# Stiffness Calculation for Negative Moment Region of Steel-Concrete Composite Beams considering the Influence of Cracking and Interface Slip

Authors: Wenlong Cao; Jianqing Bu; Cunbao Zhao; Xueyan Wang; Liping Zhang; Jiren Zhang
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

Concrete cracking and interface slip decrease the bending stiffness of steel-concrete composite beams (SCCBs) in the negative moment region. To evaluate this decrease in the stiffness of SCCBs, a static differential equation is used to derive analytical expressions of interface slip and deflection. Based on the static load test results for SCCBs, the influence of concrete cracking and interface slip on the stiffness is analyzed. A linear stiffness reduction method is proposed for calculating the stiffness of SCCBs in the negative moment region after the concrete undergoes cracking. The proposed method can be used in engineering applications as it enables the facile and accurate calculation of the deflection and slip and the changes in stiffness of SCCBs in the negative moment region. The method can also be used for predicting and evaluating performance in the field of structural design.


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## Body

## 1. Introduction

Steel-concrete composite beams (SCCBs) have been extensively used as continuous beams on highways. In the negative bending moment region of a continuous beam, the concrete slab is subjected to tension while the steel beam is subjected to compression. Because the tensile strength of concrete is considerably low, the concrete in the negative bending moment region is brittle and thus vulnerable to cracking. Relative slip occurs at the interface due to different longitudinal strains between the bottom surface of the concrete slab and the top surface of the steel beam. Cracking and interface slip reduce the bending stiffness of the section. When calculating the deflection of a continuous SCCB, the reduction of the stiffness must be considered to ensure that the section stiffness is not exceedingly large and that the calculated deflection value is not exceedingly small.Interface slip of simply supported beams has been investigated in several studies. Considering that interface slip and shear deformation are caused by shear (shear lag effect), Zhou et al. [1] derived differential equations for calculating the deflection and slip of composite beams through the variational method and the principle of minimum potential energy. They also derived analytical expressions for the deflection and interface slip of composite beams under concentrated and uniformly distributed loads. Zhang [2] used the energy variational principle and displacement superposition method to evaluate the stress, slip, and deflection in a steel-concrete composite box girder during normal operation, considering the influence of both interface slip and shear lag. Based on the principle of virtual work and the variational method, Dezi et al. [3] proposed a theoretical model of a steel-concrete composite I-beam considering the influence of slip and shear lag. The viscoelasticity of concrete was also considered in the model, which was solved through the step-by-step integration method and finite difference method. Zhu et al. [4] applied the model proposed by Dezi et al. to a steel-concrete composite box girder. They developed independent functions for the concrete slab and steel beam for shear lag warping degree of freedom, which are used separately. Wang [5] performed a mechanical analysis of a microsegment unit and discussed the sliding mechanism of the composite beam. These studies have been focused on the positive bending moment region of an SCCB, while the negative bending moment region has been focused on in a few studies.For the negative moment region of SCCBs, a few experimental studies have been conducted on the mechanism of stiffness development [6–11]. Rossi et al. [12] numerically analyzed the buckling and postbuckling using the ABAQUS software to evaluate the behavior of an SCCB under the influence of hogging moment. Nie et al. [13, 14] proposed the use of effective stiffness to calculate stiffness reduction caused by interface slip and investigated the superposition effect of simply supported beams to evaluate the deflection of continuous beams. Li et al. [15] established a model for calculating the bearing capacity of the negative bending moment section of composite beams with uplift-restricted and slip-permitted (URSP) connections. In terms of materials, Xing [16] applied recycled tire rubber-filled concrete (RRFC) to steel-concrete composite beams and studied its static and fatigue characteristics. Wang et al. [17–20] studied the mechanical properties and durability of concrete.The stiffness of composite beams caused by concrete cracking is reduced through different methods in different countries. As per European standards [21], a bending stiffness with 15% of the span length of a beam is determined by the stiffness of a cracked section and that of the remaining beam is determined by the stiffness of the beam before cracking. This method can be used to approximately simulate the reduction of the bending stiffness of concrete after cracking, while the stiffness changes rapidly. However, the results obtained using this method are not consistent with those obtained under practical conditions. The Building Code Requirements for Structural Concrete (ACI318M-05) [22] considers the tensile rigidification effect of concrete and adopts a method involving the effective moment of inertia, which is employed to calculate the deflection of reinforced concrete members. The effective moment of the inertia method considers stiffness degradation at different stages; however, calculations using this method are complicated because the bending moment of all sections with the cracking moment must be compared, so the calculation is more complicated.The existing calculation method for stiffness is focused on positive simply supported beams with concrete slabs on top and steel beams at the bottom. However, the stiffness in the negative moment region of composite beams is simultaneously influenced by concrete cracking and interface slip. Therefore, both cracking and interface slip should be considered for the calculation of stiffness in the negative moment region of SCCBs. A method for calculating the bending stiffness of composite beams in the negative moment region is proposed in this study considering the influence of both concrete cracking and interface slip.


## 2. Calculation of Stiffness considering the Influence of Concrete Cracking

Figure1 shows the model of an overhanging SCCB. Without considering the influence of concrete cracking, the differential equation for beam deflection based on material mechanics can be expressed as follows:(1)d2fxdx2=MxEsI0.Figure 1
Model of overhanging steel-concrete composite beam.The analytical solution of the above equation can be expressed as follows:(2)−b≤x≤0,fx=aFx23b+x6bEsI0+abFx3EsI0,0≤x≤a,fx=Fx23a−x6EsI0+abFx3EsI0,(3)Ieff=McrM3I0+1−McrM3Icr≤I0.A linear reduction method is proposed herein for calculating stiffness in the negative bending moment region of composite beams. The moment of inertiaIeff of the section near the support can be calculated according to (3). The moment of inertia of the section at the contraflexure point is denoted by I0, and the moment of inertia between the two points is linear interpolation. The moment of inertia of the test beam is expressed as follows:(4)−b≤x≤0,Icr=Ieff−I0−Ieffxb,0≤x≤a,Icr=Ieff+I0−Ieffxa.The differential equation for the deflection of the beam is as follows:(5)−b≤x≤0,d2fxdx2=ab+xFbEsIeff−I0−Ieffx/b,0≤x≤a,d2fxdx2=Fa−xEsIeff+I0−Ieffx/a.The above equation is integrated twice to derive the analytic formula of beam deflection as follows:(6)−b≤x≤0,fx=abFI0EsI0−Ieff2D1x−I0−Ieff2bI0x2+bIeffI0−Ieff−xlnbIeffI0−Ieff−x+D2,0≤x≤a,fx=−a2FI0EsI0−Ieff2D3x+I0−Ieff2aI0x2−x+aIeffI0−Iefflnx+aIeffI0−Ieff+D4.The following boundary conditions are considered for the calculations:(7)fxx=0=0,dfxdx|x=0−=dfxdx|x=0+,fxx=−b=0.Ifξeff=Ieff/I0−Ieff and ξ0=I0/I0−Ieff, the deflection in the negative bending moment region considering the influence of concrete cracking is expressed as follows:(8)−b≤x≤0,fx=abFξ02EsI0ξ0lnbξ0−ξefflnbξeff−12ξ0x−12bξ0x2+bξeff−xlnbξeff−x−bξefflnbξeff,0≤x≤a,fx=−a2Fξ02EsI0−bξ0alnξ0ξeff+lnaξeff+a+ba+b2aξ0x+12aξ0x2−x+aξefflnx+aξeff+aξefflnaξeff.


## 3. Calculation of the Stiffness considering the Influence Interface Slip

The interface slip between the steel beam and concrete slab causes a reduction in the stiffness of the composite beam, which is referred to as additional deflection. The interface slip is analytically calculated according to the static differential equation in the negative moment region of composite beams; the additional deflection caused by the slip is also calculated.


### 3.1. Fundamental Assumptions

(1)
The steel beam, steel bar, and concrete are ideally elastic under compression and tension. The tensile stress-strain curve of concrete is a straight line before the tensile strain is maximum. The concrete cracks cannot sustain the load after maximum tensile strain is exceeded.(2)
The slip between the concrete slab and steel beam is proportional to the shear force of the shear studs:(9)pτx=KSx.(3)
The steel beam and concrete slab have the same curvature(10)φx=Msx+McxEsI0.


### 3.2. Relative Slip at the Interface

Figure2 shows the internal force distribution in a microsegment of length dx of the overhanging beam model.Figure 2
Internal force distribution in a microsegment of the model.Assuming static equilibrium in the microsegment, the following equation is obtained:(11)∑Fcx=0⇒dTcxdx=−τx⇒Tcx=−∫0xτxdx,∑Fsx=0⇒dCsxdx=−τx⇒Csx=−∫0xτxdx.A longitudinal strain, which is a superposition of bending moment and axial force, is exerted at the bottom of the concrete slab and is expressed as follows:(12)εcbx=−Mcxhc2EcIc+TcxEcAc=−hc2φx+TcxEcAc.The longitudinal strain at the top of the steel beam is also a superposition of bending moment and axial force on the beam.(13)εstx=Msxhs2EsIs−CsxEsAs=hs2φx−CsxEsAs.The relative slip strain on the contact surface of the concrete slab and steel beam is the difference between the longitudinal strain on the bottom of the concrete slab and the top of the steel beam.(14)εsx=dSxdx=εstx−εcbx=h2φx+1EsAs∫0xτxdx+1EcAc∫0xτxdx.The axial force is expressed asNx=∫0xτxdx.From the fundamental assumption 2, the following equation is obtained.(15)dSxdx=pK⋅dN2xdx2From (14), the following equation is obtained:(16)dSxdx=pK⋅dN2xdx2=h2φx+1EsAs∫0xτxdx+1EcAc∫0xτxdx.The differential equation of axial force is obtained by calculating the value of bending moment and considering the fundamental assumption 3.(17)−b≤x≤0,dN2xdx2−Kp1EsAs+1EcAcNx=hKFab+x2pbEsI0,0≤x≤a,dN2xdx2−Kp1EsAs+1EcAcNx=hKFa−x2pEsI0.Ifλ2=K/p1/EsAs+1/EcAc, (17) is substituted as follows:(18)−b≤x≤0,dN2xdx2−λ2Nx=hKFab+x2pbEsI0,0≤x≤a,dN2xdx2−λ2Nx=hKFa−x2pEsI0.The ordinary solution of (18) is as follows:(19)−b≤x≤0,Nx=A1sinhλx+B1coshλx−hKFab+x2pbλ2EsI0,0≤x≤a,Nx=A2sinhλx+B2coshλx−hKFa−x2pλ2EsI0.The following boundary conditions are considered:(20)Sxx=3−3b3=0,Nxx=a=0,Nxx=−b=0,Sxx=0+=Sxx=0−.The solution for the unknown parameter is expressed as follows:(21)A1=hKFa2pbλ3EsI0⋅coshλbcoshλbcosh0.423λb−sinhλbsinh0.423λb,B1=A1sinhλbcoshλb,A2=A1−hFKa+b2pbλ3EsI0,B2=−A2sinhλacoshλa.The slip of the overhanging beam is expressed as follows:(22)−b≤x≤0,Sx=pKdNxdx=pλA1Kcoshλx+pλB1Ksinhλx−hFa2bλ2EsI0,0≤x≤a,Sx=pKdNxdx=pλA2Kcoshλx+pλB2Ksinhλx+hF2λ2EsI0.According to (14), the relative slip strain at the interface is expressed as follows:(23)−b≤x≤0,εsx=pλ2A1Ksinhλx+pλ2B1Kcoshλx,0≤x≤a,εsx=pλ2A2Ksinhλx+pλ2B2Kcoshλx.


### 3.3. Additional Deflection due to Relative Slip

The additional deflection caused by relative slip is expressed as follows:(24)Δφx=εsxh.From (10), the following equation is obtained.(25)Δφx=ΔMxEsI0.Therefore,(26)d2Δfxdx2=ΔMxEsI0=εsxh.Substituting (23) into the above equation and integrating it twice, the following equation is obtained.(27)−b≤x≤0,Δfx=pKhA1sinhλx+B1coshλx+E1x+E2,0≤x≤a,Δfx=pKhA2sinhλx+B2coshλx+E3x+E4.The following boundary conditions are considered.(28)Δfxx=0=0,dΔfxdx|x=0−=dΔfxdx|x=0+,Δfxx=−b=0.Substituting boundary conditions into (27), the following equation is obtained.(29)E1=−A1sinhbλ+B1coshbλ−B1b,E2=−B1,E3=λA1−A2+E1=λA1−A2+−A1sinhbλ+B1coshbλ−B1b.E4=−B2,The deflection of the composite beam is a sum of the deflection due to concrete cracking and the additional deflection due to interface slip and can be expressed as follows:(30)f′x=fx+Δfx,where fx and Δfx are calculated according to (8) and (27), respectively.


## 3.1. Fundamental Assumptions

(1)
The steel beam, steel bar, and concrete are ideally elastic under compression and tension. The tensile stress-strain curve of concrete is a straight line before the tensile strain is maximum. The concrete cracks cannot sustain the load after maximum tensile strain is exceeded.(2)
The slip between the concrete slab and steel beam is proportional to the shear force of the shear studs:(9)pτx=KSx.(3)
The steel beam and concrete slab have the same curvature(10)φx=Msx+McxEsI0.


## 3.2. Relative Slip at the Interface

Figure2 shows the internal force distribution in a microsegment of length dx of the overhanging beam model.Figure 2
Internal force distribution in a microsegment of the model.Assuming static equilibrium in the microsegment, the following equation is obtained:(11)∑Fcx=0⇒dTcxdx=−τx⇒Tcx=−∫0xτxdx,∑Fsx=0⇒dCsxdx=−τx⇒Csx=−∫0xτxdx.A longitudinal strain, which is a superposition of bending moment and axial force, is exerted at the bottom of the concrete slab and is expressed as follows:(12)εcbx=−Mcxhc2EcIc+TcxEcAc=−hc2φx+TcxEcAc.The longitudinal strain at the top of the steel beam is also a superposition of bending moment and axial force on the beam.(13)εstx=Msxhs2EsIs−CsxEsAs=hs2φx−CsxEsAs.The relative slip strain on the contact surface of the concrete slab and steel beam is the difference between the longitudinal strain on the bottom of the concrete slab and the top of the steel beam.(14)εsx=dSxdx=εstx−εcbx=h2φx+1EsAs∫0xτxdx+1EcAc∫0xτxdx.The axial force is expressed asNx=∫0xτxdx.From the fundamental assumption 2, the following equation is obtained.(15)dSxdx=pK⋅dN2xdx2From (14), the following equation is obtained:(16)dSxdx=pK⋅dN2xdx2=h2φx+1EsAs∫0xτxdx+1EcAc∫0xτxdx.The differential equation of axial force is obtained by calculating the value of bending moment and considering the fundamental assumption 3.(17)−b≤x≤0,dN2xdx2−Kp1EsAs+1EcAcNx=hKFab+x2pbEsI0,0≤x≤a,dN2xdx2−Kp1EsAs+1EcAcNx=hKFa−x2pEsI0.Ifλ2=K/p1/EsAs+1/EcAc, (17) is substituted as follows:(18)−b≤x≤0,dN2xdx2−λ2Nx=hKFab+x2pbEsI0,0≤x≤a,dN2xdx2−λ2Nx=hKFa−x2pEsI0.The ordinary solution of (18) is as follows:(19)−b≤x≤0,Nx=A1sinhλx+B1coshλx−hKFab+x2pbλ2EsI0,0≤x≤a,Nx=A2sinhλx+B2coshλx−hKFa−x2pλ2EsI0.The following boundary conditions are considered:(20)Sxx=3−3b3=0,Nxx=a=0,Nxx=−b=0,Sxx=0+=Sxx=0−.The solution for the unknown parameter is expressed as follows:(21)A1=hKFa2pbλ3EsI0⋅coshλbcoshλbcosh0.423λb−sinhλbsinh0.423λb,B1=A1sinhλbcoshλb,A2=A1−hFKa+b2pbλ3EsI0,B2=−A2sinhλacoshλa.The slip of the overhanging beam is expressed as follows:(22)−b≤x≤0,Sx=pKdNxdx=pλA1Kcoshλx+pλB1Ksinhλx−hFa2bλ2EsI0,0≤x≤a,Sx=pKdNxdx=pλA2Kcoshλx+pλB2Ksinhλx+hF2λ2EsI0.According to (14), the relative slip strain at the interface is expressed as follows:(23)−b≤x≤0,εsx=pλ2A1Ksinhλx+pλ2B1Kcoshλx,0≤x≤a,εsx=pλ2A2Ksinhλx+pλ2B2Kcoshλx.


## 3.3. Additional Deflection due to Relative Slip

The additional deflection caused by relative slip is expressed as follows:(24)Δφx=εsxh.From (10), the following equation is obtained.(25)Δφx=ΔMxEsI0.Therefore,(26)d2Δfxdx2=ΔMxEsI0=εsxh.Substituting (23) into the above equation and integrating it twice, the following equation is obtained.(27)−b≤x≤0,Δfx=pKhA1sinhλx+B1coshλx+E1x+E2,0≤x≤a,Δfx=pKhA2sinhλx+B2coshλx+E3x+E4.The following boundary conditions are considered.(28)Δfxx=0=0,dΔfxdx|x=0−=dΔfxdx|x=0+,Δfxx=−b=0.Substituting boundary conditions into (27), the following equation is obtained.(29)E1=−A1sinhbλ+B1coshbλ−B1b,E2=−B1,E3=λA1−A2+E1=λA1−A2+−A1sinhbλ+B1coshbλ−B1b.E4=−B2,The deflection of the composite beam is a sum of the deflection due to concrete cracking and the additional deflection due to interface slip and can be expressed as follows:(30)f′x=fx+Δfx,where fx and Δfx are calculated according to (8) and (27), respectively.


## 4. Experimental Verification

### 4.1. Flexural Performance Test of SCCBs under Negative Bending Moment

#### 4.1.1. Experimental Design

The flexural performance of an SCCB in the negative moment region was tested to reveal the mechanism of stiffness degradation. An overhanging beam similar to an actual continuous beam was used to simulate the negative moment region of a continuous beam. Figure3 illustrates the dimensions of the test beam. The total length of the beam was 4 m, the distance between the two supports was 2.4 m, and the overhanging length was 1.4 m. The concrete slab and steel beam were attached using bolts, separated by a longitudinal distance of 140 mm and a transverse distance of 70 mm.Figure 3
Structure of the overhanging SCCB.Table1 shows the performance of concrete, and Table 2 shows the performance of the steel bar and steel beam.Table 1
Mechanical properties of concrete.
ComponentMaterialElastic modulusEC (MPa)Cube compressive strengthfcu (MPa)Bending and tensile strengthft (MPa)ConcreteC504.113 × 10453.954.728Table 2
Mechanical properties of the steel bar and steel beam.
ComponentMaterialYield strengthfy (MPa)Ultimate tensile strengthfu (MPa)Elongation (%)Steel beamQ355D41553630Steel barHRB400451.4668.321For the performance test, loads are applied to the end of the overhanging beam. Figure4 illustrates the loading device at the test site.Figure 4
Experimental setup.


#### 4.1.2. Loading Setup

The test beam is supported by two points and loaded at a single point at the end of the overhanging beam. The loading process involves two stages: preloading and formal loading. The load increment in each stage of preloading was 0.2 times the maximum elastic load. After loading to 0.4 times the maximum elastic load, the beam was unloaded. The maximum elastic load was estimated using finite element software. In the formal loading stage, the initial load increment each stage is 20 kN. A load of 10 kN was used in each stage till the test beam reached its elastic limit and was destroyed. In each stage, the load was held for 5 min, and the value of the load, slip, and strain gauge reading was noted.


#### 4.1.3. Experimental Results

At the initial stage of loading, the deflection of the overhanging beam increased gradually with the increase in load. When the load was increased to 34 kN, the first crack appeared on top of the movable support. When the load was increased to 81 kN, an abnormal sound was observed in the test beam, and a bond failure was observed at the interface between steel beam and concrete. With the increase in load, the length and number of cracks increases. When the load increased to 200 kN, the length of the crack in the concrete slab increases to 1.6 mm at the support. Furthermore, the displacement of the overhanging beam increased even when the load was unchanged. The bottom flange and web of the steel beam bent under pressure, and the structure of the bean was destroyed. Figure5 shows the yielding of the steel beam, and Figure 6 shows the crack distribution at the end of loading.Figure 5
Bending of steel beam.Figure 6
Crack distribution in test beam.


### 4.2. Comparison of Test Results and Calculated Values for the Overhanging Beam

The loading of the overhanging beam occurred in three stages: elastic, yielding, and strengthening stages. In the elastic stage, no cracks are observed in the concrete slab, and the corresponding load-displacement characteristics were linear. In the yielding stage, cracks occur and gradually increase, the load and deflection are nonlinear, and the stiffness of the composite beams gradually decreases. In the strengthening stage, the steel beam begins to bend and deflection increases rapidly. Figure7 shows the relationship between the load and the deflection of the overhanging end.Figure 7
Deflection of overhanging beam under loads.According to the Specifications for Design and Construction of Highway Steel-Concrete Composite Bridge (JTG/T D64-01-2015) of China [23], the shear stiffness of a single bolt is kss=13dssEcfck. Figure 7 shows a comparison of the deflection value calculated according to equation (30) and the test value. Figure 8 shows a comparison between the test and calculated values of deflection when the overhanging composite beam was loaded at 140 kN. Before the load value was 180 kN, the calculated value was consistent with the test value. After the load was increased to 180 kN, the test and calculated values were significantly different because after 180 kN, the composite beam entered the strengthening stage and gradual bending of the steel beam occurred. Meanwhile, the nonlinear characteristics of the composite beam were more significant and were not considered for calculations. The proposed method can be used to accurately calculate the deflection of composite beams during normal operation. Figure 9 shows a comparison of the interface slip calculated according to equation (22) and the test value. When the load is less than 100 kN, the calculated value is in good agreement with the experimental value. But after 100 kN, the difference between the two curves increases. The main reason is that the stiffness reduction caused by concrete cracking is not considered in the calculation.Figure 8
Deflection of the overhanging end of the beam loaded at 140 kN.Figure 9
Interface slip of overhanging beam under loads.


### 4.3. Comparison of Test Results and Calculated Values for Inverted Beams

Static load tests were performed on the composite beam in the negative moment region by inverting the test beam. During the tests, the concrete slab was placed at the bottom and the steel beam on the top, and the load was applied on the midspan of the beam. Simple supports were applied at both ends of the beam. Figure10 shows a schematic of the static load test of the inverted beam.Figure 10
Schematic of the static load test of the inverted beam.The accuracy of the proposed method was verified by applying it to calculate the slip and deflection of inverted beams. The deflection in the negative bending moment region under the influence of concrete cracks can be expressed as follows:(31)−a≤x≤0,fx=a2Fξ02EsI01+lnaξeffx−12aξ0x2+aξeff−xlnaξeff−x+3aξ0+2aξeffξ0+alnaξeff−aξ0lnaξ0,0≤x≤a,fx=−a2Fξ02EsI01+lnaξeffx+12aξ0x2−x+aξefflnx+aξeff−3aξ0−2aξeffξ0−alnaξeff+aξ0lnaξ0.The interface slip of the beam is expressed as follows:(32)−a≤x≤0,Sx=hF4λ2EsI0coshλx+sinhλacoshλasinhλx−1,0≤x≤a,Sx=−hF4λ2EsI0coshλx−sinhλacoshλasinhλx−1.The additional deflection caused by the slip is expressed as follows:(33)−a≤x≤0,Δfx=F4λ3EsI0sinhλx+sinhλacoshλacoshλx−λx−λa,0≤x≤a,Δfx=−F4λ3EsI0sinhλx−sinhλacoshλacoshλx−λx+λa.Table3 shows the calculated parameters of the inverted steel-concrete beam. The calculated results were compared with the experimental data reported in the literature [2, 6, 24], as shown in Figures 11 and 12. The calculated values were consistent with the experimental values before the yielding of the steel beams, as shown in Figure 11. Thus, the proposed method can be used to calculate the deflection of SCCBs during normal operation. In Figure 12, the calculated slip curve is consistent with the experimental value in shape, but there are some differences between them, which may be caused because there is inaccuracy in the measurement of slip value and the method proposed in this study does not consider the reduction of stiffness caused by cracks in concrete slab when calculating interface slip. We note that although there is some difference between the calculated value and the test value, the maximum difference is no more than 20%, which is considered to be an acceptable range for engineering. Figure 12 indicates that the calculated value of the interface slip was consistent with the experimental value, thus indicating that the proposed method can be used to accurately calculate the interface slip.Table 3
Parameters of the inverted steel-concrete beam.
ParameterSCB-1 of literature [2]SCB-3 of literature [2]Literature [24]CBS-1 of literature [6]a (mm)1800180017502000K (N/mm)2.44E + 052.44E + 055.55E + 055.70E + 05p (mm)120120100150As (mm2)23962396978828000Ac (mm2)420004200090000200000Es (N/mm2)2.06E + 052.06E + 052.06E + 052.06E + 05Ec (N/mm2)3.28E + 043.28E + 043.47E + 043.35E + 04I0 (mm4)4.38E + 074.79E + 077.00E + 085.89E + 09λ0.0023635610.0023635610.0018971680.001106847F (N)1000001000007000002000000h (mm)250250500891Figure 11
Comparison of deflection data of inverted beams.Figure 12
Comparison of slip data of inverted beams. (a) Literature [2]. (b) Literature [11].
(a)(b)


## 4.1. Flexural Performance Test of SCCBs under Negative Bending Moment

### 4.1.1. Experimental Design

The flexural performance of an SCCB in the negative moment region was tested to reveal the mechanism of stiffness degradation. An overhanging beam similar to an actual continuous beam was used to simulate the negative moment region of a continuous beam. Figure3 illustrates the dimensions of the test beam. The total length of the beam was 4 m, the distance between the two supports was 2.4 m, and the overhanging length was 1.4 m. The concrete slab and steel beam were attached using bolts, separated by a longitudinal distance of 140 mm and a transverse distance of 70 mm.Figure 3
Structure of the overhanging SCCB.Table1 shows the performance of concrete, and Table 2 shows the performance of the steel bar and steel beam.Table 1
Mechanical properties of concrete.
ComponentMaterialElastic modulusEC (MPa)Cube compressive strengthfcu (MPa)Bending and tensile strengthft (MPa)ConcreteC504.113 × 10453.954.728Table 2
Mechanical properties of the steel bar and steel beam.
ComponentMaterialYield strengthfy (MPa)Ultimate tensile strengthfu (MPa)Elongation (%)Steel beamQ355D41553630Steel barHRB400451.4668.321For the performance test, loads are applied to the end of the overhanging beam. Figure4 illustrates the loading device at the test site.Figure 4
Experimental setup.


### 4.1.2. Loading Setup

The test beam is supported by two points and loaded at a single point at the end of the overhanging beam. The loading process involves two stages: preloading and formal loading. The load increment in each stage of preloading was 0.2 times the maximum elastic load. After loading to 0.4 times the maximum elastic load, the beam was unloaded. The maximum elastic load was estimated using finite element software. In the formal loading stage, the initial load increment each stage is 20 kN. A load of 10 kN was used in each stage till the test beam reached its elastic limit and was destroyed. In each stage, the load was held for 5 min, and the value of the load, slip, and strain gauge reading was noted.


### 4.1.3. Experimental Results

At the initial stage of loading, the deflection of the overhanging beam increased gradually with the increase in load. When the load was increased to 34 kN, the first crack appeared on top of the movable support. When the load was increased to 81 kN, an abnormal sound was observed in the test beam, and a bond failure was observed at the interface between steel beam and concrete. With the increase in load, the length and number of cracks increases. When the load increased to 200 kN, the length of the crack in the concrete slab increases to 1.6 mm at the support. Furthermore, the displacement of the overhanging beam increased even when the load was unchanged. The bottom flange and web of the steel beam bent under pressure, and the structure of the bean was destroyed. Figure5 shows the yielding of the steel beam, and Figure 6 shows the crack distribution at the end of loading.Figure 5
Bending of steel beam.Figure 6
Crack distribution in test beam.


## 4.1.1. Experimental Design

The flexural performance of an SCCB in the negative moment region was tested to reveal the mechanism of stiffness degradation. An overhanging beam similar to an actual continuous beam was used to simulate the negative moment region of a continuous beam. Figure3 illustrates the dimensions of the test beam. The total length of the beam was 4 m, the distance between the two supports was 2.4 m, and the overhanging length was 1.4 m. The concrete slab and steel beam were attached using bolts, separated by a longitudinal distance of 140 mm and a transverse distance of 70 mm.Figure 3
Structure of the overhanging SCCB.Table1 shows the performance of concrete, and Table 2 shows the performance of the steel bar and steel beam.Table 1
Mechanical properties of concrete.
ComponentMaterialElastic modulusEC (MPa)Cube compressive strengthfcu (MPa)Bending and tensile strengthft (MPa)ConcreteC504.113 × 10453.954.728Table 2
Mechanical properties of the steel bar and steel beam.
ComponentMaterialYield strengthfy (MPa)Ultimate tensile strengthfu (MPa)Elongation (%)Steel beamQ355D41553630Steel barHRB400451.4668.321For the performance test, loads are applied to the end of the overhanging beam. Figure4 illustrates the loading device at the test site.Figure 4
Experimental setup.


## 4.1.2. Loading Setup

The test beam is supported by two points and loaded at a single point at the end of the overhanging beam. The loading process involves two stages: preloading and formal loading. The load increment in each stage of preloading was 0.2 times the maximum elastic load. After loading to 0.4 times the maximum elastic load, the beam was unloaded. The maximum elastic load was estimated using finite element software. In the formal loading stage, the initial load increment each stage is 20 kN. A load of 10 kN was used in each stage till the test beam reached its elastic limit and was destroyed. In each stage, the load was held for 5 min, and the value of the load, slip, and strain gauge reading was noted.


## 4.1.3. Experimental Results

At the initial stage of loading, the deflection of the overhanging beam increased gradually with the increase in load. When the load was increased to 34 kN, the first crack appeared on top of the movable support. When the load was increased to 81 kN, an abnormal sound was observed in the test beam, and a bond failure was observed at the interface between steel beam and concrete. With the increase in load, the length and number of cracks increases. When the load increased to 200 kN, the length of the crack in the concrete slab increases to 1.6 mm at the support. Furthermore, the displacement of the overhanging beam increased even when the load was unchanged. The bottom flange and web of the steel beam bent under pressure, and the structure of the bean was destroyed. Figure5 shows the yielding of the steel beam, and Figure 6 shows the crack distribution at the end of loading.Figure 5
Bending of steel beam.Figure 6
Crack distribution in test beam.


## 4.2. Comparison of Test Results and Calculated Values for the Overhanging Beam

The loading of the overhanging beam occurred in three stages: elastic, yielding, and strengthening stages. In the elastic stage, no cracks are observed in the concrete slab, and the corresponding load-displacement characteristics were linear. In the yielding stage, cracks occur and gradually increase, the load and deflection are nonlinear, and the stiffness of the composite beams gradually decreases. In the strengthening stage, the steel beam begins to bend and deflection increases rapidly. Figure7 shows the relationship between the load and the deflection of the overhanging end.Figure 7
Deflection of overhanging beam under loads.According to the Specifications for Design and Construction of Highway Steel-Concrete Composite Bridge (JTG/T D64-01-2015) of China [23], the shear stiffness of a single bolt is kss=13dssEcfck. Figure 7 shows a comparison of the deflection value calculated according to equation (30) and the test value. Figure 8 shows a comparison between the test and calculated values of deflection when the overhanging composite beam was loaded at 140 kN. Before the load value was 180 kN, the calculated value was consistent with the test value. After the load was increased to 180 kN, the test and calculated values were significantly different because after 180 kN, the composite beam entered the strengthening stage and gradual bending of the steel beam occurred. Meanwhile, the nonlinear characteristics of the composite beam were more significant and were not considered for calculations. The proposed method can be used to accurately calculate the deflection of composite beams during normal operation. Figure 9 shows a comparison of the interface slip calculated according to equation (22) and the test value. When the load is less than 100 kN, the calculated value is in good agreement with the experimental value. But after 100 kN, the difference between the two curves increases. The main reason is that the stiffness reduction caused by concrete cracking is not considered in the calculation.Figure 8
Deflection of the overhanging end of the beam loaded at 140 kN.Figure 9
Interface slip of overhanging beam under loads.


## 4.3. Comparison of Test Results and Calculated Values for Inverted Beams

Static load tests were performed on the composite beam in the negative moment region by inverting the test beam. During the tests, the concrete slab was placed at the bottom and the steel beam on the top, and the load was applied on the midspan of the beam. Simple supports were applied at both ends of the beam. Figure10 shows a schematic of the static load test of the inverted beam.Figure 10
Schematic of the static load test of the inverted beam.The accuracy of the proposed method was verified by applying it to calculate the slip and deflection of inverted beams. The deflection in the negative bending moment region under the influence of concrete cracks can be expressed as follows:(31)−a≤x≤0,fx=a2Fξ02EsI01+lnaξeffx−12aξ0x2+aξeff−xlnaξeff−x+3aξ0+2aξeffξ0+alnaξeff−aξ0lnaξ0,0≤x≤a,fx=−a2Fξ02EsI01+lnaξeffx+12aξ0x2−x+aξefflnx+aξeff−3aξ0−2aξeffξ0−alnaξeff+aξ0lnaξ0.The interface slip of the beam is expressed as follows:(32)−a≤x≤0,Sx=hF4λ2EsI0coshλx+sinhλacoshλasinhλx−1,0≤x≤a,Sx=−hF4λ2EsI0coshλx−sinhλacoshλasinhλx−1.The additional deflection caused by the slip is expressed as follows:(33)−a≤x≤0,Δfx=F4λ3EsI0sinhλx+sinhλacoshλacoshλx−λx−λa,0≤x≤a,Δfx=−F4λ3EsI0sinhλx−sinhλacoshλacoshλx−λx+λa.Table3 shows the calculated parameters of the inverted steel-concrete beam. The calculated results were compared with the experimental data reported in the literature [2, 6, 24], as shown in Figures 11 and 12. The calculated values were consistent with the experimental values before the yielding of the steel beams, as shown in Figure 11. Thus, the proposed method can be used to calculate the deflection of SCCBs during normal operation. In Figure 12, the calculated slip curve is consistent with the experimental value in shape, but there are some differences between them, which may be caused because there is inaccuracy in the measurement of slip value and the method proposed in this study does not consider the reduction of stiffness caused by cracks in concrete slab when calculating interface slip. We note that although there is some difference between the calculated value and the test value, the maximum difference is no more than 20%, which is considered to be an acceptable range for engineering. Figure 12 indicates that the calculated value of the interface slip was consistent with the experimental value, thus indicating that the proposed method can be used to accurately calculate the interface slip.Table 3
Parameters of the inverted steel-concrete beam.
ParameterSCB-1 of literature [2]SCB-3 of literature [2]Literature [24]CBS-1 of literature [6]a (mm)1800180017502000K (N/mm)2.44E + 052.44E + 055.55E + 055.70E + 05p (mm)120120100150As (mm2)23962396978828000Ac (mm2)420004200090000200000Es (N/mm2)2.06E + 052.06E + 052.06E + 052.06E + 05Ec (N/mm2)3.28E + 043.28E + 043.47E + 043.35E + 04I0 (mm4)4.38E + 074.79E + 077.00E + 085.89E + 09λ0.0023635610.0023635610.0018971680.001106847F (N)1000001000007000002000000h (mm)250250500891Figure 11
Comparison of deflection data of inverted beams.Figure 12
Comparison of slip data of inverted beams. (a) Literature [2]. (b) Literature [11].
(a)(b)


## 5. Conclusion

In this study, a method for the calculation of stiffness of SCCBs in the negative moment region is established and verified through theoretical and experimental analyses. The main conclusions drawn from the results are as follows:(1)
The stiffness in the negative moment region of an SCCB is reduced because of concrete cracking. A facile and accurate linear reduction method is proposed for the evaluation of the stiffness in the negative moment region.(2)
Based on the test model of the overhanging beam, a mechanical model of a composite beam of the negative moment region is established, considering that interface slip occurs. An analytical solution of interface slip in the negative bending moment region and additional deflection caused by the slip is calculated. The accuracy of the method is verified by comparing the calculated results with the experimental results.(3)
The proposed method is applied to the calculation of slip and deflection in the negative moment region of simply supported inverted SCCBs, and the calculated results are compared with the experimental data in references, which are in good agreement.The proposed method is convenient for engineering applications as it can be used to easily and accurately calculate the interface slip and vertical deflection in the negative moment region of SCCBs. Static load tests are performed in this study. The load on continuous beams in bridges is generated by vehicles mainly as a time-varying spectrum. Under such loading, steel-concrete composite bridges are vulnerable to fatigue failure. Studies are being conducted to further investigate the mechanical properties and fatigue failure of SCCBs in the negative moment region.



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*Source: 5171416-2022-08-28.xml*