Analysis of Stud Spacing on the Mechanical Behaviour of Steel-Concrete Composite Beam under Uniform Load

The stud spacing significantly affects the bond strength and interracial slip between steel and concrete of thin-walled steel-concrete composite beams, and it interacts with shear lag and shear deformation to affect the structural deflection and sectional stresses. In this paper, a new interracial slip model considering the influence of stud spacing and stud shear stiffness is proposed first, then a potential energy equation of a simply supported steel-concrete composite beam under vertical uniform load considering the coupled effect of interracial slip, shear lag and shear deformation is obtained, and the calculation formulae for structural deflections and sectional stresses are derived, in which the parameter of stud spacing is explicitly shown. The comparison between the observed data in the completed experiment with the theoretical values using the presented formulae and the design code formulae are fulfilled, showing that the presented formulae hold agreeable calculation precise. The proposed formulae can be used to estimate the influence of stud spacing on the mechanical behaviour of steel-concrete composite beams in the preliminary design.


Introduction
The steel-concrete composite (SCC) beam, composed of a concrete slab and a thin-walled steel beam by steel connectors, bears agreeable mechanical performance and excellent economic efficiency for both the high compressive strength of concrete and high tensile strength of steel can be utilized fully.Although SCC beams have been used widely in the buildings and bridges, the engineers still confront some issues to be solved in the design, for example, how to directly judge the precise effect of the arrangement of shear connectors on the deflection and stress of SCC beams in the design [1][2][3][4][5].
In a thin-walled SCC beam under loads, the interracial slip between the concrete and steel occurs inevitably [5,6], resulting in the reduction of the cross-sectional stiffness [3,7,8] and the change in sectional stresses [5][6][7][9][10][11].In addition, it is coupled with shear lag of flange slabs and shear deformation to affect the mechanical behaviour of SCC beam [3-5, 10, 12].Due to the complexity of structural analysis considering all the above factors, researchers have resorted to simplified analysis [7][8][9][10] or finite element based numerical analysis [13] as alternatives.
Although the finite element approach can comprehensively fulfil the structural analysis of SCC beams considering the effect of steel connector spacing, it is difficult to guide the preliminary design directly.For the presented calculation formulas, the influence of connector spacing still cannot be explicitly demonstrated.In this paper, a simplified approach to analysis the effect of steel stud spacing on deflections and sectional stresses of SCC beams by using energy variation method is established, and the theoretical values are compared with the completed experimental results.In the analysis, following assumptions are adopted: (1) the interracial slip between steel and concrete is proportional to the shear rigidity of connectors, and the slip between steel and concrete at the interface is distributed according to the sectional depth; (2) the longitudinal displacement of the flange plate varies in a cubic parabola along the transverse direction; (3) the steel and the concrete are bonded without lifting under loading.

Describe of Longitudinal Displacement and Slip Strain
For an SCC beam subjected to uniformly distributed load (figure 1), set b as the maximum value of the half width of the middle part of the concrete slab, the cantilever width, and the half width of the upper and lower flanges of the steel beam, then the width of each part can be expressed as ( 1,2,3,4) i bi  = .Unless otherwise stated, the subscript "c" stands for concrete and "s" stands for steel.
According to the assumptions, the longitudinal displacement at any position of the cross-section can be described by: where: ( , ) U x y denotes the longitudinal displacement of the section; () wx is the vertical deflection of the SCC beam; () Sx is the generalized interracial slip; () ux is the maximum difference in shear angle; h , c h and s h are the depths of the overall section, concrete slab, and steel beam, respectively; c Z is the distance from the centroid of the concrete slab to that of SCC cross-section; st(b)

Z
is the distance from the centroid of upper or lower plate to that of the SCC cross-section.The relation of relative slip and slip strain can be established using the similar approach adopted for SCC beam under concentrated load when the basis of Goodman's assumption of an elastic interlayer is used [7]: ( ) where: ( )

AK dE I    =  
( ) A and c A are the sectional areas of steel beam and concrete slab, respectively; K is the shear stiffness of the shear connector; s n is the number of stud on a transvers section (equals to the number of longitudinal rows of steel studs); c v N is the shear strength of a single stud; s

E and c
E are the elastic moduli of steel and concrete, respectively; 1 n is the modulus ratio of steel to concrete; s I and c I are the moments of inertia of steel beam and concrete slab with respect to itself centroid, respectively; q is the uniformly distributed load; l is the calculation span; d is the longitudinal stud spacing.
The longitudinal stud spacing d is explicitly shown in coefficients  and  , which are included in the following proposed formulas of deflection and stress.

Potential Energy and Governing Differential Equation
Analytical approaches on structural analysis of SCC beams considering the effect of shear lag mainly include elastic theory, analogy rod method, and energy variation method, among which the energy variation method is widely used [13][14][15][16].According to the principle of virtual work, the total potential energy of a SCC beam under vertical uniform load can be obtained: where: w I is the moment of inertia of the steel web with respect to its centroid;  By using the principle of minimum potential energy, =0   , the following equations can be derived: where: ( ) () Mx is the bending moment function when interracial slip and shear lag are not considered; F () Mx is the additional bending moment caused by the interracial slip and shear lag effect.
From equations ( 6) to ( 8), the governing differential equation for () ux can be obtained: where:

Experimental Specimens
Three simply supported SCC beams with a steel box beam and three simply supported SCC beams with a steel I-shape beam were made to test the effect of stud spacing on the deformation and sectional stresses.The longitudinal distances of 90mm, 180mm, and 270mm were designed (2 rows, figure 2), respectively, which correspond to shear connection degree of 1.0, 0.7, and 0.4, respectively.The specimen dimension is shown in figure 2. The applied uniformly distributed load is 8.9 kN/m .C40 concrete with c 3.45 Gpa E = (tested value) for concrete slab and Q235 steel with s 2.06 Gpa E = for steel beam are adopted.The cylindrical head welded stud with  =12.8mm and L=45mm is adopted for shear connector.The deflections and sectional stresses in the concrete slab and steel plate at midspan, L/4, 3L/4 and supports are observed (figures 2 and 3).

Comparison of Deflection
Tables 1 and 2 show the observed midspan deflections of the specimens, and the calculated values using the presented formulas and the formulas provided in the GB50917-2013 [1] and GB50017-2017 [2] codes are compared.The "Difference" in tables 1 and 2 denotes the difference of the theoretical values comparing to the observed data.
From tables 1 and 2, following conclusions can be obtained: (1) the observed midspan deflection of SCC beam increases obviously as the stud spacing increases; (2) the proposed equation ( 11) gives agreeable calculation accuracy; (3) the GB50917-2013 code formulas generally underestimate the deflections; (4) the GB50017-2017 code formulas overestimate the deflections by 10% approximately.

Effect of Stud Spacing on Shear Lag Effect
Define the shear lag coefficient as the ratio of the observed (or theoretical) stress to the stress obtained by the classic elastic beam theory which doesn't consider the effect of interracial slip between the concrete and steel.The observed stress is equal to the observed strain multiplied by the elastic modulus.figures 4 and 5 illustrate the comparison of shear lag coefficient obtained from the experimental data at measure points (BT1-BT3 and IT1-IT3 as shown in figure 3) and using the proposed equation ( 12).Figures 4 and 5 show that the shear lag coefficient at the same measure point obviously increases as the stud spacing increases; for example, the observed shear lag coefficient changes from 1.116 5(a)) and changes from 1.070 to 1.217 at quarter span (figure 5(b)) as the stud spacing varies from 90mm to 270mm for I-shape steel SSC beams.In addition, the shear lag coefficients based on the proposed equation ( 12) are close to that based on the observed strain (stress), demonstrating that the proposed equation ( 12) give good calculation accuracy.

Effect of Stud Spacing on Slip
Figure 6 illustrates the relation between the interfacial slip and the stud spacing in the SCC beams under the uniformly distributed load, showing that the interlayer slip is 0 in the midspan section and it reaches the maximum at two beam ends, and the slip increases sharply as the stud spacing increases.For boxsteel SCC beams, the interfacial slip changes from 0.066mm to 0.140mm at the end of the beams as the stud spacing varies from 90mm to 270mm (figure 6(a)), while the interfacial slip changes from 0.059mm to 0.126mm for I-shape steel SCC beams (figure 6

Conclusions
Specific analysis was conducted on the bond slip, shear lag effect, and mechanical behaviour of SCC beam for different stud spacing.The conclusions are drawn as follows: (1) A set of calculation formulas for deflection and stress of simply supported SCC beams under uniform load considering the coupling effect of interlayer slip and shear lag is derived, in which the parameter 'stud spacing' is explicitly shown; this allows engineers to directly determine the impact of stud spacing on the mechanical behaviour of SCC beams in design based on these formulas.
(2) Both the observed and the calculated midspan deflection of SCC beams increases obviously as the stud spacing increases; the proposed formulas give agreeable calculation accuracy on deflection, while the GB50917-2013 code formulas generally underestimate the deflections and the GB50017-2017 code formulas overestimate the deflections.
(3) The shear lag coefficient at the same position in the flange obviously increases as the stud spacing increases; meanwhile, the interfacial slip increases sharply as the stud spacing increases.

Figure 1 .
Figure 1.Schematic diagram of steel concrete composite box girder.
G are the shear moduli of concrete and steel, respectively; C is the Constant of integration.
Qx is the shear function.

Figure 4 .
Figure 4. Schematic diagram of the relationship between shear lag coefficient and stud spacing of steel box beams (E-experimental data, T-theroetical data).

Figure 5 .
Figure 5. Schematic diagram of the relationship between shear lag coefficient and stud spacing of Ishaped steel beams (E-experimental data, T-theroetical data).

Figure 6 .
Figure 6.Schematic diagram of the relationship between slip amount and stud spacing.

Table 1 .
Mid-span deflection of SCC beams with a steel box section.

Table 2 .
Mid-span deflection of SCC beams with a steel I section.
above web at support section as the stud spacing varies from 90mm to 270mm (figure 4(a)) and from 1.061 to 1.206 at quarter span (figure 4(b)) for box-steel SCC beams, and changes from 1.123 to 1.301 at the position above web at support section (figure