figure 5.10 Load-bearing behaviour in compression (upper left), tension (upper right) and shear (lower left) of a cubic element using the ANSYS concrete material option (CONC), a combination of the concrete material option and bilinear isotropic hardening plasticity (CONC + BISO) and a combination of the concrete material option and multilinear isotropic hardening plas-ticity (CONC + MISO), all without reinforcement and with 1 % of reinforcement in each di-rection (X, Y, Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Figure 5.11 Left: Single element model to test shear load-deformation behaviour Right: Load-slip behaviour of a shear interface element with different key options . . . . 46

Figure 5.12 Refined model (left: finite element model, right: load-deflection diagram) . . . . . . . . . . 48Figure 5.13 Load-slip curve for shear connectors from push tests (left) and generic ones (right) . . . 49Figure 5.14 Load-slip curve for shear connectors from push tests [38] used in beam tests [39]. . . . . 49Figure 5.15 Load-slip curve for shear connectors RT3 (left) and parametric studies (right) . . . . . . . 50Figure 5.16 Load-slip curve for parametric studies groups elX (left) and epX (right) . . . . . . . . . . . . . 50Figure 5.17 Results of simulations on beam test A-01 with connector behaviour from push tests . . . 51Figure 5.18 Results of simulations on beam test A-02 with connector behaviour from push tests . . . 51Figure 5.19 Results of simulations with adapted connector behaviour (A-01 left, A-02 right) . . . . . 52Figure 5.20 Results of simulation on beam tests performed at Hilti . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 5.21 Geometry of beams simulated in series A200. The sectional geometry is identical to that in

beam tests A-01 and A-02 (cf. Figure 5.4). For each span the beam overhangs the support by 150 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Figure 5.22 Results of simulations on beam test B-01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 5.23 Series A200: comparison between A200_072m_02 and A200_072m_A02 . . . . . . . . . . 57Figure 5.24 Results of series A200: comparison with results from analytical calculations . . . . . . . . 57Figure 5.25 Comparison of bending resistances from numerical and analytical models for the cases of

numerical simulations C3P_N_IPE270_2_72_x_xx_1_S300 (left) and C3P_N_IPE270_D_72_x_xx_1_S300 (right). Curve parameter is the deformation capacity of the shear connectors δu (varying from 1 mm to 15 mm). . . . . . . . . . . . . . . . . . . . . . . 60

Figure 5.26 Symmetrical and asymmetrical failure in tests and simulations. In buildings and in laborato-ry tests asymmetrical failure occurs while the finite element model assumes symmetrical failure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 6.1 Distribution of bending resistance between the support (x/L = 0) and mid-span of the beam

(x/L = 0.5) assuming rigid-ideal plastic behaviour of the interface in numerical simulations series C3, group S235. Composite beams with steel sections IPE 270 (left) and IPE 550 (right) η between 0 and 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Figure 6.2 Position of the critical cross-section in the composite beams used for finite element simula-tions series C3 for 2 concentrated loads (left) and for uniform load (right). For composite beams subjected to concentrated loads the critical cross-section is located in the position

95

where the load is introduced. For composite beams subjected to uniform load the location of the critical cross-section is described by the simplified model shown in equation (6.1). 64

Figure 6.3 Position of the critical cross-section in a composite beam with elastic (left) and rigid-plastic (right) interface behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Figure 6.4 Position of the critical cross-section in a composite beam with η = 2.0 and rigid-ideal plastic interface behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Figure 6.5 Longitudinal distribution of shear forces with rigid-brittle shear connectors (left), Real and 3 examples of characteristic load-slip behaviour of a shear connector (cf. Figure 2.1) (right). 68

Figure 6.6 Longitudinal distribution of slip in the beam mentioned in Figure 2.2 for k = 104 N/mm² (left) and k = 10-4 N/mm² (right) determined using the differential equation for linear elastic composite action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Figure 6.7 Elastic and plastic behaviour of a composite beam. The plastic behaviour is characterised by plastic hinges (assumed as discrete) at the critical cross-sections, while the other parts re-main in an elastic state. Slip due to the plastic part of deformation is zero between the critical cross-sections at mid-span and constant between the critical cross-sections and the supports. 71

Figure 6.8 Longitudinal distribution of bending moment and slip for concentrated load at L/3 (left) and for uniform load (right). Results from analytical and numerical models. The longitudinal dis-tribution of slip at yield according to the analytical model agrees well with the results from the numerical simulations series C3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Figure 6.9 End slip at the ultimate load in numerical simulation series C3, groups S235, for two con-centrated loads at L/3 (left) and uniform load (right) with deformation capacities δu varying between 1 mm and 15 mm. The curves show that slip at the ultimate load is governed by the deformation capacity δu of the shear connector. For low degrees of partial shear connection some simulations were aborted prematurely. Therefore, end slip at the ultimate load is also too small. In the simulations of composite beams without shear connection, the bending moment is in-creased further as the deformation is increased, because bilinear material behaviour was as-sumed for the concrete slab and the steel beam. Thus the ultimate load is reached at the end of the simulation and the slip at the ultimate load is the slip at the end of the simulation. 75

Figure 6.10 Design diagram for the minimum degree of partial shear connection. Values from results of numerical simulations series C3 (left) and C4 (right). PRk/Pu = 0.9. All evaluations were per-formed at the critical cross-section according to equation (6.1). In the numerical simulations with two concentrated loads the shear connectors in the middle part of the composite beam (between the critical cross-sections) make a considerable contribution to the shear resistance of the interface. Therefore, the design model for ηmin in equation (6.21) is more conservative for these beams. In simulation groups the three outer shear connectors were placed in the outer thirds only. These beams agree well with the analytical model. For θ-values below ap-proximately θ = 0.2 ηmin is greater than predicted with equation (6.21). . . . . . . . . . . . . 76

Figure 6.11 Design diagram for the minimum degree of partial shear connection. Values from the results of simulation groups C3 elX (left) and C3 epX (right). PRk/Pu = 0.9. All evaluations were performed at the critical cross-section according to equation (6.1). For small θ-values ηmin is greater than predicted with equation (6.21).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Figure 6.12 Comparison of the rigid-ideal plastic and the advanced design method. As an example, a composite beam with three shear connectors is compared. Connectors number 2 and 3 are considered for design, connector number 1 is placed in the middle part of the beam to pre-vent uplift of the concrete slab. According to equation (6.4) the three connectors exhibit slips of d1, d2 and d3. With the rigid-ideal plastic design method the axial force in the chords amounts to N = 2 * PRk and thus with PRk/Pu = 0.9 to N = 1.8 * Pu. With the advanced de-sign method, the axial force in the chords amounts to N = (0.72 + 0.96 + 0.93) * Pu = 2.61 * Pu.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Figure 6.13 Plastic redistribution of bending moment in a continuous composite beam. The amount of plastic redistribution may be chosen by the designer if the rotation capacity of the chords is sufficient. The elastic distribution of bending moment shows bending moments at the sup-port which are twice as much as in the field. In concrete structures bending moments are of-ten redistributed such that the bending moments at the support and in the field are of the same magnitude. For composite structures it is often economical if the bending moments are

96

further redistributed, such that the bending moment in the field is greater than the bending moment at the support. The amount of plastic redistribution affects the equivalent span (6.27) and the locations at which the bending moment is zero ((6.25) and (6.26)).. . . . . 79

Figure 6.14 Geometry for example of application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 6.15 Partial interaction diagram for the composite beam to be designed. . . . . . . . . . . . . . . . . 83

7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

97

Index of Tables

1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Composite Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Analytical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Experimental Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Table 4.1 Overview on specimens tested . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Numerical Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Table 5.1 Material properties for STATIK-N simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Table 5.2 Results of 2D beam models for composite beam without shear connection . . . . . . . . . . . . 34Table 5.3 Results of 2D beam models for full, rigid shear connection . . . . . . . . . . . . . . . . . . . . . . . . 35Table 5.4 Results of 2D beam models for beam test A-01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Table 5.5 Results of 2D beam models for beam test A-02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Table 5.6 Element types, key options and formulations used in the 3D-finite element model . . . . . . 37Table 5.7 Real constant sets used for concrete material tests and beam simulations. . . . . . . . . . . . . . 41Table 5.8 Results of test simulations for concrete material models and combinations. With real constant

sets # 5 and #6 the results are identical in all three directions because both material options and reinforcement are identical in all three directions. With real constant sets #1 to #4 the re-sults for the three directions show slight differences because the amount of reinforcement is not the same in all three directions. With the combined material options accurate results are obtained, while with the pure CONC options some inconsistencies are observed. . . . . . . 43

Table 5.9 Orthotropic material properties for the shear interface and for the soft layers at the load dis-tribution blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Table 5.10 Overview on beam tests used for verification of the numerical model . . . . . . . . . . . . . . . . 48Table 5.11 Materials used for verification of finite element model . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Table 5.12 Shear connectors used with finite element model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Table 5.13 Parametric load-slip behaviours of shear interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Table 5.14 Parameters and results of the simulations performed in series A200 . . . . . . . . . . . . . . . . . 54Table 5.15 Overview of parameters for series C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Table 5.16 Overview of parameters for series C3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Table 5.17 Explanation of nomenclature for simulation series C3 and C4 . . . . . . . . . . . . . . . . . . . . . . 59Table 5.18 Overview of simulation groups for series C3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6 Synthesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Table 6.1 End slip determination from strain for special load distributions . . . . . . . . . . . . . . . . . . . . 72Table 6.2 Data sheet for the Ribdeck 80 profiled steel sheeting (cf. http://www.holorib.com) . . . . . 80Table 6.3 Loads considered for construction state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Table 6.4 Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Table 6.5 Load-slip behaviour of the shear connector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

98

99

Notation

• Indices

• Latin uppercase letters

a related to the steel beamc related to the concrete slabeff effectiveel elasticinf inferior, on the undersidek characteristicpl plasticR Resistances related to the reinforcementsup superior, on the uppersidet tensileu ultimatey yield

A1, 2, 3 Scalar parameters used for the design method by Aribert (section 3.5.3 on page 24) -Aa Cross-sectional area of steel beam mm²Ac Cross-sectional area of concrete slab mm²C1, 2, ... Parameters -E Modulus of elasticity of the respective material N/mm²Ea Modulus of elasticity of steel N/mm²Ec Modulus of elasticity of concrete N/mm²Et Tangent modulus of elasticity after yield N/mm²EaAa Axial stiffness of steel section NEcAc Axial stiffness of concrete slab NEaIa Flexural stiffness of steel section Nmm²EcIc Flexural stiffness of concrete slab Nmm²EIeff Effective flexural stiffness of composite beam considering the flexibility of the shear connection Nmm²H(x) Heaviside step function -Ia Moment of inertia of steel beam mm4

Ic Moment of inertia of concrete slab mm4

L Span of a beam mmLe Equivalent span: Length of the part of a composite beam with concrete in compression mmM(x) Bending moment in a beam at position x kNmMa Bending moment in steel beam kNmMc Bending moment in concrete slab kNmMpl,N Plastic resistance of a steel beam against bending under a given axial force N kNm

100

• Latin lowercase letters

MR Resistance of a beam against bending moment kNmMR,el Elastic resistance of a composite beam against bending moment kNm

N Sum of the shear resistance of all shear connectors within one shear length. With ductile shear connectors N equals the axial force in the chords at the critical cross-section. -

N(x) Axial force in a beam at position x kNNa,pl Resistance of steel beam against axial force kNNc,pl Resistance of concrete slab with its reinforcement against compressive axial force kNNc,t Resistance of concrete slab with its reinforcement against tensile axial force kNNf Axial force in the chords required for full shear connection -Ny Axial force in steel beam at yield kNP(x) Longitudinal shear force in the interface at position x kNPRk Characteristic resistance of a shear connector against longitudinal shear kNPu Resistance of a shear connector against longitudinal shear kNQ(x) Vertical shear force in a beam at position x kNQi Concentrated load number i kNS Part of bending stiffness of composite beam due to axial stiffness of the chords Nmm²Si Initial elastic stiffness of a shear connector kN/mmW Section modulus mm³Wel,a Elastic section modulus of the steel beam: mm³

a0Scalar parameter used in equation (3.38), a0 = 0.5 for propped construction, a0 = 0.3 for unpropped construction -

a1, a2 Scalar parameters (e.g. a1 = 0.1, a2 = 100) -b(x, z) Width of a section at position (x, z) mmbc,eff Total effective width of the concrete slab mm∆c Connector spacing: longitudinal distance between two shear connectors mmd Slip in the interface mmd(x) Slip at position x mme Vertical distance between the resulting axial forces in the two chords (used for plastic design) mmeel Vertical distance between the centres of inertia in the two chords (used for elastic analysis) mmfc Compressive strength of concrete N/mm²fct Tensile strength of concrete N/mm²fy Yield strength N/mm²h Overall depth of composite section mmha Depth of steel beam mmk Stiffness of the shear interface, k = Si/(connector spacing) kN/mm²q(x) Vertical load on the beam at position x kN/m‘n Ratio of moduli of elasticity n = Ea/Ec -n Number of shear connectors per shear length (used in [26]) -nf Number of shear connectors per shear length required for full shear connection (used in [26]) -nmin Minimum number of shear connectors per shear length (used in [26]) -r Fillet radius of the steel beam mm

Wel a,Ia

ha 2⁄------------=

101

• Greek uppercase letters

• Greek lowercase letters

tf Flange thickness of the steel beam mmtw Web thickness of the steel beam mmt(x) Shear flow in interface kN/mm‘

Mean shear flow in the interface (averaged over shear length Ls) kN/mm‘

Maximum shear flow in the interface (peak value) kN/mm‘

Maximum shear flow in the interface due to uniform load kN/mm‘

Maximum shear flow in the interface due to concentrated loads kN/mm‘

w(x) Vertical displacement of beam at position x mmw‘‘(x) Second derivative of the vertical displacement, curvature of a beam at position x 1/mm‘w Deflection of a beam: Maximum vertical displacement, usually w = w(L/2) mmweff Deflection of a beam considering the flexibility of the interface (cf. equation (3.38)) mmwa Deflection of a beam considering the stiffness of the steel beam only (cf. equation (3.38)) mmwrigid Deflection of a composite beam assuming rigid shear connection (cf. equation (3.38)) mmx Longitudinal position in the beam mmz Vertical position in a cross-section, from lower side of steel section mmz0(x) Vertical position where strain is zero at position x mmz0,a(x) Vertical position where strain is zero in steel beam at position x mmz0,c(x) Vertical position where strain is zero in concrete slab at position x mmz0,a,0(x) Vertical position where strain is zero in steel beam at position x with γ = 0 mmz0,c,0(x) Vertical position where strain is zero in concrete slab at position x with γ = 0 mmz0,a,r(x) Vertical position where strain is zero in steel beam at position x with γ = 1.0 mmz0,c,r(x) Vertical position where strain is zero in concrete slab at position x with γ = 1.0 mm

∆c Connector spacing: longitudinal distance between two shear connectors mm

∆ε Difference in longitudinal strain between the underside of the concrete slab (or at the under-side of a gap of height ∆, if present) and on top of the steel beam -

∆(x) Vertical distance between top of steel beam and underside of concrete slab at position x mm

α Scalar parameter for the longitudinal load distribution -β Parameter used in the differential equation 1/mmγ Degree of partial interaction, varying in a range of -δ(x) Dirac δ-function -

δi

Initial slip of a shear connector:Slip of a shear connector when reaching PRk on the ascending branch of the load-slip curve

mm

δPu Slip at ultimate load of a shear connector: Slip of a shear connector when reaching Pu mm

δu

Deformation capacity of a shear connector: Slip of a shear connector when reaching PRk on the descending branch of the load-slip curve

mm

δukCharacteristic deformation capacity of a shear connector according to Eurocode 4 [26] mm

t

t̂

t̂ q( )

t̂ Q( )

0 γ 1≤ ≤

102

ε(x, z) Strain at position (x, z) -εa,sup(x) Longitudinal strain on top of steel beam at position x -εc,inf(x) Longitudinal strain at underside of concrete slab at position x -εct Ultimate strain for concrete in tension -εcu Ultimate strain for concrete in compression -

∆ε Difference in longitudinal strain between the underside of the concrete slab (or at the un-derside of a gap of height ∆(x), if present) and on top of the steel beam -

ϕ Sum of the bending stiffnesses of the chords, used in the differential equation Nmm²η Degree of partial shear connection, -

ηminMinimum degree of partial shear connection for application of the rigid-plastic design method according to Eurocode 4 [26]

ηf Degree of partial shear connection required for full shear connectionν Poisson ratio -π Constant, π = 3.14159236 -θ Minimum ratio, cf. section 6.4 on page 69 -σ Axial stress N/mm²σ(x, z) Stress at position (x, z) N/mm²τ Shear stress N/mm²ω Descriptor of the axial stiffness of the chords in the differential equation 1/mm

0 η≤

103

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[27] European Code EN 10002 (1991): Metallic materials; Tensile testing; Part 1: Method of testing, April 1991

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[39] Hilti (1988): Hilti Verbunddübel HVB 100 in 3 Verdübelungsanordnungen mit Profil-blech Hi-Bond 55/0.75 auf IPE 220 Stahlträger in Normalbeton, Biegebalken-Versuche, Hilti-Report IB 04/88, March 1988

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[42] Johnson, R. P., Anderson, D. (1993): Designer’s Handbook to Eurocode 4, Part 1-1: De-sign of composite steel and concrete structures, London, June 1993

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[47] Lehner, M. (2004): Schubtragverhalten von mittels Setzbolzen befestigten Schub-verbindern, Diplomarbeit in Konstruktion, Institute of Structural Engineering, Group of Steel, Timber and Composite Structures, ETH Zurich, January 2004

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108

A-1

Appendix

A-2

A-3

Contents

1 Appendix: Moment and Axial Force Results from 2D Numerical Simulations for Rigid Shear Connection . . . . . . . . . . . . . . . . . . . 5

2 Appendix: Results of Simulation Series A200 . . . . . . . . . . . . . . 73 Appendix: Results of Simulation Series C3 . . . . . . . . . . . . . . . . 9

3.1 Bending resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93.1.1 Steel section IPE 270, Span = 7.2 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.2 Steel section IPE 270, Span = 10.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.3 Steel section IPE 270, Span = 15.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1.4 Steel section IPE 550, Span = 7.2 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.5 Steel section IPE 550, Span = 10.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.6 Steel section IPE 550, Span = 15.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.7 Steel section IPE 550, Span = 20.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Slip at yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .183.2.1 Steel section IPE 270, Span = 7.2 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.2 Steel section IPE 270, Span = 10.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.3 Steel section IPE 270, Span = 15.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.4 Steel section IPE 550, Span = 7.2 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.5 Steel section IPE 550, Span = 10.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.6 Steel section IPE 550, Span = 15.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.7 Steel section IPE 550, Span = 20.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Slip at ultimate limit state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .273.3.1 Steel section IPE 270, Span = 7.2 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.2 Steel section IPE 270, Span = 10.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3.3 Steel section IPE 270, Span = 15.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.4 Steel section IPE 550, Span = 7.2 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.5 Steel section IPE 550, Span = 10.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.6 Steel section IPE 550, Span = 15.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.7 Steel section IPE 550, Span = 20.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Appendix: Results of Simulation Series C4 . . . . . . . . . . . . . . . 374.1 Bending resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37

4.1.1 Steel section IPE 270, Span = 7.2 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1.2 Steel section IPE 270, Span = 15.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1.3 Steel section IPE 550, Span = 7.2 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1.4 Steel section IPE 550, Span = 15.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Slip at yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .384.2.1 Steel section IPE 270, Span = 7.2 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2.2 Steel section IPE 270, Span = 15.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2.3 Steel section IPE 550, Span = 7.2 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2.4 Steel section IPE 550, Span = 15.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Slip at ultimate limit state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .394.3.1 Steel section IPE 270, Span = 7.2 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3.2 Steel section IPE 270, Span = 15.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3.3 Steel section IPE 550, Span = 7.2 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3.4 Steel section IPE 550, Span = 15.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

A-4

A-5

1 Appendix: Moment and Axial Force Results from 2D Numerical Simulations for Rigid Shear Con-nection

Figure 1.1 Distribution of bending moment for beam model (N = 0 kN)

Figure 1.2 Distribution of bending moments and normal forces for slash-shaped truss

Figure 1.3 Distribution of bending moments and normal forces for cross-shaped truss

Figure 1.4 Distribution of bending moments and normal forces for K-shaped truss

Appendix: Moment and Axial Force Results from 2D Numerical Simulations for Rigid Shear

A-6

Figure 1.5 Distribution of bending moments and normal forces for Z-shaped truss

Figure 1.6 Distribution of bending moments and normal forces for H-shaped frame

Figure 1.7 Distribution of bending moments and normal forces for I-shaped frame

A-7

2 Appendix: Results of Simulation Series A200

Figure 2.1 Results from series A200 for spans of 2.4 m (left) and 3.6 m (right)

Figure 2.2 Results from series A200 for spans of 4.8 m (left) and 6.0 m (right)

Figure 2.3 Results from series A200 for spans of 7.2 m (left) and 8.4 m (right)

0

100

200

300

400

500

0 20 40 60 80 100Deflection [mm]

Load

per

jack

[kN

]

024m_01 024m_02 024m_03

024m_04 024m_05 024m_06

full con no con0

50100150200250300350

0 50 100 150Deflection [mm]

Load

per

jack

[kN

]

036m_01 036m_02 036m_03

036m_04 036m_05 036m_06

full con no con

0

50

100

150

200

0 50 100 150 200 250Deflection [mm]

Load

per

hal

f spa

n [k

N]

060m_01 060m_02 060m_03

060m_04 060m_05 060m_06

full con no con0

50

100

150

200

250

0 50 100 150 200Deflection [mm]

Load

per

jack

[kN

]

048m_01 048m_02 048m_03

048m_04 048m_05 048m_06

full con no con

020406080

100120140

0 100 200 300Deflection [mm]

Load

per

hal

f spa

n [k

N]

084m_01 084m_02 084m_03

084m_04 084m_05 084m_06

full con no con0

20406080

100120140160

0 100 200 300Deflection [mm]

Load

per

hal

f spa

n [k

N]

072m_01 072m_02 072m_03072m_04 072m_05 072m_06072m_A01 072m_A02 full conno con

Appendix: Results of Simulation Series A200

A-8

Figure 2.4 Results from series A200 for spans of 9.6 m (left) and 10.8 m (right)

Figure 2.5 Results from series A200 for spans of 12.0 m (left) and 13.2 m (right)

Figure 2.6 Results from series A200 for spans of 14.4 m (left)

0

20

40

60

80

100

0 100 200 300 400 500Deflection [mm]

Load

per

jack

[kN

]

108m_01 108m_02 108m_03

108m_04 108m_05 full con

no con0

20

40

60

80

100

120

0 100 200 300 400Deflection [mm]

Load

per

jack

[kN

]

096m_01 096m_02 096m_03

096m_04 096m_05 096m_06

full con no con

01020304050607080

0 200 400Deflection [mm]

Load

per

jack

[kN

]132m_01 132m_02 132m_03

132m_04 132m_05 132m_06

full con no con0

102030405060708090

0 100 200 300 400 500Deflection [mm]

Load

per

jack

[kN

]

120m_01 120m_02 120m_03

120m_04 120m_05 120m_06

full con no con

010203040506070

0 200 400 600Deflection [mm]

Load

per

jack

[kN

]

144m_01 144m_02 144m_03

144m_04 144m_05 144m_06

full con no con

Bending resistance

A-9

3 Appendix: Results of Simulation Series C3

3.1 Bending resistance

3.1.1 Steel section IPE 270, Span = 7.2 m

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_S235

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_S235

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_S460

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_S460

Appendix: Results of Simulation Series C3

A-10

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_outer3

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_outer3

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_bc2400

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_bc2400

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_hc200

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_hc200

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_hc300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_hc300

Bending resistance

A-11

3.1.2 Steel section IPE 270, Span = 10.0 m

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_fc50

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_fc50

0.8

0.9

1.0

1.1

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

el1el2el4el3el5

C3P_N_IPE270_2_72_x_xx_1_elx

0.8

0.9

1.0

1.1

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

el1el2el3el4el5

C3P_N_IPE270_D_72_x_xx_1_elx

0.8

0.9

1.0

1.1

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

ep1ep2ep3ep4ep5

C3P_N_IPE270_2_72_x_xx_1_epx

0.8

0.9

1.0

1.1

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

ep1ep2ep3ep4ep5

C3P_N_IPE270_D_72_x_xx_1_epx

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_100_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_100_x_xx_1_S300

Appendix: Results of Simulation Series C3

A-12

3.1.3 Steel section IPE 270, Span = 15.0 m

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_150_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_150_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_150_x_xx_1_S460

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_150_x_xx_1_S460

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_150_x_xx_1_outer3

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_150_x_xx_1_outer3

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_150_x_xx_1_fc50

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_150_x_xx_1_fc50

Bending resistance

A-13

3.1.4 Steel section IPE 550, Span = 7.2 m

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_S235

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_S235

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_S460

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_S460

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_outer3

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_outer3

Appendix: Results of Simulation Series C3

A-14

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_bc2400

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_bc2400

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_hc200

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_hc200

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_hc300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_hc300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_fc50

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_fc50

Bending resistance

A-15

3.1.5 Steel section IPE 550, Span = 10.0 m

3.1.6 Steel section IPE 550, Span = 15.0 m

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_100_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_100_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_S235

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_S235

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_S460

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_S460

Appendix: Results of Simulation Series C3

A-16

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_outer3

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_outer3

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_bc2400

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_bc2400

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_hc200

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_hc200

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_hc300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_hc300

Bending resistance

A-17

3.1.7 Steel section IPE 550, Span = 20.0 m

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_fc50

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_fc50

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_200_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_200_x_xx_1_S300

Appendix: Results of Simulation Series C3

A-18

3.2 Slip at yield

3.2.1 Steel section IPE 270, Span = 7.2 m

0

2

4

6

8

10

12

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_S235

0

1

2

3

4

5

6

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_S235

02468

101214

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_S300

012345678

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_S300

0

5

10

15

20

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_S460

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_S460

012345678

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_outer3

01234567

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_outer3

Slip at yield

A-19

0

2

4

6

8

10

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_bc2400

012345678

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_bc2400

0

5

10

15

20

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_hc200

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_hc200

0

5

10

15

20

25

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_hc300

0

5

10

15

20

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_hc300

02468

101214

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_fc50

0

2

4

6

8

10

12

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_fc50

Appendix: Results of Simulation Series C3

A-20

3.2.2 Steel section IPE 270, Span = 10.0 m

012345678

0.0 0.5 1.0 1.5 2.0 2.5 3.0N/Nf [-]

d end

,y(F

EM) [

mm

]el1el2el3el4el5

C3P_N_IPE270_2_72_x_xx_1_elx

012345678

0.0 0.5 1.0 1.5 2.0 2.5 3.0N/Nf [-]

d end

,y(F

EM) [

mm

]

el1el2el3el4el5

C3P_N_IPE270_D_72_x_xx_1_elx

012345678

0.0 0.5 1.0 1.5 2.0 2.5 3.0N/Nf [-]

d end

,y(F

EM) [

mm

]

ep1ep2ep3ep4ep5

C3P_N_IPE270_2_72_x_xx_1_epx

01234567

0.0 0.5 1.0 1.5 2.0 2.5 3.0N/Nf [-]

d end

,y(F

EM) [

mm

]

ep1ep2ep3ep4ep5

C3P_N_IPE270_D_72_x_xx_1_epx

0

5

10

15

20

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_100_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_100_x_xx_1_S300

Slip at yield

A-21

3.2.3 Steel section IPE 270, Span = 15.0 m

0

5

10

15

20

25

30

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_150_x_xx_1_S300

02468

1012141618

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_150_x_xx_1_S300

05

101520253035

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_150_x_xx_1_S460

05

101520253035

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_150_x_xx_1_S460

02468

101214

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_150_x_xx_1_outer3

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_150_x_xx_1_outer3

0

5

10

15

20

25

30

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_150_x_xx_1_fc50

0

5

10

15

20

25

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_150_x_xx_1_fc50

Appendix: Results of Simulation Series C3

A-22

3.2.4 Steel section IPE 550, Span = 7.2 m

00.5

11.5

22.5

33.5

44.5

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_S235

00.5

11.5

22.5

33.5

44.5

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_S235

0

1

2

3

4

5

6

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_S300

0

1

2

3

4

5

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_S300

0123456789

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_S460

0123456789

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_S460

0

1

2

3

4

5

6

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_outer3

0

1

2

3

4

5

6

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_outer3

Slip at yield

A-23

0

1

2

3

4

5

6

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_bc2400

0

1

2

3

4

5

6

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_bc2400

01234567

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_hc200

0

1

2

3

4

5

6

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_hc200

02468

101214

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_hc300

012345678

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_hc300

0

1

2

3

4

5

6

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_fc50

0

1

2

3

4

5

6

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_fc50

Appendix: Results of Simulation Series C3

A-24

3.2.5 Steel section IPE 550, Span = 10.0 m

3.2.6 Steel section IPE 550, Span = 15.0 m

012345678

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_100_x_xx_1_S300

012345678

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_100_x_xx_1_S300

0123456789

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_S235

0

2

4

6

8

10

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_S235

0

2

4

6

8

10

12

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_S300

0

2

4

6

8

10

12

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_S300

0

5

10

15

20

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_S460

02468

1012141618

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_S460

Slip at yield

A-25

0

2

4

6

8

10

12

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_outer3

0

2

4

6

8

10

12

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_outer3

0

2

4

6

8

10

12

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_bc2400

0

2

4

6

8

10

12

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_bc2400

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_hc200

02468

101214

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_hc200

05

101520253035

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_hc300

0

5

10

15

20

25

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_hc300

Appendix: Results of Simulation Series C3

A-26

3.2.7 Steel section IPE 550, Span = 20.0 m

0

2

4

6

8

10

12

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_fc50

0

2

4

6

8

10

12

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_fc50

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_200_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_200_x_xx_1_S300

Slip at ultimate limit state

A-27

3.3 Slip at ultimate limit state

3.3.1 Steel section IPE 270, Span = 7.2 m

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_S235

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_S235

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_S460

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_S460

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_outer3

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_outer3

Appendix: Results of Simulation Series C3

A-28

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_bc2400

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_bc2400

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_hc200

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_hc200

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_hc300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_hc300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_72_x_xx_1_fc50

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_72_x_xx_1_fc50

Slip at ultimate limit state

A-29

3.3.2 Steel section IPE 270, Span = 10.0 m

02468

10121416

0.0 0.5 1.0 1.5 2.0 2.5 3.0N/Nf [-]

d end

,u(F

EM) [

mm

]el1el2el3el4el5C3P_N_IPE270_2_72_x_xx_1_elx

02468

10121416

0.0 0.5 1.0 1.5 2.0 2.5 3.0N/Nf [-]

d end

,u(F

EM) [

mm

]

el1el2el3el4el5C3P_N_IPE270_D_72_x_xx_1_elx

02468

10121416

0.0 0.5 1.0 1.5 2.0 2.5 3.0N/Nf [-]

d end

,u(F

EM) [

mm

]

ep1ep2ep3ep4ep5

C3P_N_IPE270_2_72_x_xx_1_epx

02468

10121416

0.0 0.5 1.0 1.5 2.0 2.5 3.0N/Nf [-]

d end

,u(F

EM) [

mm

]

ep1ep2ep3ep4ep5

C3P_N_IPE270_D_72_x_xx_1_epx

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_100_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_100_x_xx_1_S300

Appendix: Results of Simulation Series C3

A-30

3.3.3 Steel section IPE 270, Span = 15.0 m

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_150_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_150_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_150_x_xx_1_S460

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_150_x_xx_1_S460

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_150_x_xx_1_outer3

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_150_x_xx_1_outer3

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_2_150_x_xx_1_fc50

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE270_D_150_x_xx_1_fc50

Slip at ultimate limit state

A-31

3.3.4 Steel section IPE 550, Span = 7.2 m

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_S235

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_S235

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_S460

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_S460

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_outer3

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_outer3

Appendix: Results of Simulation Series C3

A-32

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_bc2400

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_bc2400

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_hc200

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_hc200

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_hc300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_hc300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_72_x_xx_1_fc50

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_72_x_xx_1_fc50

Slip at ultimate limit state

A-33

3.3.5 Steel section IPE 550, Span = 10.0 m

3.3.6 Steel section IPE 550, Span = 15.0 m

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_100_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_100_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_S235

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_S235

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_S460

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_S460

Appendix: Results of Simulation Series C3

A-34

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_outer3

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_outer3

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_bc2400

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_bc2400

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_hc200

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_hc200

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_hc300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_hc300

Slip at ultimate limit state

A-35

3.3.7 Steel section IPE 550, Span = 20.0 m

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_150_x_xx_1_fc50

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_150_x_xx_1_fc50

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_2_200_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C3P_N_IPE550_D_200_x_xx_1_S300

Appendix: Results of Simulation Series C3

A-36

Bending resistance

A-37

4 Appendix: Results of Simulation Series C4

4.1 Bending resistance

4.1.1 Steel section IPE 270, Span = 7.2 m

4.1.2 Steel section IPE 270, Span = 15.0 m

4.1.3 Steel section IPE 550, Span = 7.2 m

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE270_2_72_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE270_D_72_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE270_2_150_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE270_D_150_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE550_2_72_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE550_D_72_x_xx_1_S300

Appendix: Results of Simulation Series C4

A-38

4.1.4 Steel section IPE 550, Span = 15.0 m

4.2 Slip at yield

4.2.1 Steel section IPE 270, Span = 7.2 m

4.2.2 Steel section IPE 270, Span = 15.0 m

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE550_2_150_x_xx_1_S300

0.8

0.9

1.0

1.1

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

MR

(FEM

)/MR

(Ana

lytic

al) [-

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE550_D_150_x_xx_1_S300

02468

101214

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE270_2_72_x_xx_1_S300

012345678

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE270_D_72_x_xx_1_S300

0

5

10

15

20

25

30

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE270_2_150_x_xx_1_S300

02468

1012141618

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE270_D_150_x_xx_1_S300

Slip at ultimate limit state

A-39

4.2.3 Steel section IPE 550, Span = 7.2 m

4.2.4 Steel section IPE 550, Span = 15.0 m

4.3 Slip at ultimate limit state

4.3.1 Steel section IPE 270, Span = 7.2 m

0

1

2

3

4

5

6

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE550_2_72_x_xx_1_S300

0

1

2

3

4

5

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE550_D_72_x_xx_1_S300

0

2

4

6

8

10

12

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE550_2_150_x_xx_1_S300

0

2

4

6

8

10

12

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,y(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE550_D_150_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE270_2_72_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE270_D_72_x_xx_1_S300

Appendix: Results of Simulation Series C4

A-40

4.3.2 Steel section IPE 270, Span = 15.0 m

4.3.3 Steel section IPE 550, Span = 7.2 m

4.3.4 Steel section IPE 550, Span = 15.0 m

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE270_2_150_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE270_D_150_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE550_2_72_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE550_D_72_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE550_2_150_x_xx_1_S300

02468

10121416

0.00 0.25 0.50 0.75 1.00 1.25 1.50N/Nf [-]

d end

,u(F

EM) [

mm

]

1 2 34 5 67 8 910 11 1213 14 15

C4P_N_IPE550_D_150_x_xx_1_S300