Analytical solution for the resistance of composite beams subjected to bending

The paper deals with the resistance of steel and concrete composite beams, named BH beams, subjected to bending. They are structurally connected with prefabricated or cast in situ slabs, forming floor slab system. The beams under consideration consist of the reinforced concrete (RC) rectangular core placed inside a reversed TT welded profile. The stress-strain relationship for concrete in compression of the RC core is assumed for nonlinear analysis according to Eurocode 2. For reinforcing and profile steels linear elastic – ideal plastic model is applied. The normalized ultimate bending moment determining the resistance of the BH beam is derived by integrating the equilibrium equations of the bending moments about the horizontal axis of the RC core rectangle, taking into account the physical and geometrical relationships. The presented model was verified by tests carried out on two BH beams subjected to bending. The comparisons made indicated good convergence between the analytical solution and the experimental results in ultimate bending moments.


Introduction
In recent years, newly designed floor slab systems have been the subject of particular interest of civil engineers and researchers due to their importance in the engineering practice. The systems under consideration consist of steel and concrete composite beams, named BH beams, structurally connected with prefabricated or cast in situ slabs [1][2][3][4][5]. General rules for the design of these structures are given in the relevant codes [6 -8]. The considerations of this paper are focused on the analytical solution for determining the resistance of the BH beams subjected to bending. Two types of the BH beams designated as BH 27-350 and BH 20-300 are considered. They consist of the reinforced (RC) rectangular core placed inside a reversed TT welded profile, as shown in Fig. 1 and Fig. 4. The floor slabs are supported on lower flanges of the steel profile part of the composite beam to provide a flat lower surface of the finished floor. In order to ensure the demanded bond between these components, a number of reinforced concrete studs are used. They were designed as a set of horizontal rebars passing through the perforated webs of the beam and anchored in the slabs. In the case of separate BH beams, steel studs are applied with the spacing of 0.3 m. The results of FEM modelling of failure behaviour of BH beams revealed that significant differences in the ultimate bending moments occur compared to the bending test results. Therefore, the crosssectional analysis is proposed and employed. It adequately describes BH behaviour at failure. This approach consists in the derivation of analytical formulae for the normalized ultimate bending moment determining the resistance of the BH beam.

Derivation of formulae
In the presented considerations, the following assumptions are introduced: o plane sections remain plane, o elasto-plastic stress-strain relationship for concrete and reinforcing and profile steels are used, o the tensile strength of concrete is ignored, o the demanded bond between the RC core and the steel profile is ensured.
The stress-strain relationship for concrete σcεc in the compression of the RC core is assumed for nonlinear structural analysis according to Eurocode 2 ( Fig. 2), [9]: where: η = εc/εc1 , εc1 -the strain at peak stress on the σcεc diagram, fcm -the mean compressive strength of concrete, k = 1,05 Ecm ⏐εc1⏐/ fcm, Ecm -secant modulus of elasticity of concrete. This stress-strain relation adequately represents the behavior of concrete by introducing four parameters: fcm, εc1 , εcu and Ecm. For reinforcing and profile steels, characterized by yield stresses fyk , fHyk , respectively, linear elastic -ideal plastic model is applied and described by the following formulae: where Es -modulus of elasticity of profile or reinforcing steels. The above given formulae also apply to profile steel by replacing fyk with fHyk . The resistance of the composite cross-section is reached when either ultimate compressive strain in concrete εcu1 or ultimate tensile strain in steel εsu , or ultimate tensile strain in steel profile εHsu is reached anywhere in that section. The equilibrium equation of the bending moments about the horizontal axis of the RC core rectangle takes the following form: where: t -height, b -width of the RC rectangular core, respectively, Fa1 , Fa2 -areas of reinforcing steels in compression and in tension, respectively, t1, t2 -coordinates describing the locations of compressive and tensile rebars, respectively, σs1 , σs2 -stresses in steels in compression and in tension, respectively, FHf , tf -area and thickness of the lower flange, FHs, ts -area and thickness of the web, x0 -coordinate of the upper edge of web, σHf -stress referring to the middle thickness of the lower flange, σHs -stress function referring to the web, x -coordinate describing the location of neutral axis, x' -vertical coordinate of any point of the section, 0 ≤ x, x' ≤ t. Coordinates x, x' are measured from the upper edge of the section.
The formulae (8) -(11) describe the resistance mHRm of the considered steel and concrete composite cross-section subjected to bending.

Experimental study
In order to verify the obtained analytical solution, the bending tests were carried out in the Building Research Institute on separate BH 27-350 and BH 20-300 beams. For the RC rectangular core, the reinforcing steel with yield stress fyk = 500 MPa was used, while for the profile steel with yield stress fHyk = 460 MPa. The BH beams were loaded up to failure. The setup of the tests is presented in Fig. 3. The scope of the tests covered the determination of the failure loads and the corresponding strains. In each load step, the strains in concrete εc, reinforcing steel εs1 in compression and profile steel εHf were measured using strain gauges located in the middle section of the BH beam (Fig. 4). The strains in reinforcing steel in tension εs2 were determined by taking into account the assumption that plane sections remain plane.

Beam BH 27-350
Beam BH 27-350 was manufactured from concrete with the mean compressive strength fcm = 58 MPa (concrete grade C 50/60; Ecm = 37 GPa), which was approved by standard testing. This beam had the following geometric characteristics:  The failure mechanism of the BH 27-350 beam occurred in the form of crushing of the concrete (Fig. 5). In Table 1, the mean measured values of strains and bending moment at failure are collected. The compressive strain in concrete reached the ultimate value εc = εcu1 = -3.5 ‰ which is combined with |εs1|= |-2.6 ‰| > εss = 2.5‰. This in turn means that plastic strains in the rebars in compression may have occurred. Based on the derived formulae (8) -(11) the value MHRm = 634.7 kNm was calculated which is close to the failure bending moment Mu = 635 kNm. This shows very good convergence between the analytical solution MHRm and the test result in ultimate bending moment Mu (relative difference 0.05%).

Beam BH 20-300
On the basis of the standard testing performed by the manufacturer, the mean compressive strength of the beam's concrete was determined as fcm = 53 MPa (concrete grade C45/55; Ecm = 36 GPa The BH 20-300 beam underwent the same form of failure, i.e. crushing of the concrete (Fig.  5). The mean measured values of strains and bending moment at failure are given in Table 2.
The compressive strain in concrete reached the ultimate value εc = εcu1 = -3.5 ‰. It is worth noting that plastic strains in WKH ORZHU IODQJH may have occurred as εHf = 2.42 ‰ > εss = 2.3‰. According to the derived formulae (8) -(11) the value MHRm = 366.4 kNm was calculated which is close to the failure bending moment Mu = 374 kNm. This confirms good convergence between the analytical solution and the test results in ultimate bending moment as well (relative difference: 2%).

Summary and conclusions
The analytical solution was developed for the resistance of composite steel and concrete beams subjected to bending, based on the nonlinear stress-strain relationship for concrete. The solution closely reflects the actual behavior of concrete in compression. This solution shows good convergence with the test results in ultimate bending moment. Such a formulation enables the analysis of the behavior of the cross-section of composite steel and concrete beams in the post-critical phase, and in this respect, it can be regarded as a valuable solution in the theory of composite structures. Thus, it can be very useful as far as the prediction and verification of test results are concerned.