# Shear Capacity of Open Sandwich Steel Plate-Concrete Composite Slab: Experimental and Analytical Studies

Authors: Lili Wu; Lipei An; Jiawei Li
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

Considering that the fixed crack model by default of the general finite element software was unable to simulate the shear softening behavior of concrete in the actual situation, a rotational crack model based on the modified compression field theory developed by UMAT (user material) of ABAQUS software was proposed and applied to the nonlinear analysis, and a numerical simulated model for the steel-concrete composite slab was built for shear analysis. Experimental studies and numerical analyses were used to investigate the shear load-carrying capacity, deformation, and crack development in steel plate-concrete composite slab, as well as the effects of the shear span ratio and shear stud spacing on the shear performance and the contribution of the steel plate and the concrete to the shear performance. Shear capacity tests were conducted on three open sandwich steel plate-concrete composite slabs and one plain concrete slab without a steel plate. The results indicated that the shear-compression failure mode occurred primarily in the steel plate-concrete composite slab and that the steel plate sustained more than 50% of the total shear force. Because of the combination effect of steel plate, the actual shear force sustained by the concrete in the composite slab was 1.27 to 2.22 times greater than that of the calculated value through the Chinese Design Code for Concrete Structures (GB 50010-2010). Furthermore, the shear capacity of the specimen increases by 37% as the shear stud spacing decreases from 250 mm to 150 mm. By comparing the shear capacity, the overall process of load deformation development, and the failure mode, it was shown that the simulation results corresponded with the experimental results. Furthermore, the numerical simulation model was applied to analyze the influence of some factors on composite slab, and a formula of shear bearing capacity of slab was obtained. The results of the formula agreed with the test result, which could provide references to the design and application of steel plate-concrete composite slab.


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## Body

## 1. Introduction

A steel plate-concrete composite component is a combination of steel and concrete (Figure1). Anchor shear studs are welded onto a steel plate to connect the steel plate to a concrete slab.Figure 1
Steel plate-concrete composite component.Due to the simplicity of these structures, steel plate-concrete composite components, including composite beams, slabs, or shear walls, have been recently used for strengthening purposes [1, 2] and as components of steel plate-concrete composite shear walls [3–6]. Other applications of steel plate-concrete composite slab, such as in railway and subway bridge structures, are expected to emerge, such as bridge structures of railway viaducts [7, 8].Link and Elwi [4] conducted an experimental study and numerical simulation of a dual-layer steel plate-concrete composite shear wall. The transverse and vertical load-carrying capacity and ductility were analyzed, and the effect of the friction and contact between the steel plate and concrete was simulated. The results showed that it was easy to understand the behavior of the walls in light of failure modes, bearing mechanism, stress redistribution, and postpeak response. Takeuchi et al. [9] explored the performance of the steel plate-concrete composite shear walls used in nuclear power plants by experiments, in which the bending and shear performance of the composite shear walls were analyzed. As demonstrated through the compression tests conducted in that study, local buckling of the steel plate had no influence on the load-carrying capacity of the steel plate-concrete composite shear wall. Liang et al. [10] developed a finite element analysis on the performance of a dual-layer steel plate-concrete composite shear wall subjected to compression and in-plane shear. The effects of the local buckling and spacing of the studs on the ultimate load-carrying capacity of the composite shear wall were analyzed. The results indicated that when the height-to-thickness ratio of the composite shear wall was small, the ultimate load-carrying capacity of the shear wall was determined by the shear bearing capacity of the shear studs and the local buckling capacity of the steel plate. Furthermore, when the height-to-thickness ratio of the composite shear wall was large, the shear wall was able to produce postbuckling strength. Varma et al. [11] put forward a mechanical model for a steel plate-concrete composite shear wall, which was primarily used to analyze the in-plane shear force and out-of-plane bending moment. A composite wall designed to prevent failure modes such as shear damage at the related plane of action and local buckling of the steel plate was proposed. Arabzadeh et al. [12] conducted experimental studies of one- and three-story steel plate-composite shear walls with the scales of 1 : 3 and 1 : 4. Their results showed that bolt spacing to plate thickness ratio was directly related to system ductility, whereas the plate yield load decreased with the increase of the ratio. Through that study, plate stiffening requirements were obtained with minimum reinforcement for reinforced concrete. Arabzade et al. [13] analyzed the steel plate-composite shear walls through experimental and theoretical studies and addressed the buckling load of the steel plate. The elastic buckling coefficients of steel plates with various aspect ratios were obtained. Zhang et al. [14] studied steel plate-concrete composite walls used for nuclear power plants, consisting of thick concrete walls with exterior steel plates serving as reinforcement. Considering the significant influence on the behavior of steel-concrete composite walls, the shear connectors should be designed accordingly. The maximum steel plate slenderness (i.e., the ratio of shear connector spacing to plate thickness) to prevent local buckling was derived.Some studies have focused on dual-layer steel plate-concrete composites. Yang et al. [15] studied open sandwich steel plate-concrete composite slabs, mostly focusing on bending performance. In practical engineering, open sandwich steel plate-concrete composite slabs have been widely used for composite strengthening and in railway bridges. The amount of shear load individually borne by the steel plate and concrete slab components of a single steel plate-concrete composite slab is not clear. Meanwhile, accurate indication of the effect of the material performance of the steel plate and concrete components and the degree of shear connection on the interface is urgently needed for shear strengthening projects.At present, common finite element software incorporates a fixed crack model in numerical simulations by default. Application of a rotating crack model, which is more practical, requires secondary analysis by the finite element software, and there are few systematic studies in this aspect.In this study, an experimental evaluation of the shear resistance of a simply supported open sandwich steel plate-concrete composite slab was conducted, using three steel plate-concrete composite slabs and one concrete slab. The effects of the shear span ratio and shear stud spacing on the shear performance of the composite slab, the damage mode and the mechanism of the shear failure of the steel plate-concrete composite slab, and the shear sustained individually by the steel plate and concrete were analyzed. A secondary analysis using the UMAT subprogram of ABAQUS was conducted, and the modified compression field theory (MCFT) was applied to analyze the shear capacity of the concrete. The experimental results were compared with the results of the numerical simulation in order to validate the model. Furthermore, the influence of some factors on composite slab was analyzed by the numerical simulation model and a formula of shear bearing capacity of slab was obtained, which could provide references to the design and application of steel plate-concrete composite slab.


## 2. Experimental Program

### 2.1. Specimen

Three steel plate-concrete composite slabs, identified as SCS-1, SCS-2, and SCS-3, were designed as shown in Figure2 and the main parameters of each specimen are shown in Table 1. The shear stud spacing and shear span ratio, λ (=a/h), where a is the distance between the loading point and support point and h is the height of the composite slab, varied among the different composite slabs. bs is the thickness of the steel plate. All composite slabs had height of 400 mm, thickness of 150 mm, and distance between the support point and slab ends of 100 mm.Figure 2
Sectional dimensions of specimens.Table 1
Test matrix of steel plate-concrete composite deep slabs.
Specimens
λ
Span L0 (mm)
a (mm)
bs (mm)
Stud spacing (mm)
Steel mesh (mm)
SCS-1
2.63
2100
1050
6
150
Φ6-150
SCS-2
1.88
1500
750
6
150
Φ6-150
SCS-3
1.88
1500
750
6
250
Φ6-150
SCS-4
1.88
1500
750
—
—
Φ6-150The steel material used for the steel plate had the yield strength of 316.7 MPa, the ultimate strength of 448.8 MPa, and the percentage elongation of 28.8%. The concrete used in the composite samples had the averaged compression strength of 28.66 MPa based on three standard concrete cube specimens of 150 × 150 × 150 mm. Shear studs, made of ML-15 steel, had the diameter of 13 mm and the ultimate tensile strength offus = 400 MPa. For comparison, a plain concrete slab without a steel plate, identified as SCS-4, was cast using the same material and reinforcement as shown in Figure 2. In order to demonstrate typical shear failure and prevent local damage at the loading zone and supports, two layers of steel mesh were placed at the loading zone, and small steel plates of 6 mm thick were placed on the sides of the concrete at the loading zone (short studs were welded at the inner surface).


### 2.2. Test Setup and Instruments

Three-point static loading tests were conducted on the four specimens. The ends of the specimens were simply supported. Cushion blocks were established at the midspan loading zones of the composite slabs to ensure that the vertical displacement of the steel plate and concrete at the loading zone remained identical, to sustain the load and prevent local damage at the loading zone together. Step loading which was driven by displacement was used according to the predicted failure load in order to measure the displacement and collect the strain value when the deformation was fully developed. The test setup is shown in Figure3.Figure 3
Test setup.
(a)
(b)The test measurements included deflection at the midspan and 350 mm around it, transverse and side displacement of the support, and strain of each section. A labelled layout of the measuring points is provided in Figure4, in which Hi denotes the displacement gauges, Si denotes the strain gauges and the strain rosettes on the steel plate, and Ci denotes the strain gauges on the concrete. Strain gauges were placed in the general area of both sides of the plain concrete slab and the steel plate.Figure 4
Layout of displacement gauges and strain gauges. (a) Layout of displacement gauges. (b) Layout of strain gauges on steel plate. (c) Layout of strain gauges on concrete slabs.
(a)
(b)
(c)


## 2.1. Specimen

Three steel plate-concrete composite slabs, identified as SCS-1, SCS-2, and SCS-3, were designed as shown in Figure2 and the main parameters of each specimen are shown in Table 1. The shear stud spacing and shear span ratio, λ (=a/h), where a is the distance between the loading point and support point and h is the height of the composite slab, varied among the different composite slabs. bs is the thickness of the steel plate. All composite slabs had height of 400 mm, thickness of 150 mm, and distance between the support point and slab ends of 100 mm.Figure 2
Sectional dimensions of specimens.Table 1
Test matrix of steel plate-concrete composite deep slabs.
Specimens
λ
Span L0 (mm)
a (mm)
bs (mm)
Stud spacing (mm)
Steel mesh (mm)
SCS-1
2.63
2100
1050
6
150
Φ6-150
SCS-2
1.88
1500
750
6
150
Φ6-150
SCS-3
1.88
1500
750
6
250
Φ6-150
SCS-4
1.88
1500
750
—
—
Φ6-150The steel material used for the steel plate had the yield strength of 316.7 MPa, the ultimate strength of 448.8 MPa, and the percentage elongation of 28.8%. The concrete used in the composite samples had the averaged compression strength of 28.66 MPa based on three standard concrete cube specimens of 150 × 150 × 150 mm. Shear studs, made of ML-15 steel, had the diameter of 13 mm and the ultimate tensile strength offus = 400 MPa. For comparison, a plain concrete slab without a steel plate, identified as SCS-4, was cast using the same material and reinforcement as shown in Figure 2. In order to demonstrate typical shear failure and prevent local damage at the loading zone and supports, two layers of steel mesh were placed at the loading zone, and small steel plates of 6 mm thick were placed on the sides of the concrete at the loading zone (short studs were welded at the inner surface).


## 2.2. Test Setup and Instruments

Three-point static loading tests were conducted on the four specimens. The ends of the specimens were simply supported. Cushion blocks were established at the midspan loading zones of the composite slabs to ensure that the vertical displacement of the steel plate and concrete at the loading zone remained identical, to sustain the load and prevent local damage at the loading zone together. Step loading which was driven by displacement was used according to the predicted failure load in order to measure the displacement and collect the strain value when the deformation was fully developed. The test setup is shown in Figure3.Figure 3
Test setup.
(a)
(b)The test measurements included deflection at the midspan and 350 mm around it, transverse and side displacement of the support, and strain of each section. A labelled layout of the measuring points is provided in Figure4, in which Hi denotes the displacement gauges, Si denotes the strain gauges and the strain rosettes on the steel plate, and Ci denotes the strain gauges on the concrete. Strain gauges were placed in the general area of both sides of the plain concrete slab and the steel plate.Figure 4
Layout of displacement gauges and strain gauges. (a) Layout of displacement gauges. (b) Layout of strain gauges on steel plate. (c) Layout of strain gauges on concrete slabs.
(a)
(b)
(c)


## 3. Test Results and Discussion

### 3.1. Failure Mode

Vertical cracks were first observed at midspan when the loading force approached 40%Pu, where Pu is the peak load. These vertical cracks at midspan further propagated and widened with the increase of the applied load. When the loading force reached 85% Pu, diagonal shear cracks simultaneously appeared on the side of the concrete and peeling of the steel plate appeared near the loading area on the top. The pattern of the cracks is shown in Figure 5, where the number denotes the load at the time of crack development at the respective location.Figure 5
Crack pattern of steel plate-concrete composite slabs. (a) Crack pattern diagram for SCS-1 (λ=2.63). (b) Crack pattern diagram for SCS-2 (λ=1.88). (c) Crack pattern diagram for SCS-3 (λ=1.88). (d) Crack pattern diagram for SCS-4 (λ=1.88).
(a)
(b)
(c)
(d)As is evident from the damage to SCS-2 and SCS-3 whenλ≤2.0, the mode of failure appears to be typical shear-compression failure. The failure process was rapid. The failure load and the load at the time of development of diagonal cracks were similar. The development of cracks in the diagonal section was more sudden, and the brittleness was more obvious in the specimen with smaller shear stud spacing. Due to the increased shear span ratio of specimens SCS-1 (λ=2.63) and SCS-4, the concrete failure appeared to be bending one, and the concrete failure of SCS-4 was the result of the failure of the normal section of the bending member, which was sparsely reinforced. Therefore, the mode of failure of the steel plate-concrete composite slab was related not only to the shear stud spacing but also to the shear span ratio of the composite slab.


### 3.2. Load-Displacement Relations

The load-displacement relations of the three composite slabs and the plain concrete slab are shown in Figure6, where the x-coordinate denotes the midspan deflection δ and the y-coordinate denotes the load of the specimen. As shown in Figure 6, the loading process of the steel plate-concrete composite slabs could be separated into three stages:(a)
Elastic stage: the stage lasted from the start of loading until the elastic shear capacity stage. All parts of the specimen were in the elastic state, and the relationship between the shear stress and midspan deflection was essentially linear for all specimens.(b)
Elastoplastic stage: when cracks appeared on the concrete, the stiffness of the specimen decreased and deflection development was accelerated. The load-displacement curve of the composite slab was nonlinear. Vertical cracks appeared on the concrete at the midspan and the cracks widened as the load increased.(c)
Descending stage: as the load was continually increased, once a certain load-carrying capacity of the steel plate-concrete composite slab was reached, diagonal cracks appeared in the concrete. At this point, the load-carrying capacity was at its maximum and shear failure occurred at the side of the concrete. Due to the increase in shear span ratio and decrease in shear stud spacing, the descending branch of the curve was smooth. The three composite slabs demonstrated outstanding ductility compared to SCS-4.Figure 6
Load-displacement diagrams: (a) SCS-1, (b) SCS-2, (c) SCS-3, and (d) SCS-4.
(a)
(b)
(c)
(d)


### 3.3. Shear Capacity of the Specimens

In the Chinese Design Code for Concrete Structures (GB 50010-2010)), the reinforced concrete shear capacity,Vc, is stipulated by(1)Vc=αcvftbh0,where αcv=1.75/λ+1, λ is shear span ratio, b and h0 are, respectively, the width and effective height of the section. The contribution of the reinforcement mesh bars is ignored here.All data, as well as test results, are summarized in Table2. In this table, Pj is the measured ultimate load value and Pjs is the ultimate load sustained by the steel plate that can be obtained by the strain gauges on the steel plate. Therefore, Pjc is the ultimate load sustained by the concrete that can be found by subtracting Pjs from Pj. Vc is the ultimate load of the concrete calculated through the standard of the concrete. The ratio Pjs/Vc represents the strengthening function of the steel plates in the steel plate-concrete composite slabs for the concrete part.Table 2
Experimental load value (kN).
Specimens
λ
P
j
P
jc
P
js
P
jc
/
P
j
V
c
P
jc
/
V
c
SCS-1
2.63
344.3
155.3
189.0
0.45
87.7
1.77
SCS-2
1.88
416.9
195.0
221.9
0.47
87.7
2.22
SCS-3
1.88
304.4
111.7
192.7
0.37
87.7
1.27
SCS-4
1.88
71.9
71.9
0
1.0
87.7
—The ultimate load sustained by the steel platePjs could be calculated from ε1, ε2, and ε3 measured by the strain gauges on the steel plate. The strain analysis process for the steel plate was as follows. When the material was still elastic, two principal stresses σ1 and σ2 and the vertical stress τxy could be derived according to the mechanics of the material. The following formulas were obtained:(2)σ1,2=Eε1+ε321−ν±E21−ν2ε1−ε22+2ε2−ε32,(3)τxy=E21+νε1+ε2−2ε3.According to the von Mises yield condition, when the material equivalent stressσn (calculated by formula (4)) was greater than the material yield strength fw, the material came into the yield stage. On the basis of the incremental theory in plastic mechanics, when the material was in the plastic state, the expressions for calculating the two principal stresses and shear stress could be obtained from the following formulas:(4)σn=12σ1−σ22+σ12+σ22,(5)σ1′=2β+13β2+β+1fw,(6)σ2′=β+23β2+β+1fw,(7)τxy′=12σ1−σ2sin2α0.Table2 indicates that the shear force sustained by the concrete ranged from approximately 37% to 47%, and the steel plate contributed more than 50% of the shear capacity. As shown by comparison of SCS-1 and SCS-2, the ultimate shear capacity decreased as the shear span ratio increased. Furthermore, as shown by comparison of SCS-2 and SCS-3, the ultimate shear capacity decreased as the shear stud spacing increased. As evident from comparisons of SCS-4 to SCS-1 and SCS-3, the steel plate improved the ultimate shear capacity of the composite slab and the loading capacity of the composite slab was three times greater than that of the plain concrete. The actual shear force sustained by the concrete in the composite slab was 1.27 to 2.22 times greater than that of the calculated value through the Chinese Design Code for Concrete Structures (GB 50010-2010). Moreover, the load-carrying capacity of the concrete was higher with smaller shear stud spacing. Bending failure with a low load-carrying capacity occurred in the plain concrete specimen, whereas the failure mode for the composite slabs was shear failure owing to the presence of the steel plate.


## 3.1. Failure Mode

Vertical cracks were first observed at midspan when the loading force approached 40%Pu, where Pu is the peak load. These vertical cracks at midspan further propagated and widened with the increase of the applied load. When the loading force reached 85% Pu, diagonal shear cracks simultaneously appeared on the side of the concrete and peeling of the steel plate appeared near the loading area on the top. The pattern of the cracks is shown in Figure 5, where the number denotes the load at the time of crack development at the respective location.Figure 5
Crack pattern of steel plate-concrete composite slabs. (a) Crack pattern diagram for SCS-1 (λ=2.63). (b) Crack pattern diagram for SCS-2 (λ=1.88). (c) Crack pattern diagram for SCS-3 (λ=1.88). (d) Crack pattern diagram for SCS-4 (λ=1.88).
(a)
(b)
(c)
(d)As is evident from the damage to SCS-2 and SCS-3 whenλ≤2.0, the mode of failure appears to be typical shear-compression failure. The failure process was rapid. The failure load and the load at the time of development of diagonal cracks were similar. The development of cracks in the diagonal section was more sudden, and the brittleness was more obvious in the specimen with smaller shear stud spacing. Due to the increased shear span ratio of specimens SCS-1 (λ=2.63) and SCS-4, the concrete failure appeared to be bending one, and the concrete failure of SCS-4 was the result of the failure of the normal section of the bending member, which was sparsely reinforced. Therefore, the mode of failure of the steel plate-concrete composite slab was related not only to the shear stud spacing but also to the shear span ratio of the composite slab.


## 3.2. Load-Displacement Relations

The load-displacement relations of the three composite slabs and the plain concrete slab are shown in Figure6, where the x-coordinate denotes the midspan deflection δ and the y-coordinate denotes the load of the specimen. As shown in Figure 6, the loading process of the steel plate-concrete composite slabs could be separated into three stages:(a)
Elastic stage: the stage lasted from the start of loading until the elastic shear capacity stage. All parts of the specimen were in the elastic state, and the relationship between the shear stress and midspan deflection was essentially linear for all specimens.(b)
Elastoplastic stage: when cracks appeared on the concrete, the stiffness of the specimen decreased and deflection development was accelerated. The load-displacement curve of the composite slab was nonlinear. Vertical cracks appeared on the concrete at the midspan and the cracks widened as the load increased.(c)
Descending stage: as the load was continually increased, once a certain load-carrying capacity of the steel plate-concrete composite slab was reached, diagonal cracks appeared in the concrete. At this point, the load-carrying capacity was at its maximum and shear failure occurred at the side of the concrete. Due to the increase in shear span ratio and decrease in shear stud spacing, the descending branch of the curve was smooth. The three composite slabs demonstrated outstanding ductility compared to SCS-4.Figure 6
Load-displacement diagrams: (a) SCS-1, (b) SCS-2, (c) SCS-3, and (d) SCS-4.
(a)
(b)
(c)
(d)


## 3.3. Shear Capacity of the Specimens

In the Chinese Design Code for Concrete Structures (GB 50010-2010)), the reinforced concrete shear capacity,Vc, is stipulated by(1)Vc=αcvftbh0,where αcv=1.75/λ+1, λ is shear span ratio, b and h0 are, respectively, the width and effective height of the section. The contribution of the reinforcement mesh bars is ignored here.All data, as well as test results, are summarized in Table2. In this table, Pj is the measured ultimate load value and Pjs is the ultimate load sustained by the steel plate that can be obtained by the strain gauges on the steel plate. Therefore, Pjc is the ultimate load sustained by the concrete that can be found by subtracting Pjs from Pj. Vc is the ultimate load of the concrete calculated through the standard of the concrete. The ratio Pjs/Vc represents the strengthening function of the steel plates in the steel plate-concrete composite slabs for the concrete part.Table 2
Experimental load value (kN).
Specimens
λ
P
j
P
jc
P
js
P
jc
/
P
j
V
c
P
jc
/
V
c
SCS-1
2.63
344.3
155.3
189.0
0.45
87.7
1.77
SCS-2
1.88
416.9
195.0
221.9
0.47
87.7
2.22
SCS-3
1.88
304.4
111.7
192.7
0.37
87.7
1.27
SCS-4
1.88
71.9
71.9
0
1.0
87.7
—The ultimate load sustained by the steel platePjs could be calculated from ε1, ε2, and ε3 measured by the strain gauges on the steel plate. The strain analysis process for the steel plate was as follows. When the material was still elastic, two principal stresses σ1 and σ2 and the vertical stress τxy could be derived according to the mechanics of the material. The following formulas were obtained:(2)σ1,2=Eε1+ε321−ν±E21−ν2ε1−ε22+2ε2−ε32,(3)τxy=E21+νε1+ε2−2ε3.According to the von Mises yield condition, when the material equivalent stressσn (calculated by formula (4)) was greater than the material yield strength fw, the material came into the yield stage. On the basis of the incremental theory in plastic mechanics, when the material was in the plastic state, the expressions for calculating the two principal stresses and shear stress could be obtained from the following formulas:(4)σn=12σ1−σ22+σ12+σ22,(5)σ1′=2β+13β2+β+1fw,(6)σ2′=β+23β2+β+1fw,(7)τxy′=12σ1−σ2sin2α0.Table2 indicates that the shear force sustained by the concrete ranged from approximately 37% to 47%, and the steel plate contributed more than 50% of the shear capacity. As shown by comparison of SCS-1 and SCS-2, the ultimate shear capacity decreased as the shear span ratio increased. Furthermore, as shown by comparison of SCS-2 and SCS-3, the ultimate shear capacity decreased as the shear stud spacing increased. As evident from comparisons of SCS-4 to SCS-1 and SCS-3, the steel plate improved the ultimate shear capacity of the composite slab and the loading capacity of the composite slab was three times greater than that of the plain concrete. The actual shear force sustained by the concrete in the composite slab was 1.27 to 2.22 times greater than that of the calculated value through the Chinese Design Code for Concrete Structures (GB 50010-2010). Moreover, the load-carrying capacity of the concrete was higher with smaller shear stud spacing. Bending failure with a low load-carrying capacity occurred in the plain concrete specimen, whereas the failure mode for the composite slabs was shear failure owing to the presence of the steel plate.


## 4. Modeling of Shear Capacity of the Composite

Numerical simulation using finite element analysis (FEA) was conducted to further explore the shear capacity of the composite. The commercially available FEA software ABAQUS was used in this study.


### 4.1. Material Properties of Steel Plate

A shell element, S4R, was used to simulate the steel plate, and a rectangular beam section element, B32, was used to simulate the concrete. Twenty-five integral points were placed in the beam section, and modified compression field theory was applied to each integral point. The constitutional relationship was defined by a bilinear elastic hardening model (bilinear model). The yield strength and elastic modulus were given according to the experimental results given in Section2.1 Young’s modulus of the steel was 194 GPa, and Poisson’s ratio was 0.3.


### 4.2. Material Properties of Concrete Using Fixed Crack Model and Rotation Crack Model

Here, two models were used to simulate the concrete cracking: (a) the rotation crack model using the developed UMAT subprogram of the concrete rotation crack model (denoted as RCM) and (b) a fixed crack model (denoted as FCM), which is presented in the next section.


#### 4.2.1. Fixed Crack Model

According to the fixed crack model, the direction of the cracks that appeared after the first was identical to that of the first. In other words, the shear stress at the crack surface increased until shear locking occurred. In practice, crack direction angles vary as the load increases, and the shear stress at the crack surface remains steady or even decreases. However, the fixed crack model cannot simulate the shear softening problem of concrete in practical situations and it overestimates the analytical result of shear capacity. The rotation crack model can demonstrate the variation in angle direction, as presented in the next section. The stiffness matrix can vary at any time under different crack directions, and the shear stress at the crack surface can be better simulated.


#### 4.2.2. Rotation Crack Model Using Modified Compression Field Theory

Vecchio and Collins proposed the MCFT [16] based on compression field theory [17, 18]. This theory suggests that when a crack appears in concrete, the principal tensile stress exists on the crack’s surface and the shear force is sustained by both principal tensile and compressive stresses of the concrete. The equilibrium condition, deformation compatibility condition, and constitutional condition (including the compressive and tensile constitution of the concrete) are established according to the mean stress and strain values. Thereby, the shear strength of reinforced concrete can be better represented.The UMAT in ABAQUS was used to define the subroutine of the MCFT for describing the rotation crack model for the concrete. When the state variant and strain increment were input by the main program, the UMAT solved the stress increment according to the strain variant, solved and returned the Jacobian matrix to develop the global stiffness matrix, and then temporarily stored the state variant for the next increment. The key was to solve the Jacobian matrix.Because the coordinates and positive direction were different before and after the appearance of cracks in the concrete, the Jacobian matrix was also different. Whenε1≤εcr, where ε1 is the principal tensile strain, εcr refers to the cracking strain, the uniaxial tensile stress of concrete is equal to the value before cracking, and the shear modulus of concrete is represented by G=f1−f2/2ε1−ε2, as recommended by Zhu et al. [19]. The derived stiffness matrix Dc is shown in equation (8); when ε1>εcr, the uniaxial tensile stress is equal to the value after cracking, and the other parameters are not varied. According to the transformation of coordinates, the transformation matrix Tθ is obtained in equation (9) and the stiffness matrix Dc′ after the appearance of cracks is shown in equation (10). In these formulas, f1 is the average tensile stress, f2 is the average compressive stress, ε2 is the principal compressive strain, θ is the directional angle of principal strain, ν12 is the average shear stress in the 1-2 coordinate, and γ12 is the average shear strain in the 1-2 coordinate(8)Dc=∂σ∂ε=∂f1∂ε1∂f1∂ε2∂f1∂γ12/2∂f2∂ε1∂f2∂ε2∂f2∂γ12/2∂ν12∂ε1∂ν12∂ε2∂ν12∂γ12/2,(9)Tθ=cos2θsin2θ2sinθcosθsin2θcos2θ−2sinθcosθ−sinθcosθsinθcosθcos2θ−sin2θ,(10)Dc′=T−θDcTθ.The detailed framework of the program is shown in Figure7. As to belonging to nonlinear physical properties, the Newton–Raphson method was used in the iteration calculation process. The iterative convergence criterion was based on force, and the convergence tolerance was set to the default values in the software.Figure 7
Flowchart of the rotation crack modeling using ABAQUS subroutine.In accordance with the MCFT, the cracked concrete was treated as a new type of material. The reinforcement and cracks were smeared within the material so that no crack existed. The MCFT model used in this study required modification in order to cancel crack detection. The constitutional relationship of the MCFT was used as the compressive constitutional relationship of the concrete; the tensile constitutional relationship is represented by equations (11) and (12) in accordance with the modification of coefficient due to the cancellation of crack detection.The tensile stress-strain relationship of concrete is written as(11)fc1=Ecε1,ε1≤εcr,(12)fc1=fcr1+200ε1,ε1>εcr,where Ec is the elastic modulus of concrete, fc1 is the principal tensile stress, fcr is the cracking stress, ε1 is the principal tensile strain, and εcr is the cracking strain.


### 4.3. Modeling of Steel-Concrete Interface

The stud was the connecting element between steel plate and concrete. The relative slippage of the steel plate and concrete should be taken into account when modeling with ABAQUS software. Therefore, a nonlinear spring element was used to simulate the role of studs. The spring element adopted the shear slip formula for the slip curve proposed by Ollgaard et al. [20], as shown in the following formulas:(13)Vu=0.43AsEcfc≤0.7Asfu,V=Vu1−e−nsm,where V is the shear force of the stud, fu is the ultimate tensile strength of the stud, Vu is the ultimate shear capacity of the stud, Ec is the elastic modulus of concrete, fc is the compressive strength of concrete, s is the slippage, mandn are the shear correlation coefficients, and the selected values in this paper are m=0.558,n=1mm−1.


## 4.1. Material Properties of Steel Plate

A shell element, S4R, was used to simulate the steel plate, and a rectangular beam section element, B32, was used to simulate the concrete. Twenty-five integral points were placed in the beam section, and modified compression field theory was applied to each integral point. The constitutional relationship was defined by a bilinear elastic hardening model (bilinear model). The yield strength and elastic modulus were given according to the experimental results given in Section2.1 Young’s modulus of the steel was 194 GPa, and Poisson’s ratio was 0.3.


## 4.2. Material Properties of Concrete Using Fixed Crack Model and Rotation Crack Model

Here, two models were used to simulate the concrete cracking: (a) the rotation crack model using the developed UMAT subprogram of the concrete rotation crack model (denoted as RCM) and (b) a fixed crack model (denoted as FCM), which is presented in the next section.


### 4.2.1. Fixed Crack Model

According to the fixed crack model, the direction of the cracks that appeared after the first was identical to that of the first. In other words, the shear stress at the crack surface increased until shear locking occurred. In practice, crack direction angles vary as the load increases, and the shear stress at the crack surface remains steady or even decreases. However, the fixed crack model cannot simulate the shear softening problem of concrete in practical situations and it overestimates the analytical result of shear capacity. The rotation crack model can demonstrate the variation in angle direction, as presented in the next section. The stiffness matrix can vary at any time under different crack directions, and the shear stress at the crack surface can be better simulated.


### 4.2.2. Rotation Crack Model Using Modified Compression Field Theory

Vecchio and Collins proposed the MCFT [16] based on compression field theory [17, 18]. This theory suggests that when a crack appears in concrete, the principal tensile stress exists on the crack’s surface and the shear force is sustained by both principal tensile and compressive stresses of the concrete. The equilibrium condition, deformation compatibility condition, and constitutional condition (including the compressive and tensile constitution of the concrete) are established according to the mean stress and strain values. Thereby, the shear strength of reinforced concrete can be better represented.The UMAT in ABAQUS was used to define the subroutine of the MCFT for describing the rotation crack model for the concrete. When the state variant and strain increment were input by the main program, the UMAT solved the stress increment according to the strain variant, solved and returned the Jacobian matrix to develop the global stiffness matrix, and then temporarily stored the state variant for the next increment. The key was to solve the Jacobian matrix.Because the coordinates and positive direction were different before and after the appearance of cracks in the concrete, the Jacobian matrix was also different. Whenε1≤εcr, where ε1 is the principal tensile strain, εcr refers to the cracking strain, the uniaxial tensile stress of concrete is equal to the value before cracking, and the shear modulus of concrete is represented by G=f1−f2/2ε1−ε2, as recommended by Zhu et al. [19]. The derived stiffness matrix Dc is shown in equation (8); when ε1>εcr, the uniaxial tensile stress is equal to the value after cracking, and the other parameters are not varied. According to the transformation of coordinates, the transformation matrix Tθ is obtained in equation (9) and the stiffness matrix Dc′ after the appearance of cracks is shown in equation (10). In these formulas, f1 is the average tensile stress, f2 is the average compressive stress, ε2 is the principal compressive strain, θ is the directional angle of principal strain, ν12 is the average shear stress in the 1-2 coordinate, and γ12 is the average shear strain in the 1-2 coordinate(8)Dc=∂σ∂ε=∂f1∂ε1∂f1∂ε2∂f1∂γ12/2∂f2∂ε1∂f2∂ε2∂f2∂γ12/2∂ν12∂ε1∂ν12∂ε2∂ν12∂γ12/2,(9)Tθ=cos2θsin2θ2sinθcosθsin2θcos2θ−2sinθcosθ−sinθcosθsinθcosθcos2θ−sin2θ,(10)Dc′=T−θDcTθ.The detailed framework of the program is shown in Figure7. As to belonging to nonlinear physical properties, the Newton–Raphson method was used in the iteration calculation process. The iterative convergence criterion was based on force, and the convergence tolerance was set to the default values in the software.Figure 7
Flowchart of the rotation crack modeling using ABAQUS subroutine.In accordance with the MCFT, the cracked concrete was treated as a new type of material. The reinforcement and cracks were smeared within the material so that no crack existed. The MCFT model used in this study required modification in order to cancel crack detection. The constitutional relationship of the MCFT was used as the compressive constitutional relationship of the concrete; the tensile constitutional relationship is represented by equations (11) and (12) in accordance with the modification of coefficient due to the cancellation of crack detection.The tensile stress-strain relationship of concrete is written as(11)fc1=Ecε1,ε1≤εcr,(12)fc1=fcr1+200ε1,ε1>εcr,where Ec is the elastic modulus of concrete, fc1 is the principal tensile stress, fcr is the cracking stress, ε1 is the principal tensile strain, and εcr is the cracking strain.


## 4.2.1. Fixed Crack Model

According to the fixed crack model, the direction of the cracks that appeared after the first was identical to that of the first. In other words, the shear stress at the crack surface increased until shear locking occurred. In practice, crack direction angles vary as the load increases, and the shear stress at the crack surface remains steady or even decreases. However, the fixed crack model cannot simulate the shear softening problem of concrete in practical situations and it overestimates the analytical result of shear capacity. The rotation crack model can demonstrate the variation in angle direction, as presented in the next section. The stiffness matrix can vary at any time under different crack directions, and the shear stress at the crack surface can be better simulated.


## 4.2.2. Rotation Crack Model Using Modified Compression Field Theory

Vecchio and Collins proposed the MCFT [16] based on compression field theory [17, 18]. This theory suggests that when a crack appears in concrete, the principal tensile stress exists on the crack’s surface and the shear force is sustained by both principal tensile and compressive stresses of the concrete. The equilibrium condition, deformation compatibility condition, and constitutional condition (including the compressive and tensile constitution of the concrete) are established according to the mean stress and strain values. Thereby, the shear strength of reinforced concrete can be better represented.The UMAT in ABAQUS was used to define the subroutine of the MCFT for describing the rotation crack model for the concrete. When the state variant and strain increment were input by the main program, the UMAT solved the stress increment according to the strain variant, solved and returned the Jacobian matrix to develop the global stiffness matrix, and then temporarily stored the state variant for the next increment. The key was to solve the Jacobian matrix.Because the coordinates and positive direction were different before and after the appearance of cracks in the concrete, the Jacobian matrix was also different. Whenε1≤εcr, where ε1 is the principal tensile strain, εcr refers to the cracking strain, the uniaxial tensile stress of concrete is equal to the value before cracking, and the shear modulus of concrete is represented by G=f1−f2/2ε1−ε2, as recommended by Zhu et al. [19]. The derived stiffness matrix Dc is shown in equation (8); when ε1>εcr, the uniaxial tensile stress is equal to the value after cracking, and the other parameters are not varied. According to the transformation of coordinates, the transformation matrix Tθ is obtained in equation (9) and the stiffness matrix Dc′ after the appearance of cracks is shown in equation (10). In these formulas, f1 is the average tensile stress, f2 is the average compressive stress, ε2 is the principal compressive strain, θ is the directional angle of principal strain, ν12 is the average shear stress in the 1-2 coordinate, and γ12 is the average shear strain in the 1-2 coordinate(8)Dc=∂σ∂ε=∂f1∂ε1∂f1∂ε2∂f1∂γ12/2∂f2∂ε1∂f2∂ε2∂f2∂γ12/2∂ν12∂ε1∂ν12∂ε2∂ν12∂γ12/2,(9)Tθ=cos2θsin2θ2sinθcosθsin2θcos2θ−2sinθcosθ−sinθcosθsinθcosθcos2θ−sin2θ,(10)Dc′=T−θDcTθ.The detailed framework of the program is shown in Figure7. As to belonging to nonlinear physical properties, the Newton–Raphson method was used in the iteration calculation process. The iterative convergence criterion was based on force, and the convergence tolerance was set to the default values in the software.Figure 7
Flowchart of the rotation crack modeling using ABAQUS subroutine.In accordance with the MCFT, the cracked concrete was treated as a new type of material. The reinforcement and cracks were smeared within the material so that no crack existed. The MCFT model used in this study required modification in order to cancel crack detection. The constitutional relationship of the MCFT was used as the compressive constitutional relationship of the concrete; the tensile constitutional relationship is represented by equations (11) and (12) in accordance with the modification of coefficient due to the cancellation of crack detection.The tensile stress-strain relationship of concrete is written as(11)fc1=Ecε1,ε1≤εcr,(12)fc1=fcr1+200ε1,ε1>εcr,where Ec is the elastic modulus of concrete, fc1 is the principal tensile stress, fcr is the cracking stress, ε1 is the principal tensile strain, and εcr is the cracking strain.


## 4.3. Modeling of Steel-Concrete Interface

The stud was the connecting element between steel plate and concrete. The relative slippage of the steel plate and concrete should be taken into account when modeling with ABAQUS software. Therefore, a nonlinear spring element was used to simulate the role of studs. The spring element adopted the shear slip formula for the slip curve proposed by Ollgaard et al. [20], as shown in the following formulas:(13)Vu=0.43AsEcfc≤0.7Asfu,V=Vu1−e−nsm,where V is the shear force of the stud, fu is the ultimate tensile strength of the stud, Vu is the ultimate shear capacity of the stud, Ec is the elastic modulus of concrete, fc is the compressive strength of concrete, s is the slippage, mandn are the shear correlation coefficients, and the selected values in this paper are m=0.558,n=1mm−1.


## 5. Results and Discussion of Numerical Simulation

### 5.1. Calibration of Numerical Simulation Using the Shear Capacity of Experimental Results

The steel plate-concrete composite slabs were modeled, and the boundary condition was established as a simple support at both ends. Loading was controlled through displacement. The finite element model of the composite slab is shown in Figure8.Figure 8
FEM model of composite slab.The results of simulation and the experimental data are listed in Table3, in which Pj is the experimental shear capacity, PR is the shear capacity calculated by the RCM, and PF is the shear capacity calculated by the FCM. As shown in Table 3, in comparison to the results obtained by the FCM, the shear capacity acquired by the RCM better matches that in the experimental results, validating the selection of the element and material properties of the shear model of the steel plate-concrete composite slab. The analytical shear capacity results were larger than the experimental results when the default concrete fixed crack model in ABAQUS was directly applied. The errors were equal to 8.7%, 9.5% 15.3%, and 40.2% for each specimen, with SCS-4 having the largest error. This was due to the use of the default fixed crack model in ABAQUS.Table 3
Comparisons of shear capacity.
Specimens
λ
P
j (kN)
P
R (kN)
P
F (kN)
P
R
/
P
j
P
F
/
P
j
SCS-1
2.63
344.3
336.6
374.4
0.98
1.09
SCS-2
1.88
416.9
408.6
456.6
0.98
1.10
SCS-3
1.88
304.4
299.4
350.8
0.98
1.15
SCS-4
1.88
71.9
67.2
100.8
0.93
1.40


### 5.2. Load-Displacement Relations

Figure9 shows the load-displacement diagram during the numerical calculation and experimental process. It also demonstrated that the analytical results obtained by the RCM fitted the experimental curve better than the results obtained by the FCM, suggesting that the rotation crack model secondarily developed from ABAQUS was more suitable for the analysis of the steel plate-concrete composite slab shear capacity.Figure 9
Comparison of experimental load-displacement curve to that of FEM: (a) SCS-1, (b) SCS-2, (c) SCS-3, and (d) SCS-4.
(a)
(b)
(c)
(d)


## 5.1. Calibration of Numerical Simulation Using the Shear Capacity of Experimental Results

The steel plate-concrete composite slabs were modeled, and the boundary condition was established as a simple support at both ends. Loading was controlled through displacement. The finite element model of the composite slab is shown in Figure8.Figure 8
FEM model of composite slab.The results of simulation and the experimental data are listed in Table3, in which Pj is the experimental shear capacity, PR is the shear capacity calculated by the RCM, and PF is the shear capacity calculated by the FCM. As shown in Table 3, in comparison to the results obtained by the FCM, the shear capacity acquired by the RCM better matches that in the experimental results, validating the selection of the element and material properties of the shear model of the steel plate-concrete composite slab. The analytical shear capacity results were larger than the experimental results when the default concrete fixed crack model in ABAQUS was directly applied. The errors were equal to 8.7%, 9.5% 15.3%, and 40.2% for each specimen, with SCS-4 having the largest error. This was due to the use of the default fixed crack model in ABAQUS.Table 3
Comparisons of shear capacity.
Specimens
λ
P
j (kN)
P
R (kN)
P
F (kN)
P
R
/
P
j
P
F
/
P
j
SCS-1
2.63
344.3
336.6
374.4
0.98
1.09
SCS-2
1.88
416.9
408.6
456.6
0.98
1.10
SCS-3
1.88
304.4
299.4
350.8
0.98
1.15
SCS-4
1.88
71.9
67.2
100.8
0.93
1.40


## 5.2. Load-Displacement Relations

Figure9 shows the load-displacement diagram during the numerical calculation and experimental process. It also demonstrated that the analytical results obtained by the RCM fitted the experimental curve better than the results obtained by the FCM, suggesting that the rotation crack model secondarily developed from ABAQUS was more suitable for the analysis of the steel plate-concrete composite slab shear capacity.Figure 9
Comparison of experimental load-displacement curve to that of FEM: (a) SCS-1, (b) SCS-2, (c) SCS-3, and (d) SCS-4.
(a)
(b)
(c)
(d)


## 6. Formula of Shear Bearing Capacity

From test analysis of Section3, it could be concluded that the shear bearing capacity of the composite was related to several parameters, such as shear span ratio λ, steel plate thickness t, steel yield strength fys, concrete strength fc, concrete thickness b, and section height h0. Exclusion of the material property influence of steel yield strength fys and concrete strength fc (material properties), the parametric analysis of shear capacity of steel plate-concrete composite slabs was carried out using the FEA model on shear span ratio, steel plate thickness, concrete thickness, and shear stud spacing. Then, based on the formula of shear capacity of ordinary concrete beams, a formula of shear bearing capacity of composite slabs was proposed.


### 6.1. Parametric Analysis

#### 6.1.1. Shear Span Ratioλ

The shear span ratioλ=a/h was affected by shear span length a and section height h. Therefore, the shear span ratio could be changed by the length of shear span or section height. In the analysis, the yield strength of steel plate is 235 MPa. The shear stud diameter is 13 mm, and the shear stud spacing is 150 mm. The strength of concrete is 30 MPa, and the concrete thickness is 150 mm. Firstly, the influence of shear span length was considered: the height of the composite is 400 mm and the length of the composite is 1400 mm, 1600 mm, 1800 mm, 2000 mm and 2200 mm, respectively, to gain different shear span ratio. Figure 10(a) showed the effect of shear span length on shear capacity, in which the x-axis a is the length of shear span and the y-axis P is the shear bearing capacity. On the other hand, under the condition that the span remained unchanged, different shear span ratios were obtained by taking the section height of 400 mm, 450 mm, 500 mm, and 550 mm, respectively. Figure 10(b) showed the relationship between the section height and shear capacity. However, considering the stability of the composite slabs, the height of the composite should be limited within 600 mm.Figure 10
Relationship between shear span ratio and shear capacity. (a) Length of shear span. (b) Section height.
(a)
(b)From Figure10(a), with the increase of the shear span length, the shear capacity showed a relatively gentle downward trend, indicating that the shear span had little effect on the shear capacity of the composite. With the increase of section height, the shear capacity showed a more significant upward trend, as shown in Figure 10(b). The comparison between the above two cases showed that the influence of section height on shear span ratio was more obvious.


#### 6.1.2. Steel Plate Thicknesst

Based on the parameters in Section6.1.1, with that the other parameters remained unchanged, the height and length of the composite were given as 400 mm and 1700 mm, respectively. Then, the thickness of steel plate was changed from 6 mm to 14 mm, in order to analyze the effect of steel plate thickness on the shear bearing capacity of steel plate-concrete composite slabs. The results are shown in Figure 11, in which the x-axis t is the thickness of steel plate and the y-axis P is the shear bearing capacity. The thickness t of steel plate was proportional to the shear capacity of composite slabs, and this parameter was one of the important factors affecting the ultimate shear capacity of steel plate-concrete composite slabs, as shown in Figure 11. The shear capacity of steel plate was an integral part of the shear capacity of composite slabs.Figure 11
Relationship between steel plate thickness and shear capacity.


#### 6.1.3. Concrete Thicknessb

As mentioned above, the shear bearing capacity of steel plate-concrete composite slabs was similar to combination of the respective shear resistance of steel plate and concrete, so concrete thicknessb was also an important factor of shear capacity. Based on the parameters of SCS-2 and under the condition of keeping other parameters unchanged, the concrete thickness b was given as 110 mm, 120 mm, 130 mm, 140 mm, and 150 mm, respectively. Then, the relationship between the concrete thickness b and the shear capacity was obtained, as shown in Figure 12. The x-axis b is the concrete thickness, and the y-axis P is the shear capacity. From the graph, the concrete thickness showed a significant proportional relationship with the shear capacity of the composite slabs, and it could be inferred that the concrete thickness was also one of main factors affecting the shear capacity.Figure 12
Relationship between concrete thickness and shear capacity.


#### 6.1.4. Shear Stud Spacingd

In model test, the difference between SCS-2 and SCS-3 was the shear stud spacingd. Therefore, in order to better consider the influence of studs, the relationship between the stud spacing d and shear capacity of the composite was studied. The results are shown in Figure 13, in which the x-axis d is the stud spacing and the y-axis P is the shear capacity of the composite slabs. Based on the model test, the stud spacing was given as 150 mm, 250 mm, 450 mm, and 500 mm, respectively. From the graph, it could be concluded that the shear bearing capacity decreased slowly with the increase of stud spacing, so the stud spacing only had a small effect on the shear capacity of the composite slabs.Figure 13
Relationship between shear stud spacing and shear capacity.In the finite element analysis, considering the slip effect between steel plate and concrete slab, the nonlinear springs were used to simulate the stud. Figure14 is the displacement diagram obtained by finite element simulation. It showed that the slip between steel plate and concrete slab of the three components was small, and the increase of stud spacing had little effect on the slip. It could be seen from the above analysis that the studs not only made the steel plate and concrete slabs work together as a whole but also played an important role in improving the shear bearing capacity of the composite by combing the two materials, but the stud spacing did not affect a lot to the shear bearing capacity of the integrity of component.Figure 14
Deformation figure of composite slab under different stud spacings: (a) 150 mm, (b) 250 mm, and (d) 500 mm.
(a)
(b)
(c)


### 6.2. Formula of Shear Bearing Capacity

#### 6.2.1. Formula Derivation

Through the parameter analysis of steel plate-concrete composite slab, the main factors of shear capacity were steel plate thicknesst, concrete thickness b, and section height h. Moreover, the shear capacity of composite slabs was proportional to steel plate thickness t and concrete thickness b and inversely proportional to section height h, which coincided with the interfering factors of shear capacity of steel plate and concrete slabs. Therefore, based on the shear bearing capacity formula of ordinary reinforced concrete beams, the shear bearing capacity formula of steel plate-concrete composite slabs was expressed as the sum of the bearing capacity of two materials, as shown in the following formula:(15)Vu=mfcbh+nfsyth,where b is the concrete thickness, h is section height, t is the steel plate thickness, and m, n are the combination coefficient of shear capacity of concrete and steel plate to be derived, respectively. The combination coefficients m and n should be determined by regression analysis. Considering that the section heights of steel plate and concrete slab were equal, the formula of shear bearing capacity could be changed into formula (16), in which the section height of component was proportional to the influence of the whole component. So, the effect of section height could not be considered in the regression analysis of combination coefficient.(16)Vuh=mfcb+nfsyt.(1) The Combination Coefficient m. When the parameters of steel plate were unchanged, only one parameter of concrete slab was changed: the concrete thickness b. According to the changes of this parameter, the influence on the shear capacity of composite slabs was analyzed, and then the combination coefficient of concrete was deduced. Six sets of example parameters are shown in Table 4. The relationship between the product of concrete thickness b and concrete strength fc and Vu/h was obtained by mechanical analysis, as shown in Figure 15, and then the combination coefficient could be obtained by further fitting analysis: m=0.1956.Table 4
The parameters of the example.
h (mm)
t (mm)
f
c (MPa)
f
sy(MPa)
b (mm)
V
u (kN)
400
6
20.1
235
150
335.63
140
321.86
130
295.65
120
287.34
110
273.21
100
256.39Figure 15
Fitting curve of concrete combination coefficient.(2) The Combination Coefficient n. When only the parameters of steel plate were changed, the combination coefficient of the steel plate was obtained by calculating some typical examples according to the change of the steel plate thickness t. Six sets of example parameters are shown in Table 5. The relationship between the product of steel plate thickness t and steel yield strength fsy and Vu/h was obtained by mechanical analysis, as shown in Figure 16, and then the combination coefficient could be obtained by further fitting analysis: n=0.1008.Table 5
The parameters of the example.
f
c (MPa)
b (mm)
h (mm)
f
ys (MPa)
t (mm)
V
u (kN)
20.1
150
400
235
6
335.63
8
366.83
10
385.78
12
396.35
14
412.68
16
438.32Figure 16
Fitting curve of steel plate combination coefficient.Based on the fitting curve of the above parameters, the general formula of shear bearing capacity of steel plate-concrete composite slabs proposed in this paper is shown in the following equation:(17)Vu=0.1956fcbh+0.1008fsyth.The limitation of formula (17) includes that the section height is h<600mm and the span is L<2300mm.


#### 6.2.2. Verification of Formula

Considering the influence on practical engineering application, the general formula of shear bearing capacity of steel plate-concrete composite slabs obtained in the preceding section needed to be validated on the basis of test data. Based on the test data of three composite slabs, the results of model test and the results obtained by the shear capacity formula (equation (17)) were compared, as shown in Table 6. It could be seen that the results obtained by the shear capacity formula in this paper agreed with the test data by comparison, which indicated that this shear bearing capacity could provide references to the design and application of steel plate-concrete composite slab.Table 6
Comparison of formula in this paper with test results.
Specimens
b (mm)
t (mm)
P
j (kN)
V
u (kN)
P
R (kN)
V
u
/
P
j
P
R
/
P
j
SCS-1
150
6
344.32
343.61
336.55
1.0
0.98
SCS-2
165
6
416.88
416.18
408.57
1.0
0.98
SCS-3
150
6
304.36
303.74
299.43
0.98
0.98


## 6.1. Parametric Analysis

### 6.1.1. Shear Span Ratioλ

The shear span ratioλ=a/h was affected by shear span length a and section height h. Therefore, the shear span ratio could be changed by the length of shear span or section height. In the analysis, the yield strength of steel plate is 235 MPa. The shear stud diameter is 13 mm, and the shear stud spacing is 150 mm. The strength of concrete is 30 MPa, and the concrete thickness is 150 mm. Firstly, the influence of shear span length was considered: the height of the composite is 400 mm and the length of the composite is 1400 mm, 1600 mm, 1800 mm, 2000 mm and 2200 mm, respectively, to gain different shear span ratio. Figure 10(a) showed the effect of shear span length on shear capacity, in which the x-axis a is the length of shear span and the y-axis P is the shear bearing capacity. On the other hand, under the condition that the span remained unchanged, different shear span ratios were obtained by taking the section height of 400 mm, 450 mm, 500 mm, and 550 mm, respectively. Figure 10(b) showed the relationship between the section height and shear capacity. However, considering the stability of the composite slabs, the height of the composite should be limited within 600 mm.Figure 10
Relationship between shear span ratio and shear capacity. (a) Length of shear span. (b) Section height.
(a)
(b)From Figure10(a), with the increase of the shear span length, the shear capacity showed a relatively gentle downward trend, indicating that the shear span had little effect on the shear capacity of the composite. With the increase of section height, the shear capacity showed a more significant upward trend, as shown in Figure 10(b). The comparison between the above two cases showed that the influence of section height on shear span ratio was more obvious.


### 6.1.2. Steel Plate Thicknesst

Based on the parameters in Section6.1.1, with that the other parameters remained unchanged, the height and length of the composite were given as 400 mm and 1700 mm, respectively. Then, the thickness of steel plate was changed from 6 mm to 14 mm, in order to analyze the effect of steel plate thickness on the shear bearing capacity of steel plate-concrete composite slabs. The results are shown in Figure 11, in which the x-axis t is the thickness of steel plate and the y-axis P is the shear bearing capacity. The thickness t of steel plate was proportional to the shear capacity of composite slabs, and this parameter was one of the important factors affecting the ultimate shear capacity of steel plate-concrete composite slabs, as shown in Figure 11. The shear capacity of steel plate was an integral part of the shear capacity of composite slabs.Figure 11
Relationship between steel plate thickness and shear capacity.


### 6.1.3. Concrete Thicknessb

As mentioned above, the shear bearing capacity of steel plate-concrete composite slabs was similar to combination of the respective shear resistance of steel plate and concrete, so concrete thicknessb was also an important factor of shear capacity. Based on the parameters of SCS-2 and under the condition of keeping other parameters unchanged, the concrete thickness b was given as 110 mm, 120 mm, 130 mm, 140 mm, and 150 mm, respectively. Then, the relationship between the concrete thickness b and the shear capacity was obtained, as shown in Figure 12. The x-axis b is the concrete thickness, and the y-axis P is the shear capacity. From the graph, the concrete thickness showed a significant proportional relationship with the shear capacity of the composite slabs, and it could be inferred that the concrete thickness was also one of main factors affecting the shear capacity.Figure 12
Relationship between concrete thickness and shear capacity.


### 6.1.4. Shear Stud Spacingd

In model test, the difference between SCS-2 and SCS-3 was the shear stud spacingd. Therefore, in order to better consider the influence of studs, the relationship between the stud spacing d and shear capacity of the composite was studied. The results are shown in Figure 13, in which the x-axis d is the stud spacing and the y-axis P is the shear capacity of the composite slabs. Based on the model test, the stud spacing was given as 150 mm, 250 mm, 450 mm, and 500 mm, respectively. From the graph, it could be concluded that the shear bearing capacity decreased slowly with the increase of stud spacing, so the stud spacing only had a small effect on the shear capacity of the composite slabs.Figure 13
Relationship between shear stud spacing and shear capacity.In the finite element analysis, considering the slip effect between steel plate and concrete slab, the nonlinear springs were used to simulate the stud. Figure14 is the displacement diagram obtained by finite element simulation. It showed that the slip between steel plate and concrete slab of the three components was small, and the increase of stud spacing had little effect on the slip. It could be seen from the above analysis that the studs not only made the steel plate and concrete slabs work together as a whole but also played an important role in improving the shear bearing capacity of the composite by combing the two materials, but the stud spacing did not affect a lot to the shear bearing capacity of the integrity of component.Figure 14
Deformation figure of composite slab under different stud spacings: (a) 150 mm, (b) 250 mm, and (d) 500 mm.
(a)
(b)
(c)


## 6.1.1. Shear Span Ratioλ

The shear span ratioλ=a/h was affected by shear span length a and section height h. Therefore, the shear span ratio could be changed by the length of shear span or section height. In the analysis, the yield strength of steel plate is 235 MPa. The shear stud diameter is 13 mm, and the shear stud spacing is 150 mm. The strength of concrete is 30 MPa, and the concrete thickness is 150 mm. Firstly, the influence of shear span length was considered: the height of the composite is 400 mm and the length of the composite is 1400 mm, 1600 mm, 1800 mm, 2000 mm and 2200 mm, respectively, to gain different shear span ratio. Figure 10(a) showed the effect of shear span length on shear capacity, in which the x-axis a is the length of shear span and the y-axis P is the shear bearing capacity. On the other hand, under the condition that the span remained unchanged, different shear span ratios were obtained by taking the section height of 400 mm, 450 mm, 500 mm, and 550 mm, respectively. Figure 10(b) showed the relationship between the section height and shear capacity. However, considering the stability of the composite slabs, the height of the composite should be limited within 600 mm.Figure 10
Relationship between shear span ratio and shear capacity. (a) Length of shear span. (b) Section height.
(a)
(b)From Figure10(a), with the increase of the shear span length, the shear capacity showed a relatively gentle downward trend, indicating that the shear span had little effect on the shear capacity of the composite. With the increase of section height, the shear capacity showed a more significant upward trend, as shown in Figure 10(b). The comparison between the above two cases showed that the influence of section height on shear span ratio was more obvious.


## 6.1.2. Steel Plate Thicknesst

Based on the parameters in Section6.1.1, with that the other parameters remained unchanged, the height and length of the composite were given as 400 mm and 1700 mm, respectively. Then, the thickness of steel plate was changed from 6 mm to 14 mm, in order to analyze the effect of steel plate thickness on the shear bearing capacity of steel plate-concrete composite slabs. The results are shown in Figure 11, in which the x-axis t is the thickness of steel plate and the y-axis P is the shear bearing capacity. The thickness t of steel plate was proportional to the shear capacity of composite slabs, and this parameter was one of the important factors affecting the ultimate shear capacity of steel plate-concrete composite slabs, as shown in Figure 11. The shear capacity of steel plate was an integral part of the shear capacity of composite slabs.Figure 11
Relationship between steel plate thickness and shear capacity.


## 6.1.3. Concrete Thicknessb

As mentioned above, the shear bearing capacity of steel plate-concrete composite slabs was similar to combination of the respective shear resistance of steel plate and concrete, so concrete thicknessb was also an important factor of shear capacity. Based on the parameters of SCS-2 and under the condition of keeping other parameters unchanged, the concrete thickness b was given as 110 mm, 120 mm, 130 mm, 140 mm, and 150 mm, respectively. Then, the relationship between the concrete thickness b and the shear capacity was obtained, as shown in Figure 12. The x-axis b is the concrete thickness, and the y-axis P is the shear capacity. From the graph, the concrete thickness showed a significant proportional relationship with the shear capacity of the composite slabs, and it could be inferred that the concrete thickness was also one of main factors affecting the shear capacity.Figure 12
Relationship between concrete thickness and shear capacity.


## 6.1.4. Shear Stud Spacingd

In model test, the difference between SCS-2 and SCS-3 was the shear stud spacingd. Therefore, in order to better consider the influence of studs, the relationship between the stud spacing d and shear capacity of the composite was studied. The results are shown in Figure 13, in which the x-axis d is the stud spacing and the y-axis P is the shear capacity of the composite slabs. Based on the model test, the stud spacing was given as 150 mm, 250 mm, 450 mm, and 500 mm, respectively. From the graph, it could be concluded that the shear bearing capacity decreased slowly with the increase of stud spacing, so the stud spacing only had a small effect on the shear capacity of the composite slabs.Figure 13
Relationship between shear stud spacing and shear capacity.In the finite element analysis, considering the slip effect between steel plate and concrete slab, the nonlinear springs were used to simulate the stud. Figure14 is the displacement diagram obtained by finite element simulation. It showed that the slip between steel plate and concrete slab of the three components was small, and the increase of stud spacing had little effect on the slip. It could be seen from the above analysis that the studs not only made the steel plate and concrete slabs work together as a whole but also played an important role in improving the shear bearing capacity of the composite by combing the two materials, but the stud spacing did not affect a lot to the shear bearing capacity of the integrity of component.Figure 14
Deformation figure of composite slab under different stud spacings: (a) 150 mm, (b) 250 mm, and (d) 500 mm.
(a)
(b)
(c)


## 6.2. Formula of Shear Bearing Capacity

### 6.2.1. Formula Derivation

Through the parameter analysis of steel plate-concrete composite slab, the main factors of shear capacity were steel plate thicknesst, concrete thickness b, and section height h. Moreover, the shear capacity of composite slabs was proportional to steel plate thickness t and concrete thickness b and inversely proportional to section height h, which coincided with the interfering factors of shear capacity of steel plate and concrete slabs. Therefore, based on the shear bearing capacity formula of ordinary reinforced concrete beams, the shear bearing capacity formula of steel plate-concrete composite slabs was expressed as the sum of the bearing capacity of two materials, as shown in the following formula:(15)Vu=mfcbh+nfsyth,where b is the concrete thickness, h is section height, t is the steel plate thickness, and m, n are the combination coefficient of shear capacity of concrete and steel plate to be derived, respectively. The combination coefficients m and n should be determined by regression analysis. Considering that the section heights of steel plate and concrete slab were equal, the formula of shear bearing capacity could be changed into formula (16), in which the section height of component was proportional to the influence of the whole component. So, the effect of section height could not be considered in the regression analysis of combination coefficient.(16)Vuh=mfcb+nfsyt.(1) The Combination Coefficient m. When the parameters of steel plate were unchanged, only one parameter of concrete slab was changed: the concrete thickness b. According to the changes of this parameter, the influence on the shear capacity of composite slabs was analyzed, and then the combination coefficient of concrete was deduced. Six sets of example parameters are shown in Table 4. The relationship between the product of concrete thickness b and concrete strength fc and Vu/h was obtained by mechanical analysis, as shown in Figure 15, and then the combination coefficient could be obtained by further fitting analysis: m=0.1956.Table 4
The parameters of the example.
h (mm)
t (mm)
f
c (MPa)
f
sy(MPa)
b (mm)
V
u (kN)
400
6
20.1
235
150
335.63
140
321.86
130
295.65
120
287.34
110
273.21
100
256.39Figure 15
Fitting curve of concrete combination coefficient.(2) The Combination Coefficient n. When only the parameters of steel plate were changed, the combination coefficient of the steel plate was obtained by calculating some typical examples according to the change of the steel plate thickness t. Six sets of example parameters are shown in Table 5. The relationship between the product of steel plate thickness t and steel yield strength fsy and Vu/h was obtained by mechanical analysis, as shown in Figure 16, and then the combination coefficient could be obtained by further fitting analysis: n=0.1008.Table 5
The parameters of the example.
f
c (MPa)
b (mm)
h (mm)
f
ys (MPa)
t (mm)
V
u (kN)
20.1
150
400
235
6
335.63
8
366.83
10
385.78
12
396.35
14
412.68
16
438.32Figure 16
Fitting curve of steel plate combination coefficient.Based on the fitting curve of the above parameters, the general formula of shear bearing capacity of steel plate-concrete composite slabs proposed in this paper is shown in the following equation:(17)Vu=0.1956fcbh+0.1008fsyth.The limitation of formula (17) includes that the section height is h<600mm and the span is L<2300mm.


### 6.2.2. Verification of Formula

Considering the influence on practical engineering application, the general formula of shear bearing capacity of steel plate-concrete composite slabs obtained in the preceding section needed to be validated on the basis of test data. Based on the test data of three composite slabs, the results of model test and the results obtained by the shear capacity formula (equation (17)) were compared, as shown in Table 6. It could be seen that the results obtained by the shear capacity formula in this paper agreed with the test data by comparison, which indicated that this shear bearing capacity could provide references to the design and application of steel plate-concrete composite slab.Table 6
Comparison of formula in this paper with test results.
Specimens
b (mm)
t (mm)
P
j (kN)
V
u (kN)
P
R (kN)
V
u
/
P
j
P
R
/
P
j
SCS-1
150
6
344.32
343.61
336.55
1.0
0.98
SCS-2
165
6
416.88
416.18
408.57
1.0
0.98
SCS-3
150
6
304.36
303.74
299.43
0.98
0.98


## 6.2.1. Formula Derivation

Through the parameter analysis of steel plate-concrete composite slab, the main factors of shear capacity were steel plate thicknesst, concrete thickness b, and section height h. Moreover, the shear capacity of composite slabs was proportional to steel plate thickness t and concrete thickness b and inversely proportional to section height h, which coincided with the interfering factors of shear capacity of steel plate and concrete slabs. Therefore, based on the shear bearing capacity formula of ordinary reinforced concrete beams, the shear bearing capacity formula of steel plate-concrete composite slabs was expressed as the sum of the bearing capacity of two materials, as shown in the following formula:(15)Vu=mfcbh+nfsyth,where b is the concrete thickness, h is section height, t is the steel plate thickness, and m, n are the combination coefficient of shear capacity of concrete and steel plate to be derived, respectively. The combination coefficients m and n should be determined by regression analysis. Considering that the section heights of steel plate and concrete slab were equal, the formula of shear bearing capacity could be changed into formula (16), in which the section height of component was proportional to the influence of the whole component. So, the effect of section height could not be considered in the regression analysis of combination coefficient.(16)Vuh=mfcb+nfsyt.(1) The Combination Coefficient m. When the parameters of steel plate were unchanged, only one parameter of concrete slab was changed: the concrete thickness b. According to the changes of this parameter, the influence on the shear capacity of composite slabs was analyzed, and then the combination coefficient of concrete was deduced. Six sets of example parameters are shown in Table 4. The relationship between the product of concrete thickness b and concrete strength fc and Vu/h was obtained by mechanical analysis, as shown in Figure 15, and then the combination coefficient could be obtained by further fitting analysis: m=0.1956.Table 4
The parameters of the example.
h (mm)
t (mm)
f
c (MPa)
f
sy(MPa)
b (mm)
V
u (kN)
400
6
20.1
235
150
335.63
140
321.86
130
295.65
120
287.34
110
273.21
100
256.39Figure 15
Fitting curve of concrete combination coefficient.(2) The Combination Coefficient n. When only the parameters of steel plate were changed, the combination coefficient of the steel plate was obtained by calculating some typical examples according to the change of the steel plate thickness t. Six sets of example parameters are shown in Table 5. The relationship between the product of steel plate thickness t and steel yield strength fsy and Vu/h was obtained by mechanical analysis, as shown in Figure 16, and then the combination coefficient could be obtained by further fitting analysis: n=0.1008.Table 5
The parameters of the example.
f
c (MPa)
b (mm)
h (mm)
f
ys (MPa)
t (mm)
V
u (kN)
20.1
150
400
235
6
335.63
8
366.83
10
385.78
12
396.35
14
412.68
16
438.32Figure 16
Fitting curve of steel plate combination coefficient.Based on the fitting curve of the above parameters, the general formula of shear bearing capacity of steel plate-concrete composite slabs proposed in this paper is shown in the following equation:(17)Vu=0.1956fcbh+0.1008fsyth.The limitation of formula (17) includes that the section height is h<600mm and the span is L<2300mm.


## 6.2.2. Verification of Formula

Considering the influence on practical engineering application, the general formula of shear bearing capacity of steel plate-concrete composite slabs obtained in the preceding section needed to be validated on the basis of test data. Based on the test data of three composite slabs, the results of model test and the results obtained by the shear capacity formula (equation (17)) were compared, as shown in Table 6. It could be seen that the results obtained by the shear capacity formula in this paper agreed with the test data by comparison, which indicated that this shear bearing capacity could provide references to the design and application of steel plate-concrete composite slab.Table 6
Comparison of formula in this paper with test results.
Specimens
b (mm)
t (mm)
P
j (kN)
V
u (kN)
P
R (kN)
V
u
/
P
j
P
R
/
P
j
SCS-1
150
6
344.32
343.61
336.55
1.0
0.98
SCS-2
165
6
416.88
416.18
408.57
1.0
0.98
SCS-3
150
6
304.36
303.74
299.43
0.98
0.98


## 7. Conclusions

Experimental studies and numerical analyses were used to investigate the shear load-carrying capacity, deformation, and crack development in steel plate-concrete composite slab, as well as the effects of the shear span ratio and shear stud spacing on the shear performance and the contribution of the steel plate and the concrete to the shear performance. A rotational crack model based on the modified compression field theory developed by UMAT (user material) of ABAQUS software was proposed and applied to the nonlinear analysis, and a numerical simulated model for the steel-concrete composite slab was built for shear analysis. The following main conclusions can be obtained:(a)
The failure modes of the steel plate-concrete composite slabs in shear included bending failure and shear-compression failure. Bending failure was primarily witnessed in the shear span ratio of equal or greater than 2.63, whereas shear-compression failure was the main mode of shear failure for the composite slabs.(b)
The shear force sustained by the concrete ranged from approximately 37% to 47%, and the steel plate contributed more than 50% of the shear capacity. As shown by comparison of SCS-1 and SCS-2, the ultimate shear capacity decreased by 21% as the shear span ratio increased from 1.88 to 2.63. Furthermore, as shown by comparison of SCS-2 and SCS-3, the ultimate shear capacity decreased by 27% as the shear stud spacing increased from 150 mm to 250 mm. The actual shear force sustained by the concrete in the composite slab was 1.27 to 2.22 times greater than that of the calculated value through the Chinese Design Code for Concrete Structures (GB 50010-2010). Moreover, the load-carrying capacity of the concrete with smaller shear stud spacing was 1.75 times higher than that of the concrete with 250 mm shear stud spacing, but the shear stud spacing did not significantly affect the failure mode of the specimen.(c)
Bending failure was witnessed in the plain concrete slab specimen. After the combination of the steel plate and the resultant formation of the composite slab, the mode of failure changed to shear failure. The shear force sustained by the steel plate in the composite slab comprised more than 50% of the ultimate shear capacity, and the shear force sustained by the concrete doubled more than the calculated shear value according to the Chinese standard, suggesting that the presence of the steel plate significantly improved the shear capacity of the member.(d)
The modified compression field theory was used for the secondary development and establishment of the numerical model for shear analysis. The ultimate shear capacity of the composite slab acquired through the numerical analysis corresponded with the experimental values and surpassed the performance of the traditional fixed crack model. Thus, the proposed finite element numerical model for shear analysis would be capable of simulating the practical shear softening of concrete.(e)
The shear capacity of steel plate-concrete composite slabs was mainly affected by three factors: steel plate thicknesst, concrete thickness b, and section height h through parameter analysis. Based on shear bearing capacity formula of ordinary reinforced concrete beams, a general formula of shear bearing capacity of steel plate-concrete composite slabs was obtained. The results of the formula agreed with the test result, which could provide references to the design and application of steel plate-concrete composite slab.



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*Source: 4726538-2019-07-10.xml*