Carina Gander
email: c.gander@unibz.it

"Counting with all children from the very beginning": One attempt to promote early arithmetical skills based on part-whole thinking

Keywords: Early arithmetic education, part-whole thinking, computation without counting, inclusive mathematics education, concepts for mathematics education

"Counting with all children from the very beginning": One attempt to promote early arithmetical skills based on part-whole thinking

Carina Gander

Preliminary remarks

Every child has the right to access education of good quality, as international organisations such as UNICEF, UNESCO or the European Union affirm. Thus, the focus regarding inclusion of all children in developed countries is now on promoting the quality of the teaching and learning situation [START_REF] Florian | INCLUSION: Special or inclusive education: future trends[END_REF]. However, there seems to be uncertainty in German-speaking countries about how to create inclusive learning environments of high quality for all children in mainstream classes, not only, but especially in early arithmetical education [START_REF] Korff | lnklusiver Mathematikunterricht in der Primarstufe: Erfahrungen, Perspektiven und Herausforderungen[END_REF]. Hence, the present research project seeks to design substantial learning environments based on part-whole thinking for First Grade.

In order to allow others to make comparisons between inclusive school education in different nations, the author of this paper draft the current situation that teachers might face regarding inclusive practices in early arithmetical education in German-speaking countries, with a focus on Austria, where the research of this paper was conducted.

With the passage of time, the Austrian education system, like that of many other countries, has made great progress as far as the issue of inclusion is concerned. In particular the ratification of the UN Convention on the Rights of Persons with Disabilities in 2008 sparked debate about inclusive education (as in many countries) and the necessary measures for the common compulsory school system. Nonetheless, even today, Austria, like many other countries in the world, still focuses on a multi-track approach: Parents are free to decide whether to send their child to a (local) mainstream school or a special school. In total, in 2020-21, approximately five percent of the children in compulsory education belong to the group denoted as having special educational needs. (Statistik Austria, 2021). It should be made clear that these data refer merely to the positioning-oriented definition of inclusion (see below).

By the time children enrol in compulsory school in German-speaking countries (between the ages of six and seven), they have received considerably different targeted early mathematical education, depending inter alia on different educational plans in kindergarten and the mathematical knowledge of the kindergarten staff. The educational plans in kindergarten differ within and between Germanspeaking countries -in Austria, the educational document consists of a few pages, and early mathematical education is dealt with in a few words. The year before school enrolment all children will already have attended kindergarten. By the end of First Grade, it seems to be widely accepted within the relevant German literature, that all students should learn how to compute without counting fluently with numbers up to 20 (see also Hasel-Weide & NUhrenborger, 2013). Therefore, it seems to be helpful if teachers first get children to grasp that numbers (at least up to 10, if not up to 20) are composed of other numbers and then, based merely on that knowledge, initiate their understanding of addition and subtraction (see also [START_REF] Schipper | Schulerinnen und Schuler mit sonderpadagogischem Forderbedarf[END_REF]Gaidoschik, 2019a;[START_REF] Gaidoschik | Didactics as a source and remedy of mathematics learning difficulties[END_REF].

The author of this paper is aware that in the field of international mathematics education, designing substantial learning environments of high quality in early arithmetical education is a central element for colleagues from other countries, where there are also likely to be challenges for teachers to teach mathematics inclusively. Thus, the following considerations concerning the implementation of partwhole thinking in early arithmetical teaching may apply to other nations as well.

Considerations on inclusive education in early arithmetical teaching

In the international inclusive education community, there is no clear working definition of inclusion. Consequently, in many nations inclusive practices are handled very differently, and these practices can differ also between schools in one country, or even within the same school (see also Booth & Ainscow, 1998). To specify inclusion, it has been defined as both a one-dimensional and/or a multidimensional oriented approach. The former means physical positioning and teaching of children in an ordinary class [START_REF] Vislie | From integration to inclusion: focusing global trends and changes in the western European societies[END_REF], the latter represents a wide plurality with other qualities (for example a positive learning environment to promote all children's learning development) in addition to positioning. Many authors agree that inclusion in developed countries is now not merely about physical positioning, but has to comprise several aspects. Empirical research has also shown that positioning alone is no guarantee of meeting the needs of all children (see also [START_REF] Mitchell | What Really Works in Special and lnclusive Education: Using Evidence-based Teaching Strategies[END_REF][START_REF] Vislie | From integration to inclusion: focusing global trends and changes in the western European societies[END_REF].

The international mathematics education community will certainly respect the multidimensional oriented approach of inclusion. To understand how pupils learn, the theory of constructivist learning is vital. The students actively construct knowledge, and of course, there is huge variation amongst children with regard to how quickly and how sophisticatedly they are able to solve mathematical problems: "Children with different developmental backgrounds may well be able to get the same answers on an arithmetical task, but how they do so might differ significantly" (for more details of the discussion between constructivist views and other views on mathematics education theory and research, see Steffe & Kieren, 1994, p. 719).

Thus, in recent decades, some considerations on general inclusive (mathematics) didactics and few substantiated didactical concepts for inclusive mathematics education in primary school have been developed in the mathematics education community, but I am not aware of any (apart from the following), that explicitly deal with the utilization of the concept of numerical part-whole thinking in First Grade.

Considerations on the TIGER concept (Gaidoschik, 2019a)

All children in First Grade classes should be aware of "numbers as compositions of other numbers" (Resnick, 1983, p. 114), also termed the "part-whole schema" (Resnick, 1983, p. 115). Resnick describes this goal as "probably the major conceptual achievement of the early school years" (1983, p. 114), therefore a preferable alternative to computing by counting. According to this concept, children should learn to divide numbers in terms of part and whole relationships for all numbers at least up to 10. Children thus can be led to understand that any whole number can in fact be divided into smaller numbers, and so the number 7 for example, can be divided into 2 and 5. When put together, the parts 5 and 2 fulfil the requirement of being equivalent to the whole number 7. With that understanding, they might be able to solve addition (2+5; 5+2) as well as subtraction problems (7-5; 7-2) -without counting.

The utilization of the concept of numerical part-whole thinking in First Grade seems to be the standard in international mathematics education as in for example the "Number framework" (New Zealand Ministry of Education, NZME, 2008) and the "Mathematics programme of study" (UK Department of Education, 2013). Quite when this concept is taught in the academic year can differ across nations. Part-whole thinking is incorporated as early as possible in the school programme in these guidelines for teachers' actions. The New Zealand Number framework explicitly states:

It is important for you to recognise that part-whole thinking is seen as fundamentally more complex and useful than counting strategies. One reason is that counting methods are strictly limited, whereas part-whole methods are more powerful. Counting strategies are an inadequate foundation for these ideas, and this means that for counters, many advanced number ideas are inaccessible. Therefore, your major objective is to assist students to understand and use part-whole thinking as soon as possible. (NZME, 2008, p. 7) Current models and learning trajectories of these and other guidelines name different numerical stages for counting strategies, and give recommendations for teachers to follow and to work with their classes from one stage to the next. However, Gaidoschik (2019a) describes in detail a structuregenetic didactical attempt as an alternative approach to the already existing ones. The concept he has named TIGER (Gaidoschik, 2019a) attempts to show how to teach children addition and subtraction on the basis of part-whole thinking. The author of this paper will provide a more detailed explanation of the TIGER concept and list some examples of the learning situations of the current project.

The TIGER concept of part-whole thinking

The TIGER ("Teile im Ganzen Erkennen und damit Rechnen") concept, created by Gaidoschik (2019a), focuses on solid number concepts in early arithmetical teaching. The 13 learning situations of the current research project are based on these ideas. Drawing on this concept, the learning situations consist of the following three fundamental aspects, whereby the researcher (Gaidoschik, 2019a, p. 424) recommends teachers "work with children in all three Fields A to C more or less concurrently" and "it would not help to stick to one single theme for too long in a row [.] the sites relate to each other": Firstly, (A) a child needs to acquire a solid understanding of counting competences and should thus master the significance of cardinality as a means to understand "how many of whatever". This includes teaching the "counting principles", defined by [START_REF] Gelman | The Child's Understanding of Number[END_REF]. The substantial learning environments that have been designed try to allow all children to consolidate their individua/ counting competences in the counting activities, for example by counting the number of children present in the class. Counting is used almost every day: Is anyone absent today? Is the number different from yesterday and if so, how has it changed? Of course, First Grade classes will be quite heterogeneous in their counting competences: Many children will already have no problems with counting (forwards and/or backwards), while some children will need more counting activities than others. As a benchmark, all children should (at an earlier or later stage) be able to count easily up to at least 10, both forwards and backwards [START_REF] Gaidoschik | Rechenschwachen vorbeugen -Erstes Schul}ahr: Vom Zahlen zum Rechnen[END_REF].

Secondly, (B) it is important that a child is able to judge very small sets without counting -through direct pattern recognition -and uses subitizing for smaller quantities (up to 4) and uses perceptua/ subitizing for quantities greater than 3 or 4. Perceptual subitizing is only possible for children and indeed us adults if bigger sets are presented in a structured way, so that counting can be avoided (see also [START_REF] Clements | Learning and Teaching Early Math. The Learning Tra}ectories Approach[END_REF]. The learning situations that the author of this paper has created include activities to teach part-whole thinking, including in particular activities using fingers, recommended by Gaidoschik (for more details of fingers as a very useful "material", see [START_REF] Gaidoschik | Rechenschwachen vorbeugen -Erstes Schul}ahr: Vom Zahlen zum Rechnen[END_REF]). Thus, for example, children should learn at the beginning to show the right number of fingers without actually counting them, and without extending them one by one. As a starting point we should focus on (at least) all numbers up to 5. In the following activities, structured dot patterns, dice, then ten-frames are useful materials in the learning situations. All in all, the use of materials in the current research project is important for children to acquire menta/ pictures and finally automatized knowledge but the use of materials is limited to a few structured arithmetical materials, as mentioned above. Note that all the activities with material should be presented in a structured way so that children are able to identify quantities (first for quantities up to 5 and later for all numbers at least up to 10) at a glance. The focus of such early activities is to teach children the interpretation of these structures, their relations to 5 and 10; and especially their part-whole compositions (5 as consisting of 2 and 3, or of 4 and 1, etc.).

Thirdly, (C) a child should compare quantities and numbers. On the basis of one-to-one matching up, children should learn and/or consolidate that no counting is needed for pairing the items for number comparison such as identifying without counting [START_REF] Gaidoschik | Rechenschwachen vorbeugen -Erstes Schul}ahr: Vom Zahlen zum Rechnen[END_REF]. The learning situations focus on activities that foster children's ability to compare quantities. Thus, inter alia in the learning activity "Throwing Tiles" they use one-to-one matching without needing to count the tiles, they create rows and then compare in pairs their quantities. In "Finger use" they should "show the right number of fingers without counting them" and then compare the fingers with other quantities and/or numbers.

Organising the created substantial learning environments in the classroom

To achieve a ba/ance between children learning individually and learning together in (early arithmetical) education, which is the general consensus within the mathematics education community in German-speaking countries (see for example, Hasel-Weide & NUhrenborger, 2015), the current learning situations are divided into: (I) Whole-class instruction; (II) Individual Work; (III) Teamwork. Each teacher decides, depending on the class situation, which organisational structures are best and for which children. Presumably, for some children it might be useful at some point to be given additional or different activities to practice and/or consolidate their arithmetical knowledge. Nevertheless, as Feuser argues (1997), all pupils should, as much as possible, be working on the same activity even if it is being carried out on a different level. The learning situations in the current paper respect these ideas and try to give specific ideas for teacher's action in classes.

The research project, research method and research question

To be able to examine the effectiveness of the learning situations created, collective case studies following the framework of design-based research approaches [START_REF] Euler | Design Research -a paradigm under development[END_REF] have been conducted. In the Alpha-cycle, conducted in the school year 2019-2020 with 45 First Grade students in two Tyrolean mainstream classes, video-recorded conclusions were drawn for the implementation and further development of the substantial learning environments. These results were processed in an additional cycle. The Beta-cycle, which is relevant for the present data analysis, was carried out in five Tyrolean primary school classes in the school year 2020-2021 with 95 students over a 10-week period in First Grade in inclusive mainstream classes. Before starting the classroom implementation, all teachers were instructed in using and adapting the learning situations. To generate insights into children's learning and their mathematical thinking, each lesson was video recorded, and framed as the basis for data analysis using "Qualitative Content Analysis" [START_REF] Mayring | Qualitative content analysis: Theoretical background and procedures[END_REF].

The pairs of children analysed are, as far as possible, unchanged, so that conclusions can be drawn regarding the progress of their learning. Furthermore, "Assessment of Teaching-Learning-Situations in Mathematics of the Early Grades" [START_REF] Steinweg | Einschatzung der Qualitat von Lehr-Lernsituationen im mathematischen Anfangsunterricht -ein Vorschlag[END_REF] is used for data analysis, as an additional level of analysis. Steinweg's idea of dimensions focuses on the teachers' possibilities for action in First and Second Grade. The combination of both gives indications for (further) development of the substantial learning environments.

The questions guiding the paper are: (1) Do the analysed transcripts of the substantial learning environments based on part-whole thinking indicate any difficulties of understanding for many children and/or a particular group of children with certain arithmetical knowledge? (2) Are there difficulties in teaching the substantial learning environments based on part-whole thinking; and if so, are these observed in severa! scenes?

Preliminary results

It should be made clear that it is not the intention of the author to draw conclusions from specific scenes or class observations. The idea is not that a learning activity should be done just once in a class or that a!! children should be able to solve the arithmetic task (at the same time). Thus, the dimensions of Steinweg (2010) -consistent with the author's intention -are considered 'competence-oriented', that is to say, the (analysing) focus is more on the teacher's achievements than on their failings. When Steinweg's dimensions are used in a way that severa! scenarios of teachers' actions in classes are analysed, conclusions can then be drawn to give teachers targeted indications for the work in daily classes.

Drawing on these ideas, preliminary outcomes of the analysis of some transcripts using "Qualitative Content Analysis" [START_REF] Mayring | Qualitative content analysis: Theoretical background and procedures[END_REF] indicates content related problematic areas in early arithmetic education: These are problematic areas of arithmetical knowledge that might be common to many children and/or a particular group of children. To illustrate by means there are some examples. In "Throwing Tiles" children take turns throwing 10 reversible tiles, which have a red side and a blue side. Then together they have to match one red with one blue tile, using one-to-one matching without counting. They create two rows to identify who has more tiles, and how many more there are of one colour compared to the other. Regarding Steinweg's idea of mathematical dimensions (2010), not all teachers were able to foster children's arithmetical competences and not all teachers were consistently able to identify these problematic areas of arithmetical knowledge. In some sequences, for example in this sequence "Throwing Tiles", some teachers did not realize that children require a solid understanding for one-to-one-matching up as a basis for further arithmetical strategies. Especially for underachieving children in mathematics and children with SEN the understanding and utilization of the one-to-one-matching process was consistently difficult in several instances on several days. Again, it is the same group of children that seem to have difficulties in the following scenarios: One of the advanced activities of "Finger Use" is to work out the 5-plus-x structures of the numbers 6 to 10. The children play in pairs, and one says a number, for example 7, and the other child, with hands underneath the table, without looking at their fingers, has to show the right number of fingers without actually counting them, and without extending their fingers one by one. Their partner then checks the solution. In further exercises, they have to describe what they would do, without actually doing it. For example, "Seven. I have to show one full hand and two fingers on the other hand; in total that makes seven". Drawing on Steinweg's idea of mathematical dimensions (2010), even in this activity not all teachers were able to foster the use of fingers. For these children it might be worthwhile to give more thought-provoking stimuli to acquire part-whole thinking. Thus, the base numbers 5 and 10 (5 fingers on one hand, 10 fingers on both hands together) should be focused on first. Thus, a preliminary indication of the analysed transcripts is that particular attention should be paid to the mathematical thinking of underachieving children and to children with SEN, and it is evident that teachers seem to find it especially difficult to meet the learning needs of those children.

Closing remarks

The current research project on early arithmetical teaching in First Grade, presented in this article, is one attempt to face challenges in teaching (arithmetic) inclusively. As has been outlined, the substantial learning environments that have been created are based on the TIGER concept (Gaidoschik, 2019a). The author of this paper agrees with the ideas of many researchers in the relevant mathematics education literature, that children who need special support in learning (mathematics) do not learn differently, nor need completely different concepts than are already recommended for all children (see Gaidoschik, 2019a[START_REF] Gaidoschik | Didactics as a source and remedy of mathematics learning difficulties[END_REF][START_REF] Moser Opitz | Zahlen, Zahlbegriff, Rechnen. Theoretische Grundlagen und eine empirische Untersuchung zum mathematischen Erstunterricht in Sonderklassen[END_REF][START_REF] Van De Walle | Elementary and Middle School Mathematics: Teaching Developmentally[END_REF]). Thus, their learning behaviour is not completely different from that of their classmates. Of course, some mathematical content might be acquired by some children at a later stage than other children, but all children should learn them at some point during their mathematical education. From the structure-genetic didactical analysis approach this idea seems logical because the mathematical content is the same for all children.

Nevertheless, empirical evidence (see for example, [START_REF] Pfister | Scaffolding for mathematics teaching in inclusive primary classrooms: a video study[END_REF], as well as some interim results of this research project indicate how difficult it is for teachers to implement already existing concepts of mathematics education in their classes, even for those who participate voluntarily in in-service development programmes.

Thus, one conclusion of this research is that it is essential to enhance teachers' knowledge of relevant pedagogical methods and content, so that teachers can then fulfil all pupils' differing learning requirements. The other conclusion is that data analysis has also shown that for teachers to get a collective picture of their students' arithmetical knowledge, it might be worthwhile to identify problematic areas of arithmetical knowledge. The awareness of these problematic areas of arithmetical knowledge is crucial for quality teaching in inclusive mathematics education. Of course, this places high demands on the teacher's abilities.

Yet, in further data analysis something that needs to be evaluated is whether such problematic areas of arithmetical knowledge involve many children and/or a particular group of children in different classes that participated; and whether it was consistently difficult for the teachers to identify these problematic areas of arithmetical knowledge and to foster children's competences. This would allow us to develop further indications for the work with teachers and children.


Julie Vangsøe Faerch
email: jvaf@edu.au.dk

Development and evaluation of primary school students' mathematical competencies via dynamic online environments

Keywords: Formative assessment, mathematics, competency, competency-oriented education

Mathematical competencies [START_REF] Niss | Mathematical competencies revisited[END_REF] have been developed and implemented in the Danish curriculum through the last two decades. At the same time, online assignment portals have gained considerable ground in the Danish elementary school. Denmark's largest portal alone, matematikfessor.dk, has 75% of the Danish primary and lower secondary schools as regular subscribers, and an average of DKK 1.5 million assignments are answered daily (EduLab, 2021). When Denmark was locked down on March 11, 2020, the number of daily completed tasks rose to approximately 6,000,000. During the lockdown, the number of unique daily logins rose from approximately 45,000 to 130,000 [START_REF] Elkjaer | Designing Tasks for a Dynamic Online Environment: Applying Research into Students' Difficulties with Linear Equations[END_REF]. Assignments in such portals, however, are mainly used for training students' basic skills. In the Danish context, a report from 2016 concludes that as many as 78% of the tasks that students in Danish primary and lower secondary schools encounter in mathematics teaching can be categorized as training tasks [START_REF] Bremholm | Elevopgaver og elevproduktion i det 21. århundrede: Praesentation af projektet-forskningsspørgsmål, metode og hovedresultater[END_REF].

The purpose of this ongoing project is to investigate to what extent it is possible to develop tasks for online assignment portals, so that students in the intermediate stage of primary school have the opportunity to develop mathematical representation competency as defined by [START_REF] Niss | Mathematical competencies revisited[END_REF]. The tasks are to be designed in a way that supports the teachers' formative assessment of their students' mathematical representation competency. As the portal works now, the teachers receive information about the number of correct responses provided by each student. In addition to these information's, the aim is to inform teachers of the level of competency for each of their students.

The methodology of design-based research (Cobb, Jackson & Sharpe 2017) is used for both continuous development and adaptation of the competency-oriented tasks. The tasks are to be designed in clusters in a way that allows for student learning, while the student's answers inform the assessment tool without the student taking an actual test. The assessment tool is formative (Black & William, 2009) as knowledge of the students' representation competency within fractions will both inform the further work in the online assignment portal and support the teacher's formative assessment through a description of the student's competency level in terms of radius of action, degree of coverage and technical level (Niss & Jensen, 2019). A minimum of three design iterations are expected to be completed during the project.

During the first iteration, development of competency-oriented tasks for online assignment portals will be based on findings from a systematic literature review regarding task design with a focus on online assignment portals and mathematical competencies, respectively. Existing knowledge of the interfaces of these areas must result in a framework for how competency-oriented tasks are to be designed for an online portal.

During the second iteration, coding categories for the assessment tool are identified based on three dimensions of possessing and developing a competency: degree of coverage; radius of action; and technical level [START_REF] Niss | Mathematical competencies revisited[END_REF]. Degree of coverage regards the extent to which the various aspects are covered. A person who can both interpret and understand relations between different representations and has knowledge about information loss and gain when shifting between representations has a higher degree of coverage-than a person who can only do the first.

Radius of action is the spectrum of contexts and situations in which the competency can be activated. The degree to which the student is possessing representation competency is to be measured by the extent to which the student can solve tasks in routine and/or new situations; in relation to internal mathematical situations and/or application-oriented situations; use fractions in different situations. The technical level deals with conceptually and technically advanced conditions and tools the person can activate the competence towards. The student's responses are used to identify development opportunities so that the student's next set of tasks can help further development the competencies radius of action.

During the third iteration, both task design and the assessment tool will be tested on a larger number of students to further inform both the task design and the assessment tool.

Matematikfessor.dk makes their online assignment portal available for testing of tasks and evaluation tool.


Clemence Lepourry-Nicollet

Piseth Heng
email: piseth.heng@insa-rennes.fr

Hugues Somja

Franck Palas

INFLUENCE OF PARTIAL SHEAR CONNECTION ON THE BEHAVIOUR OF U-SHAPED STEEL CONCRETE BEAMS WITH L-SHAPED SHEAR CONNECTORS

Keywords: Partial shear connection, L-shaped shear connector, U-shaped steel concrete beam, Flexural tests under sagging bending moment, two-layer beam element formulation

This paper presents a study on the influence of partial shear connection on the behaviour of the U-shaped steel concrete beams (USCB) with L-shaped shear connectors. An experimental campaign of three full-scale experimental tests on the USCB with a degree of shear connection ranging from 0.4 to 1 has been conducted under sagging bending moment. The test results showed high ductility of the USCB with both full and partial shear connections. The failure mode of the USCB was governed by the plastic buckling of the upper flanges of the Ushaped steel profile at the location of the shear connectors. The plastic moment resistance of the USCB with partial shear connection predicted by the full plastic analysis approach and by the simplified approach in Eurocode 4 agreed well with the experimental results. Furthermore, the flexural stiffness of the USCB with partial connections predicted by American and Australian standards fit well with the experimental results. A numerical model based on the two-layer beam element formulation taking into account the interlayer slips with continuous connection in a co-rotational framework was also adapted to determine the behaviour of the USCB. This model was validated against experimental tests as well as against the analytical approaches. It was finally used to carry out a parametrical study in order to investigate the effect of selected parameters such as the degree of shear connection, the concrete strength and the steel strength of U-shaped profile.

Introduction

Over the past years, many forms of composite beams have been proposed in order to achieve the challenging architectural demand for long-span structures such as bridges and commercial buildings. In the so-called reinforced concrete beam, the inclusion of embedded steel rebars is needed for withstanding axial tensile stresses from flexural load whereas the concrete works well to resist to axial compressive stresses. However, the RC beam is not efficient for longer spanned and tall structures due to its massive cross-section. The pre-fabricated pre-stressed concrete beams have been an alternative solution to achieve larger span-to-depth ratios, but their weight makes the handling expensive. In the same way, the detailing of the anchorage and the specific skills needed to introduce the tensioning force in the structure make post-tensioning beams complex and expensive. On the other hand, composite steel-concrete beams can reach suitable span-to-depth ratios while avoiding these inconveniences. Among other configurations, steel-concrete composite beams incorporating cold-form U-shaped steel profile as a permanent formwork have been intensively proposed in previous studies [START_REF] Oehlers | Composite profiled beams[END_REF]- [START_REF] Keo | Behaviour of a new shear connector for U-shaped steel-concrete hybrid beams[END_REF]. This new configuration of beams provides many advantages over traditional reinforced concrete beams such as more ductility, increases in shear capacities, higher span-to-depth ratio, and reduction in creep and shrinkage. The composite beams are usually composed of a steel beam, a reinforced concrete beam or slab, and shear connectors. The structural behaviour of the composite beams depends largely on the shear capacities of the connectors to transfer horizontal shear force between the steel and the concrete parts of the beams. The shear capacities can be contributed by three main sources at the interface between the steel and the concrete: chemical bonding, friction (depending on the roughness of the surfaces), and mechanical shear connection (embossments, ribs or shear connectors). The design of the shear connection can be categorized as full shear connection or partial shear connection. In the full shear connection design, sufficient strength is ensured at the steel-concrete interface, allowing the weaker between the concrete and the steel component to reach its strength. On the other hand, the maximum flexural shear capacity of the composite beams in the partial shear connection is governed by the failure at the steel-concrete interface before the full capacity of the steel or concrete beam is reached.

In most cases of medium-to-large span structures, the design is limited by the serviceability limit state requirements, i.e. the deflection limit. In such cases, the design using partial shear connection is more economical and preferable to that using full shear connection due to a substantial reduction of the number of shear connectors.

Many studies in the past were conducted to determine the effect of the partial shear connection on the flexural behaviour of the popular I-shaped steel concrete composite beams with shear studs [START_REF] Johnson | Composite Structures of Steel and Concrete, Volume 1 -Beams, slabs, columns, and frames for buildings[END_REF]- [START_REF] Nie | Experimental study of partially shear-connected composite beams with profiled sheeting[END_REF] and with channel type connectors [START_REF] Baran | Behavior of steel-concrete partially composite beams with channel type shear connectors[END_REF] [START_REF] Ozturk | Nonlinear Fiber Modeling of Steel-Concrete Partially Composite Beams with Channel Connectors[END_REF]. Oehlers et al [START_REF] Oehlers | Partial Interaction in Composite Steel and Concrete Beams with Full Shear Connection[END_REF] distinguished the difference between the partial interaction and the partial shear connection and studied the effect of partial interaction on the full-shear connection strength of an I-profile steel concrete composite beam. On the other hand, Nie et al [START_REF] Nie | Experimental study of partially shear-connected composite beams with profiled sheeting[END_REF] performed 13 experimental tests on simply supported steel-concrete composite beams subjected to positive and negative bending moments with the degree of shear connection ranging from 0.25 to 1.85. By comparing their test results with the calculations based on the equivalent method provided in Eurocode 4 [START_REF] En | Eurocode 4: Design of composite steel and concrete structures[END_REF] and AISC [START_REF]Load and resistance factor design specification for structural steel buildings[END_REF], it was shown that partial shear connections could be used in both positive and negative bending regions for continuous composite beams with good accuracy. Baran and Topkaya [START_REF] Baran | Behavior of steel-concrete partially composite beams with channel type shear connectors[END_REF] were interested in investigating the behaviour of steel-concrete partially composite beams with channel type shear connectors by performing single point bending tests on four composite beams with various degrees of connection. It was revealed that major improvement on moment capacity and stiffness was obtained even for beams with relatively low degree of partial shear connection, and that the moment inertia specified in AISC [START_REF]Load and resistance factor design specification for structural steel buildings[END_REF] specification overestimated the measured flexural stiffness of the beams for all degrees of partial degrees of connection investigated.

While U-shaped steel-concrete composite beams are considered as an interesting solution for long-span structures, their behaviour under the effect of partial shear interaction is still not well established. Among the few studies, Liu et al [START_REF] Liu | Flexural behavior of rebar truss stiffened cold-formed Ushaped steel-concrete composite beams[END_REF] performed bending tests on ten specimens of rebar stiffened cold-formed U-shaped steel-concrete composite beams considering three parameters, among which is the degree of shear connection. The shear connection in their study was ensured by a rebar truss welded to top flanges of the U profile, shear stud and tensile reinforcement. It was found out that the ultimate bending capacity of the beams could be calculated according to Eurocode 4 [START_REF] En | Eurocode 4: Design of composite steel and concrete structures[END_REF] with the consideration of three modification factors in order to take into account for effective width of the slab, effective depth of steel web and the actual concrete compressive strength. However, the effect of partial shear connection on the behaviour of the composite beams itself was not fully investigated, except that it was grossly concluded that the increase of bending capacity was less than 10 percent by improving the degree of shear connection. Liu et al [START_REF] Liu | Experimental investigation on the flexural behaviour of steel concrete composite beams with U-shaped steel girders and angle connectors[END_REF] studied the flexural behaviour of steel-concrete composite beams consisting of U-shaped steel girders and angle connectors by conducting four-point bending tests on five specimens with different connector intervals and installation locations, giving the degrees of shear connection ranging from 0.55 to 1. The results showed that the beam with angle connectors welded to the top flanges performed better than with angle connectors welded to the interior webs. Furthermore, they found that the ultimate flexural capacity of the fully composite beams could be reasonably estimated by the calculations following Eurocode 4 [START_REF] En | Eurocode 4: Design of composite steel and concrete structures[END_REF] and AISC [START_REF]Load and resistance factor design specification for structural steel buildings[END_REF] codes. Acceptable results of the ultimate bending capacity were also obtained for the partially composite beams using Eurocode 4 [START_REF] En | Eurocode 4: Design of composite steel and concrete structures[END_REF].

In a previous investigation by the authors [START_REF] Keo | Behaviour of a new shear connector for U-shaped steel-concrete hybrid beams[END_REF], a configuration of U-shaped steel-concrete beams (USCB) with L-shaped shear connectors was proposed. The composite beam is composed of a U-shaped steel profile, precast and cast-in-place slabs, encased dropdown reinforced concrete beam and L-shaped connectors welded to top flanges of the U profile, as shown in Figure 1. In this configuration, the L-shaped connectors are fully embedded in the concrete, being functional also in limiting the uplifts. As the behaviour of this new type of shear connection is not covered yet in the current norms, the first study on its behaviour and on its force transfer mechanism was carried out through five asymmetrical push-out tests. The results were used to validate a complete 3D FE model of the test, which was then used for performing a parametric study in order to propose a design formula to predict the shear capacity of the shear connectors.

The next phase of the study presented in this article is needed to verify the performance of these shear connectors in the composite beams under positive bending moment considering partial shear connection. Three full-scale flexural tests on the USCB with a degree of shear connection ranging from 0.4 to 1 have been performed under sagging bending moment in order to determine the moment resisting capacity, the slip distribution along the length of the beam, the strain distribution in the cross-section, the ductility, and the failure mode. Experimental results such as the moment resisting capacity and the flexural stiffness were then compared with the values predicted by different design approaches and standards. A numerical model based on the two-layer beam element formulation taking into account the interlayer slips with continuous connection in a co-rotational framework was also adopted to determine the behaviour of the USCB and validated against experimental tests as well as against the analytical approaches. At last, a parametrical study was carried out in order to investigate the effect of selected parameters such as the degree of shear connection, the concrete strength and the steel strength of the U-shaped profile.

Recall of asymmetrical pushout tests on L-shaped shear connectors in the USCB

A study on the behaviour of L-shaped shear connectors used in the USCB was previously performed by the authors through five large-scale pushout tests [START_REF] Keo | Behaviour of a new shear connector for U-shaped steel-concrete hybrid beams[END_REF]. The tests were to quantify the strength and the deformation capacities of shear connectors as well as to gain an insight into the force transfer mechanism in the USCB. In the tests, the specimen was placed horizontally on two vertical supports connected with slotted holes and greased PTFE plates to allow displacements (see Figure 2). The pushout force was applied through the force jack to one end of the U-shaped steel beam, whereas the opposite end surface of the concrete slab was put in contact with a rigid beam that blocked the displacement of the specimen, giving the reaction against the pushout force. A total of four tests are recalled here. The first specimen denoted by PO-L40 has three connectors with L-shaped section of 40×40×4mm. The last three specimens denoted by PO-L50a, PO-L50b and PO-L50c have three connectors with L-shaped section of 50×50×5mm. These connectors were welded on the top flanges of the U-profile.

Figure 2: Test setup and specimens in pushout tests [START_REF] Keo | Behaviour of a new shear connector for U-shaped steel-concrete hybrid beams[END_REF].

The failure mode of the specimens from the tests were the following. The failure of PO-L40 was governed by the connector fracture that was associated with the plastic deformation of the connector (see Figure 3a). A similar failure mode to PO-L40 was observed in the other three specimens PO-L50 (a,b,c) (see. Figure 3b). Figure 4: Load-mean slip/Load-mean uplift curves [START_REF] Keo | Behaviour of a new shear connector for U-shaped steel-concrete hybrid beams[END_REF].

Figure 4 shows the relation between the push-out load and the mean slip as well as the mean uplift between the concrete and the steel beams. As observed in PO-L40 and PO-L50 (a, b, c), the L-shaped shear connectors were ductile and have a sufficient capacity in resisting the vertical separation between the concrete and steel components, which meets the requirement in the design codes for partial shear connection design [START_REF] En | Eurocode 4: Design of composite steel and concrete structures[END_REF].

A complete 3D finite element model was also developed and validated against the experimental results. The FE model was then used to perform a parametric study by evaluating the effect of parameters such as the concrete strength 𝑓 , the cross-section of the L-shaped connector, the welded length of the shear connector to the upper flanges of the profile (𝐿 ), as well as the thickness of concrete slab (𝐻 ). Based on the results obtained from the parametric study, the equation for predicting the shear capacity of one L-shaped shear connector was proposed as following:

𝑃 = 2(𝐴 + 𝐴 ) 𝑓 √3 + 2𝐾 𝐴 𝑓 ( 1
)
where 𝑓 is the ultimate strength of the connector; 𝐴 , 𝐴 and 𝐴 are defined in Figure 5; and 𝐾 is the calibration factor on concrete strength due to local pressure effect, defined from a regression analysis of the parametric study by the following expression:

Experimental program

The present experimental program consists of bending tests on three full-scale specimens under positive bending moment. The main parameter for the investigation is the degree of shear connection. The first specimen, denoted by BM-1, was tested under a six-point bending configuration (see Figure 6a). The test on the second specimen was initially performed using the same configuration as BM-1; however, due to the limited capacity of the force jack, the maximum load was not reached. The specimen was then tested again but under a four-point bending configuration (Figure 6b). The second specimen is hereby denoted by BM-2(I) for the first phase of loading and BM-2(II) for the second phase. The last specimen, denoted by BM-3, was directly loaded with the four-point bending test.

Test specimens

The specimens of the USCB, as illustrated in Figure 8, were composed of a U-shaped steel profile, concrete pre-cast panels, a reinforced concrete beam encased in the U-shaped steel profile, and L-shaped shear connectors.

The dimensions of the specimens were chosen to conform to the design of the USCB in the AVRIL project. The information on the project and the design can be found in [START_REF] Lepourry | An Innovative Concrete-Steel Structural System for Long-Span Structure Allowing a Fast and Simple Erection[END_REF]. The U-shaped steel profile was made of cold formed steel with a constant thickness of 6 mm and a height of 500 mm. The bottom flange was 300 mm wide, and top flanges were 100 mm wide. The concrete slab was 200 mm high, and the total height of the USCB specimens is thus 700 mm. The concrete slab was also reinforced with the rebar mesh ST35, made of HA7 rebars with a spacing of 10 cm by 30 cm. In addition, the steel reinforcement of the concrete beam consisted of 4 HA16 rebars at the bottom, and 2 HA12 at the top, with a cover of 40 mm. The spacing of the HA8 stirrup, the dimension of the specimen, and the type and spacing of L-shaped shear connectors were different for each specimen. These details are provided in Table 1. The L-shaped shear connectors were welded to the upper flange of the U-shaped profile with the welding length of 𝐿 = 40 mm and 𝐿 = 40 mm for specimen BM-1 and 𝐿 = 40 mm and 𝐿 = 50 mm for specimens BM-2 and BM-3 (see Figure 7). The weld throat was 3 mm for the three specimens.

Material properties

The concrete has a strength class of C25/30 for the encased RC beam and C35/40 for the pre-cast panels. The concrete characteristics on the day of bending tests were determined on three cylinders samples with dimensions of 11×22 cm using cylinder concrete tests following the norm NF EN12390-3 [START_REF]NF EN12390-3, Testing harden concrete: Compressive Strength of Test Specimens[END_REF]. The results of these tests are reported in Table 2, where 𝑓 and 𝜎 are the mean resistance of the concrete and its standard deviation. The steel grades of U-shaped steel beam, L-shaped steel profile and steel rebars are S355, S235 and B500B respectively. For each steel material, a tensile test was carried out following the norm NF EN ISO 6892-1 [START_REF]NF EN ISO 6892-1, Metallic materials -Tensile testing -Part 1 : method of test at room temperature[END_REF]. 180 The results of these tests are reported in Table 3, in which 𝑓 and 𝑓 are the elastic and ultimate limits, respectively.

Test setup

The test setup was composed of a hydraulic force jack with a capacity of 1500 kN, a system of spreader beams, guiding columns, a specimen of the USCB, and a supporting system, as illustrated in Figure 9. In the test setup, the specimen of the USCB was simply supported on two supports. For one of the supports (see detail of support 1 in Figure 9), the rotation and the horizontal displacement were released with a hinge mechanism and with the use of the PTFE layers, respectively. For the other support (see detail of support 2 in Figure 9), the rotation was released, while the horizontal displacement was blocked at the upper flanges of the U-shaped steel profile (at the level of the elastic neutral axis of the composite cross-section). As already explained, the system of spreader beams had two configurations depending on the configuration of the bending tests (see Figure 10). The transversal displacement of the specimen was constantly controlled using the four guiding columns. The degree of shear connection can be defined as the ratio of the total shear capacity of the shear connectors to the yield capacity of the steel beam:

𝜂 = 𝑛𝑃 𝐴 𝑓 , ( 3
)
where 𝑃 is the ultimate resistance of one L-shaped shear connector, as being recalled in Section 2; 𝐴 is the area section of the U-shaped steel profile; 𝑓 , is the yield limit of the U-shaped steel profile; 𝑛 is the number of shear connectors placed in the USCB on the critical length 𝑙 . The critical length is defined by a distance between zero and maximum bending moment points (see Figure 6 and Figure 10). The values of the degree of shear connection and of the critical length are given in Table 4 for each test. Since two configurations were used in the second test, two values of the degree of shear connection are defined.

Loading procedure and instrumentation

The specimens were tested by applying a vertical load monotonically through the force jack with the system of spreader beams. For tests BM-1 and BM-2, the loading was split into three phases. In the first phase, two cycles of loading between zero and an estimated value of the load at serviceability limit state loading were applied. In the second phase, two more cycles of loading between zero and an estimated value of ultimate limit state loading were carried out. In the final phase, the loading was reapplied up to the failure of the specimen. For the test BM-3, the specimen was subjected to 25 cycles of loading between 5 kN and an estimated value of serviceability limit state loading, and 2 cycles of loading between 5 kN and an estimated value of ultimate limit state loading before loading incrementally up to failure.

Accepted manuscript / Final version

To measure the deflection of the USCB, five LVDT sensors and one laser sensor (see Figure 11a) were installed along the length of the specimen. The laser sensor was placed at mid-span of the beam. The distribution of the slips between the steel and the concrete along the specimen was obtained by 8 horizontal LVDT sensors in the test BM-1 and by 16 horizontal sensors in tests BM-2 and BM-3. The sensors were fixed to the top flanges of the Ushaped profile, whereas the base was fixed onto the concrete slab, as illustrated in Figure 11b. Furthermore, the distribution of the deformations on the composite cross-section at midspan was determined by 19 strain gauges. 7 strain gauges were placed on the concrete, another 6 strain gauges were located on the U-shaped steel profile, and the other 6 were positioned on the rebars (see Figure 12).

Gauges on U-shaped profile

Gauges on rebars

Gauges on concrete In addition to the analogue measurements, high-resolution photo cameras were also installed for a Digital Image

Correlation analysis (DIC) on the zones indicated in Figure 13. In the side zones, the U-shaped steel profile was drilled with 10 circular openings of 20 mm in diameter, and small marks were glued on the concrete surface inside the opening and on the U-shaped profile next to it. A series of photos of these marks were captured during the course of the test at each increment of loading. After the tests, the photos were processed in order to obtain the relative displacement of these marks, i.e. the slips between the steel and the concrete, using GOM Correlate

Professional 2016 [START_REF]Aramis v8 manual basic[END_REF].

Side zone Central zone

Test results

Moment-deflection responses and modes of failure

Figure 14 shows the moment-deflection curve at the mid-span of the specimen for the three tests. For test BM-1, a large value of the mid-span deflection superior than one fiftieth of the total span was obtained, showing the high ductility of the USCB in the case of full shear connection. In this test, a complete failure of the specimen was not however attained as the test was stopped when the verticality of the axis of the force jack was no longer ensured due to the nonlinear effects caused by the large deflections of the specimen. Figure 15 shows the deformed configuration of specimen BM-1 after the test. In the test BM-2, it can be seen from the figure that the maximum value of the moment in the second phase of loading was lower than that in the first phase. The behavior of the test BM-2 was a lot less ductile than that of the test BM-3.

The failure mode of the two specimens (BM-2 and BM-3, cases with partial shear connection) was governed by the plastic buckling of the upper flanges of the U-shaped steel profile at the location of the shear connectors, as being shown in Figure 16 and Figure 17a. This plastic buckling was provoked by localized shear forces in combination with the rotation of the beam induced by the flexural load, which could be observed more prominently at the locations of larger bending moment. This plastic buckling was accompanied by the transverse concrete cracks at the location of the connectors (see Figure 17b). The DIC analysis of the side zone (see Figure 13) allowed to determine the slip distribution between the steel and the concrete on the cross-section of the USCB. The numbering of the points where slips were measured is described in Figure 21. The evolution of the slips on the height of the cross-section is illustrated in Figure 22 for tests BM-2(II) and BM-3. It can be seen from the figure that the slips were almost constant on the height of the cross-section of the U-shaped steel profile for the tests with partial shear connection. This confirms that curvatures of the steel and of the concrete are nearly equal.

C. Strain distribution over the cross-section depth of the USCB

In order to measure the distribution of the strains along the cross-section depth of the USCB, a number of strain gauges were installed on the cross-section (steel U-shaped profile, steel reinforcement and the concrete) as shown in Figure 12. The strain distribution for test BM-1 is described in Figure 23. In this test, the results from the strain gauges on the top surface of the precast concrete, on the steel rebars, and on the top flange of the U-shaped profile were not available. However, the strain distribution for this test in the case of full shear connection seems to be almost linear and continuous from the steel to the concrete cross-section. This result confirms the hypothesis of full shear connection with insignificant effects of the slips between the two cross-sections. def. [-] 305 def. [-] Figure 24: Strain distribution on the cross-section of the USCB for test BM-2(I).

Design approach for the USCB in partial shear connection

The slips occurring at the steel-concrete interface of the USCB due to the partial shear connection provided by the mechanical shear connectors can have substantial effects on the flexural stiffness and the bearing capacity of the composite beams. In this section, the existing design approaches in the current norms taking into account the partial shear connection are investigated with regard to the new configuration of the USCB. The results are then validated against the experimental results presented in Section 3.

Ultimate flexural bearing capacity

The flexural bearing capacity of the USCB can be computed using the plastic analysis method and taking into account the effect of the partial shear connection. Two approaches are investigated in this section. The first approach was initially proposed by Chen et al [START_REF] Chen | Experimental study on mechanical performance of checkered steel-encased concrete composite beam[END_REF] for a new type of checkered steel-encased concrete composite beam. The second approach is the simplified method provided by Eurocode 4 [START_REF] En | Eurocode 4: Design of composite steel and concrete structures[END_REF] with a linear assumption based on the bearing capacity with full shear connection and without shear connection.

A. Full plastic analysis approach

The approach presented herein is based on the method proposed by Chen et al [START_REF] Chen | Experimental study on mechanical performance of checkered steel-encased concrete composite beam[END_REF], adjusted to apply to the configuration of the USCB. It includes the relative slips between the concrete and the steel. The slips are assumed constant over the depth of the cross-section of the U-shaped profile, as can be seen in Figure 26. These slips give rise to two different positions of the plastic neutral axis (PNA) of the concrete (𝑥) and of the steel (𝑦). This generates three different cases for determining the bearing capacity :

 Case 1 occurs when the PNA of the concrete is found in the concrete slab and the PNA of the steel is in the web of the U-shaped profile (see Figure 27).

 Case 2 happens when the PNA of the concrete is in the concrete slab and the PNA of the steel is in the top flange of the U-shaped profile (see Figure 27).

 Case 3 comes about when the PNA of the concrete is in the U-shaped profile and the PNA of the steel is in the web of the U-shaped profile (see Figure 28). The case in which the PNA of the concrete is in the U-shaped profile and the PNA of the steel is in the top flange of the U-shaped profile is rare and not considered for the sake of simplicity. The plastic equilibrium in the composite cross-section relies largely on the degree of shear connection 𝜂. The longitudinal shear force in the case of partial connection can be defined in relation to the degree of shear connection as:

𝑉 , = 𝜂𝑉 , ( 4
)
where 𝑉 , is the longitudinal shear force in the case of full shear connection, defined by:

𝑉 , = 𝐴 , 𝑓 , if 𝑧 < 𝐻 𝐴 , -2𝐴 , 𝑓 , if 𝑧 ≥ 𝐻 (5)
In which 𝐴 , is the cross-section area of the U-shaped profile for the compressive part. 𝑧 is the neutral axis of the composite section in full connection.

Figure 29: Parameters of the geometry of the cross-section.

The equilibrium on the composite cross-section can be decomposed into the equilibrium in the concrete crosssection and that in the steel cross-section with the load transfer mechanism by the shear connectors. The equilibrium equations of the concrete cross-section can be used to compute the position of the neutral axis of the concrete section. Two cases exist :

 𝑥 < 𝐻 (see Figure 27):

𝑏 𝑥0.85𝑓 + 𝐴′ , 𝑓 , = 𝐴 , 𝑓 , + 𝑉 , (6)
which gives 𝑥 = -𝐴′ , 𝑓 , + 𝑉 , + 𝐴 , 𝑓 , 𝑏 0.85𝑓

 𝑥 ≥ 𝐻 (see Figure 28):

(𝑏 -𝑏)𝐻 0.85𝑓 + 0.85𝑓 (𝑏 -2𝑡)𝑥 + 𝐴′ , 𝑓 , = 𝐴 , 𝑓 , + 𝑉 , (8)
which gives 𝑥 = -(𝑏 -𝑏)𝐻 0.85𝑓 -𝐴′ , 𝑓 , + 𝑉 , + 𝐴 , 𝑓 , 𝑏 0.85𝑓

where 𝑏 is the effective width of the concrete block (see Figure 29), determined according to Eurocode 4 [START_REF] En | Eurocode 4: Design of composite steel and concrete structures[END_REF].

𝐻 and 𝑏 are the height of the concrete slab and the width of the bottom flange of the U-shaped steel profile, respectively. 𝑓 is the design value of the compressive strength of the concrete and Chen et al [START_REF] Chen | Experimental study on mechanical performance of checkered steel-encased concrete composite beam[END_REF] limit the stress in concrete to 0.85𝑓 . 𝑓 , is the design value of the yield limit of the steel rebars. 𝐴′ , and 𝐴 , are the top and bottom cross-section areas of the rebars, respectively.

The equilibrium equation on the steel cross-section can be written as:

(𝐴 , -𝐴′ , )𝑓 , = 𝐴′ , 𝑓 , + 𝑉 , [START_REF] Bujnak | Analyse globale de poutres mixtes acier béton : approche analytique et modélisation nonlinéaire[END_REF] where 𝐴′ , is the cross-section area of the U-shaped profile for the compressive part, which can be deduced from Eq. ( 10) as:

𝐴′ , = 𝐴 , 𝑓 , -𝑉 , 2𝑓 , (11)
Subsequently, the position of the PNA of the steel element can be determined as following:

⎩ ⎨ ⎧ 𝑦 = 𝐴′ , 2𝑏 if 𝑦 < 𝑡 𝑦 = 𝐴′ , -2𝑏 𝑡 2𝑡 if 𝑦 ≥ 𝑡 (12)
where 𝑡 and 𝑏 are the thickness of the U-shaped profile and the width of the top flange of the U-shaped steel profile, respectively (see Figure 29).

Finally, the ultimate bending resistance of the USCB in partial shear connection (𝑀 , , ) can be determined according to the cases as following:



B. Simplified approach by Eurocode 4

The approach provided by Eurocode 4 [START_REF] En | Eurocode 4: Design of composite steel and concrete structures[END_REF] adopts the rigid plastic theory for determining the ultimate bearing capacity of the composite beam with full shear connection. In this case, the PNA of the composite cross-section is assumed to locate in the concrete slab. In the case of partial shear connection, Eurocode 4 [START_REF] En | Eurocode 4: Design of composite steel and concrete structures[END_REF] proposes a linear simplification as following:

𝑀 𝑝𝑙,𝑅𝑑,𝜂 = 𝑀 𝑝𝑙,𝑅𝑑,0 + 𝑀 𝑝𝑙,𝑅𝑑,1 -𝑀 𝑝𝑙,𝑅𝑑,0 𝜂

where 𝑀 𝑝𝑙,𝑅𝑑,1 is the ultimate bearing capacity of the composite beam with full shear connection. 𝑀 𝑝𝑙,𝑅𝑑,0 is the ultimate bearing capacity of the composite beam with zero shear connection. For conventional composite beams, i.e. I-profile steel composite beam, 𝑀 𝑝𝑙,𝑅𝑑,0 can be taken equal to the plastic bending resistance of the steel beam.

However, in the case of the USCB, the concrete contribution cannot be neglected. Consequently, 𝑀 𝑝𝑙,𝑅𝑑,0 is the combination of the plastic bending resistance of the steel beam (𝑀 , , ) and the ultimate bending resistance of the concrete beam (𝑀 , , ), which is defined as following:

𝑀 𝑝𝑙,𝑅𝑑,0 = 𝑀 𝑝𝑙,𝑅𝑑,𝑈 + 𝑀 𝑝𝑙,𝑅𝑑,𝑅𝐶

The computation of 𝑀 𝑝𝑙,𝑅𝑑,𝑅𝐶 and 𝑀 𝑝𝑙,𝑅𝑑,𝑈 can be done following the design formulations provided in Eurocode 2 [START_REF] En | Eurocode 2: Design of concrete structures[END_REF] and in Eurocode 3 [START_REF] En | Eurocode 3: Design of steel structures[END_REF], respectively.

C. Comparison between the experiment and the design approaches

The experimental results are used to evaluate the two design approaches. By adopting the empiric approach proposed by Aribert and Alain [START_REF] Aribert | Formulation de la rupture par fatigue de connecteurs acier-béton pour des sollicitations de type sismique: Application aux assemblages[END_REF] to the moment-rotation curves in Figure 14, the ultimate bending capacities (𝑀 𝑝𝑙,𝑅𝑑,𝜂 ) for test BM-1, BM-2(I), BM-2(II) and BM-3 are 1887 kN.m, 1709 kN.m, 1600 kN.m and 1578 kN.m, respectively. In order to be able to compare the results, the values of 𝑀 𝑝𝑙,𝑅𝑑,𝜂 are normalized with the ultimate bending capacities with full shear connection (𝑀 𝑝𝑙,𝑅𝑑,1 ) and presented in function of the degree of shear connection in Figure 30. As can be seen from this figure, the values obtained from the full plastic analysis approach agree well with the experimental results obtained from tests BM-1 and BM-3. However, some difference is obtained when compared with the tests BM-2(I) and BM-2(II). This might be due to the inaccurate calculation of the real degree of shear connection of the tests. In test BM-2, the actual degree of shear connection is indeed higher, as the friction between the steel and the concrete was not reduced by the so-called initial 25-cycle loading/unloading procedure prior to the monotonic loading to collapse, as having been done in test BM-3. In the case of test BM-1, this initial cycle loading procedure did not influence the resistance as full shear connection was achieved.

It can be concluded at this point that both design approaches presented here seem applicable for determining the ultimate bending capacity of the USCB; however, the full-plastic analysis approach gives more accurate results than the simplified approach provided in Eurocode 4 [START_REF] En | Eurocode 4: Design of composite steel and concrete structures[END_REF] when compared to the experimental results.

Figure 30: Normalized ultimate bending capacity of the USCB as a function of the degree of shear connection.

Flexural stiffness of the USCB

An accurate computation procedure of the deflection is needed in order to verify the requirements in serviceability limit states. For composite beams, noticeable differences exist between standards. It is indicated that the degree of shear connection can have an impact on the elastic flexural stiffness in some norms [START_REF]Structural use of steelwork in building, Part 3[END_REF][24] [START_REF]Composite structures -simply supported beams[END_REF],

whereas Eurocode 4 [START_REF] En | Eurocode 4: Design of composite steel and concrete structures[END_REF] neglects it when the degree of shear connection is superior to 0.5. The British standard [START_REF]Structural use of steelwork in building, Part 3[END_REF] provides the following expression for computing the deflection of the composite beams:

𝛿 𝜂 = 𝛿 1 1 + 0.5(1 -𝜂) 𝛿 0 𝛿 1 (18)
where 𝛿 and 𝛿 are the deflections of the USCB in the case of full shear connection and in the case of zero shear connection, respectively. Based on Eq. ( 18), Ban and Bradford [START_REF] Ban | Flexural behaviour of composite beams with high strength steel[END_REF] wrote the expression of the moment of inertia of the composite cross-section (𝐼 eff, ) in function of the degree of shear connection as following:

𝐼 eff,η = 2𝐼 0 𝐼 1 (1 + 𝜂)𝐼 0 + (1 -𝜂)𝐼 1 (19)
where 𝐼 and 𝐼 0 are the moments of inertia of cross-section of the composite beam in the case of full shear connection and in the case of zero shear connection, respectively. The American design standard AISC 360-05 [START_REF]Specification for structural steel buildings[END_REF] gives another expression for the moment of inertia as below:

𝐼 eff,η = (𝐼 0 + 𝜂(𝐼 1 -𝐼 0 )) (20)
Furthermore, the Australian standard AS 2327.1 [START_REF]Composite structures -simply supported beams[END_REF] imposes another expression for computing the moment of inertia of composite beams, as described below:

𝐼 eff,η = 𝐼 0 + 0.6(1 -𝜂)(𝐼 1 -𝐼 0 ) (21)
In order to compare the results obtained from the various expressions from the different standards with the experimental results, it is needed to determine the value of the bending moment at the SLS in the quasi-permanent combination (𝑀 𝑆𝐿𝑆,𝜂 ), which is defined here as following:

𝑀 𝑆𝐿𝑆,𝜂 = 𝑘𝑀 𝑝𝑙,𝑅𝑑,𝜂 (22)
where 𝑘 is a reduction coefficient, being the ratio between the value from the load combination at serviceability limit state (quasi permanent) and the one at ultimate limit state. As indicated in Eurocode 0 [START_REF] En | Eurocode 0: Basis of structural design[END_REF], the load combinations for SLS-quasi permanent and for ULS are given as:

𝑝 𝑆𝐿𝑆-𝑞𝑢𝑎𝑠𝑖 = 𝐺 + 0.3𝑄 (23)
𝑝 = 1.35𝐺 + 1.5𝑄 (24)
with 𝐺 and 𝑄, being the permanent and variable actions, respectively. In the case of the present study, a 60-40 distribution between the permanent and variable actions is assumed, giving the following relation:

𝐺 𝑄 = 60 40 (25)
By inserting Eq. ( 25) into Eqs. ( 23) and ( 24), the coefficient 𝑘 can be determined as

𝑘 = 𝑝 𝑆𝐿𝑆-𝑞𝑢𝑎𝑠𝑖 𝑝 = 0.51 (26)
Consequently, the values of 𝑀 𝑆𝐿𝑆,𝜂 for test BM-1, BM-2(I) and BM-3 are 970 kN.m, 903 kN.m and 913 kN.m, respectively. The deflection corresponding to the value of 𝑀 𝑆𝐿𝑆,𝜂 in the experimental tests can then be deduced from the moment-deflection curves (see Figure 14).

Numerical investigation

Nguyen [START_REF] Nguyen | Modélisation du comportement non-linéaire des poutres mixtes acier-béton avec prise en compte des effets différés[END_REF] developed finite element formulations to determine the response of steel-concrete composite beams with discrete and continuous partial interactions, taking into account the material nonlinearity. However, the account for large displacements was not included. For the large-displacement analysis of steel-concrete composite beams, a good amount of research has been devoted to the development of beam models taking into account the interlayer slips [START_REF] Battini | Non-linear finite element analysis of composite beams with interlayer slips[END_REF]- [START_REF] Hjiaj | Large displacement analysis of shear deformable composite beams with interlayer slips[END_REF] in the co-rotational framework. Battini et al [START_REF] Battini | Non-linear finite element analysis of composite beams with interlayer slips[END_REF] adopted the Bernoulli's assumption (plane sections remain plane) and the local formulation based on the exact solution of the governing equations for the composite beam with deformable shear connection. The use of the analytical local stiffness matrix 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.8

  /  1 [-] Degree of connection [-] BM-2(I) BM-3 BM-1
in their formulation permits to avoid the curvature locking that is usually encountered with low order polynomial finite elements, particularly for short elements with stiff shear connection. Keo et al [START_REF] Keo | Geometrically nonlinear analysis of hybrid beam-column with several encased steel profiles in partial interaction[END_REF] further extended the model for large displacement analysis of hybrid steel-concrete beam/column with several encased steel profiles.

On the other hand, Hjiaj et al [START_REF] Hjiaj | Large displacement analysis of shear deformable composite beams with interlayer slips[END_REF] took into account the transverse shear deformation of the layers of the composite beams by assuming that each layer of the composite beam behaved as a Timoshenko beam element. Assuming a continuously connected and partial interaction, a continuous relationship between the longitudinal shear flow and the corresponding slip at the interface between the layers was considered. The exact local stiffness derived from the closed-form solution of the governing equations of a two-layer beam was used to avoid the curvature and shear locking phenomena.

In this paper, in order to analyze the behavior of the USCB, the two-layer beam element formulation taking into account the interlayer slips with continuous connection is adopted. We adopt the governing equations of the local linear element described by Nguyen [START_REF] Nguyen | Modélisation du comportement non-linéaire des poutres mixtes acier-béton avec prise en compte des effets différés[END_REF] and the kinematics in the co-rotational framework presented by Battini et al [START_REF] Battini | Non-linear finite element analysis of composite beams with interlayer slips[END_REF].

Local linear element

In this section, the field equations for a two-layered beam element with partial shear interaction are recalled.

The concrete and the U-shaped profile are assumed to deform according to Bernoulli's assumption (plane sections remain plane). The interlayer connection is assumed to be continuously distributed. The equilibrium equations are derived by considering an infinitesimal beam segment at an arbitrary position x (see Figure 32) and are given as following:

𝜕𝑫 -𝜕 𝐷 -𝑷 = 0 [START_REF] En | Eurocode 0: Basis of structural design[END_REF] where 𝑫 = [𝑁 𝑁 𝑀] ; 𝑀 = 𝑀 + 𝑀 ; 𝑷 = [0 0 𝑝 ] and the operators 𝜕 and 𝜕 are defined by

𝜕 = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ 𝑑 𝑑𝑥 0 0 0 𝑑 𝑑𝑥 0 0 0 - 𝑑 𝑑𝑥 ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ ; 𝜕 = 1 -1 ℎ 𝑑 𝑑𝑥
in which ℎ is the distance between the elastic centroids of the layers cross-sections.

The U-shaped profile and the concrete are assumed to have the same transverse displacement, neglecting the uplift between the two members. Therefore, the rotation of the U-shaped and that of the concrete are the same, meaning that the slip is constant on the height of the cross-section. Under Bernoulli's assumption, the axial deformations (𝜀 and 𝜀 ) and the curvature 𝜅 are related to the beam displacements (see Figure 33) by :

𝜕𝒅 -𝒆 = 0 (28)
𝜕 𝒅 -𝑔 = 0

where 𝒅 = [ 𝑢 𝑢 𝑣 ] is the displacement vector and 𝒆 = [ 𝜀 𝜀 𝜅 ] is the deformation vector. The slip can also be computed by

𝑔 = 𝑢 -𝑢 -ℎ𝜃 (30)
In order to avoid curvature locking in the two nodes beam elements, 10 degrees of freedom are considered as presented in Figure 34. Thus, an extra procedure is needed to obtain a tangent operator consistent with the corotational formulation, due to the intermediate nodes. The polynomial cubic function (Hermitiens) is used to approach the transversal displacement, whereas the linear function is adopted to approximate the axial displacements.

c1, u c1 _ _ N s1, u s1 _ _ _ _ M 1, q 1 _ _ y l V 1, v 1 x l N c2, u c2 _ _ N s2, u s2 _ _ V 2, v 2 _ _ M 2, q 2 _ _ u cm _ u sm _
Neutral axis of U-shape steel section Neutral axis of concrete section

Kinematics

The structural member consists of two sub-elements: a structural U-shaped steel profile (denoted by subscript "s") and a reinforced concrete beam (denoted by subscript "c"), as illustrated in Figure 35. The co-rotational framework is applied to the structural member. The origin of the co-rotational frame is taken at node 𝑠 located at the centroid of the cross-section of the U-shaped steel. The 𝑥 -axis of the local coordinate system is defined by the line connecting node 𝑠 to node 𝑠 . The 𝑦 -axis is orthogonal to the 𝑥 -axis so that the result is right hand-sided orthogonal coordinate system. The motion of the element from the original undeformed configuration to the actual deformed configuration can be separated into two parts. The first one, which corresponds to rigid motion of the local frame, is the translation of node 𝑠 and the rotation 𝛼 of the 𝑥 -axis (see Figure 35). The second one refers to the deformations in the corotating element frame.

v 1 u c1 y l s 1 s 2 b a q 1 q 1 _ v 2 u c2 c 1 s 1 c 2 s 2 u s2 _ q 2 q 2 _ x l h b 0 x z y

Accepted manuscript / Final version

The vectors of global and local displacements are respectively defined by:

𝑷 = [𝑢 𝑣 𝜃 𝑔 𝑢 𝑣 𝜃 𝑔 ] (31)
𝑷 = [𝑢 𝑢 𝑣̅ 𝜃 ̅ 𝑢 𝑢 𝑣̅ 𝜃 ̅ ] (32)
Based on the definition of the co-rotating frame, the components of the local displacements 𝑷 are determined as following:

𝑢 = 0 (33) 𝑣̅ = 0 (34) 𝑣̅ = 0 (35) 𝑢 = 𝑙 -𝑙 (36)
𝜃 ̅ = 𝜃 -𝛼 (37)

𝜃 ̅ = 𝜃 -𝛼 (38)
𝑢 = 𝑔̅ -ℎ 𝜃 ̅ (39)
𝑢 = 𝑔̅ + 𝑢 -ℎ 𝜃 ̅ (40)
where the local slips are defined in function of the global slips (see Figure 36) by:

𝑔̅ = 𝑔 cos 𝜃 ̅ (41)
𝑔̅ = 𝑔 cos 𝜃 ̅ (42)
y l c 1 s 1 c 2 s 2 x l h x z y c 1 s 1 c 2 s 2 ɡ 1 _ ɡ 1 ɡ 2 _ ɡ 2

Material models

An integration of the appropriate uniaxial constitutive model over each cross-section of the composite beam is needed to obtain the constitutive relationships of the section. A multi-fiber approach is adopted in order to integrate along the length with an element size of approximately 50 cm for the case of test BM-1 or 8 elements with an element size of approximately 50 cm for the case of test BM-3. The case of test BM-2 is not considered in this validation due to the complication caused by the change of loading configuration. Figure 41 illustrates the comparison of slip distributions along the beam axis obtained from the experimental tests and from the numerical model for different levels of bending moment. 𝑀 is the maximum force obtained from the flexural tests (𝑀 = 1915 kN. m for BM-1 and 𝑀 = 1720 kN. m for BM-3). It can be observed that the slip distributions agree well between the results obtained from numerical model and from experiments for both cases (BM-1 and BM-3) until a load level of 80% of 𝑀 . At the load level close to 𝑀 , the difference between the two results become noticeable. However, when looking at the moment-slip curves (see Figure 42) for the slips at the extremities of the beams, the two curves correlate closely.

Extension of the validation of the analytical models

Once validated against the experimental results, the numerical model is used to compute in new configurations, in order to give more information over the consistency of the analytical models presented in section 4 for the definition of the resistance and of the bending stiffness of the USCB. As the analytical approaches do not include the hardening effect of the materials, the elastic perfectly plastic constitutive relationship is used for steel elements in the numerical model. The properties for each material are given in Table 6. The behaviour of the connector is modelled by the relationship provided in Eq. ( 43). In order to cover different values of the degree of shear connection, the resistance of the shear connection per linear meter is determined as follow:

𝑝 , = 𝑛 𝑃 = 𝜂 𝑉 , 𝑙 2 (44)
where 𝜂 is the degree of connection, taken between 0.5 and 1. 𝑉 , is computed following Eq. [START_REF] Liu | Experimental investigation on the flexural behaviour of steel concrete composite beams with U-shaped steel girders and angle connectors[END_REF]. 𝑛 is the number of shear connectors per meter length. Figure 45 provides the comparison between the results of the bending moment capacity in function of the degree of shear connection (in a range between 0.5 and 1) obtained from the analytical approach adopted from Chen et al [START_REF] Chen | Experimental study on mechanical performance of checkered steel-encased concrete composite beam[END_REF] and from the numerical model for each case of the study. It can be observed from the figure that a good agreement is obtained. The maximum difference for all the cases is six percent.

For each case, a ratio between the deflection of the beam with partial shear connection (𝛿 ) and the deflection of the beam with full shear connection (𝛿 ) is also computed using the analytical approaches of different norms (see Section 4.1.2) and using the numerical model at the serviceability load level (see Eq. ( 22)). Figure 46 shows the comparison of the evolution of the deflections in function of the degree of shear connection obtained from analytical approaches and from numerical model for each case of the study. In most of the cases, the ratios between the deflections obtained from the numerical model are smaller than the ones computed from the different approaches. The maximum difference between the results obtained from the British, the American and the Australian standards compared to the ones obtained from the numerical model are 35 percent, 7.5 percent and 8.5 percent, respectively. This once again shows the validity of the American and the Australian standards for estimating the flexural stiffness of the USCB and that the British standard is not suitable. Figure 46: Comparison of the deflections of the beam obtained from analytical approach and from numerical model.

6. Parametrical study on the bending capacity and the flexural stiffness of the USCB

Description of the parametrical study

In this section, a parametrical study using the analytical approach described in Section 4. 7, Table 8 and Table 9 for the cases P1-a, P2-a and P3, respectively:

-𝑀 , , 𝑀 , , , , , , being the ratio of the bending moment in each case to the one in the case with fy,U = 235MPa for the same concrete strength and the same degree of shear connection.

-𝑀 , , 𝑀 , , , , being the ratio of the bending moment in each case to the one in the case with fck = 25MPa for the same steel strength of U-shaped profile and the same degree of shear connection.

Figure 48: Evolution of the bending moment in function of the degree of connection for case P1-a.

f ck = 40MPa ; f y,U = 355MPa f ck = 35MPa ; f y,U = 355MPa f ck = 30MPa ; f y,U = 355MPa f ck = 25MPa ; f y,U = 355MPa f ck = 40MPa ; f y,U = 275MPa f ck = 35MPa ; f y,U = 275MPa f ck = 30MPa ; f y,U = 275MPa f ck = 25MPa ; f y,U = 275MPa f ck = 40MPa ; f y,U = 235MPa f ck = 35MPa ; f y,U = 235MPa f ck = 30MPa ; f y,U = 235MPa f ck = 25MPa ; f y,U = 235MPa
Bending moment [kN.m]

Degree of connection [-]

Case P1-a fy,U = 235MPa fy,U = 275MPa fy,U = 355MPa fy,U = 235MPa fy,U = 275MPa fy,U = 355MPa

f y,U = 355MPa f y,U = 275MPa f y,U = 235MPa
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[ fy,U = 235MPa fy,U = 275MPa fy,U = 355MPa fy,U = 235MPa fy,U = 275MPa fy,U = 355MPa

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f ck = 40MPa ; f y,U = 355MPa f ck = 35MPa ; f y,U = 355MPa f ck = 30MPa ; f y,U = 355MPa f ck = 25MPa ; f y,U = 355MPa f ck = 40MPa ; f y,U = 275MPa f ck = 35MPa ; f y,U = 275MPa f ck = 30MPa ; f y,U = 275MPa f ck = 25MPa ; f y,U = 275MPa f ck = 40MPa ; f y,U = 235MPa f ck = 35MPa ; f y,U = 235MPa f ck = 30MPa ; f y,U = 235MPa f ck = 25MPa ; f y,U = 235MPa
Bending moment [kN.m]

Degree of connection [-]

Case P2-a fy,U = 235MPa fy,U = 275MPa fy,U = 355MPa fy,U = 235MPa fy,U = 275MPa fy,U = 355MPa

f y,U = 355MPa f y,U = 275MPa f y,U = 235MPa
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[%] The main observations are as follows:

-In the three cases, the higher steel strength of the U-shaped profile increases the bending capacity. As can be seen in the figures and the tables, a raise in yield strength (fy,U) from 235 MPa to 275 MPa and from 235 MPa to 355 MPa leads to almost 13% and 40% gain of the bending capacity, respectively.

Consequently, it can be concluded that it is efficient to increase the strength of the U-shaped steel profile when the design requires higher bending capacity of the USCB;

-In cases of beam configurations with the neutral axis in the concrete slab (Cases P1-a and P2-a), a small increase of bending capacity is obtained with the increase of the concrete strength, especially when the degree of shear connection is low. A maximum increase of bending capacity of around 5 percent is achieved between the cases with concrete strength of 25 MPa and 40 MPa;

-For Case P3 (neutral axis in the beam drop), the evolution of the bending moment is different from those in the cases P1-a and P2-a. An increase of bending capacity of around 5 percent when changing the Hence, the impact of the concrete strength on the bending capacity of the USCB is less important that of the steel strength of U-shaped profile. It would be preferable economically to increase the steel grade of the U-shaped profile than the concrete strength in order to improve the bending capacity of the USCB. In the economical point of view, it would not be interesting to increase the concrete class, as the improvement of the flexural stiffness is rather limited. It is more efficient to improve the degree of shear connection.

Conclusion

This paper presents a study on the influence of the partial shear connection on the behaviour of U-shaped steel concrete beam with L-shaped shear connectors under sagging bending moment. A series of three full-scale flexural tests of the USCB with a degree of shear connection ranging from 0.4 to 1 have been performed under sagging bending moment. The experimental tests were to determine the moment resisting capacity, the slip distribution along the length of the beam, the strain distribution on the cross-section, the ductility, and the failure mode. The bearing bending capacity obtained from the experimental results were compared to the values obtained from the full plastic analysis approach and from the simplified approach by Eurocode 4 [START_REF] En | Eurocode 4: Design of composite steel and concrete structures[END_REF]. The experimental results of flexural stiffness were also compared with the values obtained from the closed-form expressions given by British, Australian and American norms. A numerical model based on the two-layer beam element formulation taking into account the interlayer slips with continuous connection in a co-rotational framework was also adapted to determine the behaviour of the USCB and validated against experimental tests as well as against the analytical approaches.

At last, a parametrical study was carried out in order to investigate the effect of selected parameters such as the 0.5 0.6 0.7 0.8 0.9 1.0 0.8 0.9 degree of shear connection, the concrete strength and the steel strength of U-shaped profile. The following outcomes can be extracted from this paper:

 The test results showed high ductility of the USCB with both full and partial shear connections. In the case of partial shear connections, the failure mode of the USCB was governed by the plastic buckling of the upper flanges of the U-shaped steel profile at the location of the critically loaded shear connectors.

 The full plastic analysis approach and the simplified approach by Eurocode 4 [START_REF] En | Eurocode 4: Design of composite steel and concrete structures[END_REF] are both applicable for determining the ultimate bending capacity of the USCB. However, the full plastic analysis approach gave more accurate results than the simplified approach provided in Eurocode 4 [START_REF] En | Eurocode 4: Design of composite steel and concrete structures[END_REF] when compared to the experimental results.

 American and Australian standards are more relevant to determine the flexural stiffness of the USCB, as compared to other discussed standards.

 The numerical model based on the two-layer beam element formulation taking into account the interlayer slips with continuous connection in a co-rotational framework was validated against experimental and analytical results.

 It would be economically preferable to increase the steel grade of the U-shaped steel profile than the concrete strength in order to improve the bending capacity of the USCB. On the other hand, to improve the flexural stiffness, it is possible to increase the concrete class and the degree of shear connection.

The steel strength of the U-shaped profile has however no influence on the flexural stiffness of the USCB.

Figure 1 :

 1 Figure 1: Configuration of U-shaped steel-concrete beams (USCB).

Figure 3 :

 3 Figure 3: Fracture of the shear connector [7].

  𝐻 and 𝐻 are the cross-section height of the L-shaped connector and the thickness of the concrete slab, respectively.

Figure 5 :

 5 Figure 5: Shear sections and concrete pressure zones [7].

  (a). Six-point configuration. (b). Four-point configuration.

Figure 6 :

 6 Figure 6: Bending test configurations used.

Figure 7 :

 7 Figure 7: Detail of test specimens.

Figure 8 :

 8 Figure 8: Detail of test specimens.

Figure 9 :

 9 Figure 9: Test setup.

Figure 10 :

 10 Figure 10: Photos of test setups.

Figure 11 :

 11 Figure 11: Vertical sensors and horizontal sensors.

Figure 12 :

 12 Figure 12: Strain gauges on the composite cross-section.

Figure 13 :

 13 Figure 13: Zones of DIC measurement.

Figure 14 :

 14 Figure 14: Moment-deflection curves.

Figure 15 :

 15 Figure 15: Photo of the specimen BM-1 after the test.

Figure 16 :

 16 Figure 16: Plastic buckling of the upper flanges of the U-shaped steel profile at locations of shear connectors.

  (a). Plastic buckling. (b). Concrete cracks.

Figure 17 :

 17 Figure 17: (a). Plastic buckling of the upper flange. (b). Concrete cracks at the location of shear connectors.

Figure 18 :

 18 Figure 18: Distribution of slips along the beam axis in test BM-1.

Figure 19 and

 19 Figure 19 and Figure 20 show the slip distribution along the beam axis for tests BM-2 and BM-3 at different load levels, respectively. It should be noted that the negative sign in the legends represent the descendent part of the bending moment-rotation curves after the peak. The maximum bending moments 𝑀 obtained in tests BM-2(I), BM-2(II) and BM-3 are 1709 kN.m, 1600 kN.m and 1720 kN.m, respectively. The distribution of the slips in the test BM-2(II) became noticeably asymmetric after the moment reached 097𝑀 . From the figures, the maximum slips for test BM-2 and BM-3 exceeded the value of 15 mm, being quite larger than the ultimate slip (around 3 to 5 mm) of L-shaped connector type L50 obtained from pushout tests.

Figure 19 :Figure 20 : 3

 19203 Figure 19: Distribution of slips along the beam axis in tests BM-2.

Figure 21 :

 21 Figure 21: Numbering of points for measured slips.

Figure 22 :

 22 Figure 22: Evolution of slips along the height of the cross-section in function of the bending moment.

  Position on the section[mm]

Figure 23 :

 23 Figure 23: Strain distribution on the cross-section of the USCB for test BM-1.

Figure 25 :

 25 Figure 25: Strain distribution on the cross-section of the USCB for test BM-3.

Figure 24 and

 24 Figure 24 and Figure 25 illustrate the strain distribution over the cross-section of the USCB for test BM-2(I)and BM-3, respectively. The discontinuity between the strains in the cross-section of the concrete slab and those in the U-shaped steel profile caused by the partial interaction is easily visible for the case of test BM-2(I). It is less intuitive for test BM-3, as the results from the strain gauges on the bottom surfaces of the concrete panels were not available.

Figure 26 :

 26 Figure 26: Strain on the cross-section of the USCB.

Figure 27 : 2 Figure 28 :

 27228 Figure 27: Model for computing the ultimate bearing capacity in Cases 1 and 2

Figure 31 illustrates

 31 Figure 31 illustrates the comparison of the normalized deflection curves obtained from the different standards with the results obtained from the experimental tests, corresponding to the value of 𝑀 𝑆𝐿𝑆,𝜂 . It can be seen from the figure that the curves obtained from the Australian standard and from the American standard fit best to the experimental results. Table5provides margins of errors of the results estimated by different standards compared to the experimental results. Consequently, it can be concluded that the American and Australian standards are suitable for computing the flexural stiffness of the USCB with partial shear connection. The British standard seems to underestimate while the European standard appears to overestimate the flexural stiffness.

Figure 31 :

 31 Figure 31: Normalized deflection curves in function of the degree of shear connection.

Figure 32 :

 32 Figure 32: Free body diagram of a two-layer beam on an infinitesimal segment.

Figure 33 :

 33 Figure 33: Kinematics of the composite beam.

N

Figure 34 :

 34 Figure 34: Local structural member.

Figure 35 :

 35 Figure 35: Co-rotational kinematic: displacements and rotations.

Figure 36 :

 36 Figure 36: Local structural member.

  525

  l

Figure 39 :

 39 Figure 39: Loading and boundary conditions for the validation cases against experimental tests.

Figure 40

 40 Figure40shows the comparison of moment-midspan displacement curves obtained from the experimental tests and from the numerical model. It can be seen from the figure that a good agreement between the two results was achieved for both experimental tests.

  (a). Test BM-1.(b). Test BM-3.

Figure 40 : 3 Figure 41 : 4 FE

 403414 Figure 40: Comparison of moment-midspan displacement curves.

Figure 42 :

 42 Figure 42: Comparison of moment-slip curves at the extremity of the beam.

For

  Figure 39.

Figure 43 :

 43 Figure 43: Beam sections for a span of 10 m.

  (a). Section P2-a. (b). Section P2-b.

Figure 44 :

 44 Figure 44: Beam sections for a span of 20 m.

Figure 45 :

 45 Figure 45: Comparison between the results obtained from analytical approach and from numerical model.

1 . 1 Figure 47 :

 1147 Figure 47: Section case P3.

Figure 50 :

 50 Figure 50: Evolution of the bending moment in function of the degree of connection for case P3.

  = 40MPa ; f y,U = 355MPa f ck = 35MPa ; f y,U = 355MPaf ck = 30MPa ; f y,U = 355MPa f ck = 25MPa ; f y,U = 355MPa f ck = 40MPa ; f y,U = 275MPa f ck = 35MPa ; f y,U = 275MPa f ck = 30MPa ; f y,U = 275MPa f ck = 25MPa ; f y,U = 275MPa f ck = 40MPa ; f y,U = 235MPa f ck = 35MPa ; f y,U = 235MPa f ck = 30MPa ; f y,U = 235MPa f ck = 25MPa ; f y,U = 235MPa Bending moment [kN.m] Degree of connection [-] Case P3 f y,U = 355MPa f y,U = 275MPa f y,U = 235MPaconcrete strength from 25 MPa to 30 MPa, and of around 12 percent when changing the concrete strength from 25 MPa to 40 MPa;

6. 3

 3 Figure 51 shows the deflection ratio as a function of the degree of shear connection for several values of different values of concrete strength fck in the cases P1-a, P2-a and P3. In this figure, the deflection ratio 𝛿 𝛿 , is the proportion between the deflection in each case and the deflection in the case with fck = 25 MPa and the degree of shear connection 𝜂 = 1. From the figure, the following observations are made: -For Cases P1-a and P3, an increase of the flexural stiffness of around 5 percent, 9 percent and 12 percent when changing the concrete strength from 25 MPa to 30 MPa, to 35 MPa and to 40 MPa, respectively; -For Case P2-a, an increase of the flexural stiffness of around 2 percent, 6 percent and 9 percent when changing the concrete strength from 25 MPa to 30 MPa, to 35 MPa and to 40 MPa, respectively; -In the three cases, by increasing the degree of shear connection from 0.5 to 1 helps to improve the flexural stiffness by about 26 percent.

Figure 51 :

 51 Figure 51: Deflection ratio in function of the degree of connection.

Table 1 :

 1 Different details of each specimen.

		Dimensions			L-shaped connector
	Specimen	Width b [mm]	Length l [mm]	Stirrup	Type	Quantity	Length [mm]	Spacing [mm]
	BM-1	2500	10610	HA8@430mm 40x40x4mm	38	380	260
	BM-2(I)
		1500	7790	HA8@350mm 50x50x5mm	12	380	700
	BM-2(II)
	BM-3	1500	7790	HA8@220mm 50x50x5mm	18	380	430

Table 2 :

 2 Properties of the concrete.

				RC beam			Pre-cast panel
	Specimen	At 28 days	On the test day	At 28 days	On the test day
		𝑓	𝜎	𝑓	𝜎	𝑓	𝜎	𝑓	𝜎
		[MPa]	[MPa]	[MPa]	[MPa]	[MPa]	[MPa]	[MPa]	[MPa]
	BM-1	26.03	0.17	27.41	1.53	48.36	1.64	48.89	0.99
	BM-2(I)			26.66	0.19	-	-	53.53	0.75
		26.66	0.19
	BM-2(II)			29.99	0.23	-	-	-	-
	BM-3	24.65	0.81	29.95	2.56	-	-	59.01	0.73

Table 3 :

 3 Properties of the steel elements.

		U-shaped profile	L-shaped connector	Steel rebar
	Specimen
		𝑓 [MPa]	𝑓 [MPa]	𝑓 [MPa]	𝑓 [MPa]	𝑓 [MPa]	𝑓 [MPa]
	BM-1	412	481	345	471	-	-
	BM-2(I)	412	481
	BM-2(II)	412	481	343	467	557	627
	BM-3	430	540

Table 4 :

 4 Degree of shear connection of the bending tests.

	Specimen	Total Span 𝑙 [mm]	𝑥 [mm]	𝑥 [mm]	Critical length 𝑙 [mm]	Degree of connection 𝜂
	BM-1	10610	2460	2100	4560	1.14
	BM-2(I)		1060	2100	3160	0.5
	BM-2(II)	7790	2845	-	2845	0.41
	BM-3		2845	-	2845	0.55

Table 5 :

 5 Margins of errors of the results estimated by different standards compared to the experimental results.

	Test		American standard British standard Australian standard European standard
	BM-1	1.14	0.01	0.01	0.01	0.01
	BM-2(I) 0.5	0.06	0.4	0.07	0.2
	BM-3	0.55	0.08	0.4	0.09	0.15

Table 6 :

 6 Properties for each material.

	U-shaped steel	Rebars		Concrete
	𝑓 , [MPa]	𝐸 [MPa]	𝑓 , [MPa]	𝐸 [MPa]	𝑓 [MPa]	𝐸 [MPa]
	235	210 000	435	210 000	16.67	25 645

Table 7 :

 7 Statistical values of ratios

	𝑀 , ,	𝑀 , , , , ,	and	𝑀 , ,	𝑀 , , ,	for case P1-a.
	𝑀 , ,				𝑀 , ,
	𝑀 , , , , ,					𝑀 , , ,

Table 9 :

 9 Statistical values of ratios

	𝑀 , ,	𝑀 , , , , ,	and	𝑀 , ,	𝑀 , , ,	for case P3.
	𝑀 , ,				𝑀 , ,
	𝑀 , , , , ,				𝑀 , , ,

Acknowledgements

The work in this paper was developed in the framework of the CIFRE convention N° 2016/0256 between INSA Rennes and INGENOVA. The authors gratefully acknowledge financial support by the ANR (Agence Nationale de la Recherche, Paris) and INGENOVA with a grant number ANR-15-LCV2-0003 through the Labcom ANR B-HYBRID project.

the non-homogeneity of the concrete cross-section. The discretization of the cross-section is illustrated in Figure 37.

(a). Concrete section.

(b). Steel section.

Figure 37: Discretization of the cross-section.

The steel materials are modeled by an elastic-perfectly-plastic model, as illustrated in Figure 38a for U-shaped steel and Figure 38b for steel rebars. For the concrete material, the uniaxial constitutive relation is constructed from the stress-strain curve (see Figure 38c) provided by EN 1992-1-1 [START_REF] En | Eurocode 2: Design of concrete structures[END_REF] and applied in the plasticity framework.

Regarding the behavior of the connector, we adopted a nonlinear force-slip relation as following:

where 𝛿 is the slip in mm and 𝑃 is the resistance of the connector defined by Eq. ( 1). The expression of Eq. ( 43) is obtained from a calibration of the mean curve of the force-slip curves for test PO-L50a, PO-L50b and PO-L50c in Figure 4.

Validation of the numerical model

Validation against experimental results

In order to validate the numerical model, the actual material properties and the geometry of the USCB in the flexural tests (see section 3) were used in the numerical applications of the model. In the model, only a half of the beam length is considered while applying a symmetrical condition to one end of the beam to save the calculation time, with the loading and boundary conditions as illustrated in Figure 39 . The beam is discretized into 11 elements

Annex: a simplified flowchart of the modelling process.

Input

Cross-section :

(concrete, U-shaped profile, rebars)

Material properties :

(concrete, U-shaped profile, rebars, connector)

Loading conditions :

(distributed and point load conditions)

Boundary conditions :

(Simple or fixed supports)

Hypotheses