1. Introduction

2. Experiment

2.2. Experiment Results

Table 2 presents the results of material tests conducted following ASTM standards[13]. For A36 4t, the yield strength was measured as 361.25MPa, and the tensile strength was 454.50Mpa. For A36 6t, the yield strength was measured as 356.75Mpa, and the tensile strength was 430.79Mpa. As for the material used for the flange (A572), the yield strength was 383.21Mpa, and the tensile strength was 554.82Mpa. The average compressive strength of the concrete was measured as 21.57Mpa.

Figure 4 depicts the experimental results of the composite beam. The initial slopes were similar for all cases, and the yield strength and tensile strength exhibited slight variations depending on the composite ratio. The specimen with the narrowest spacing of the shear connectors showed the highest yield strength and ultimate strength. However, there was little difference between the specimens with shear connector spacings of 600mm and 1000mm. This can be attributed to the non-uniform strength distribution within the concrete.

Table 3 summarizes the experimental results. The yield strength and ductility were evaluated following the method suggested by AC495[14]. The yield strengths of the specimens were measured in the order of shear connector spacing as 1165.72kN, 1135.51kN, and 1121.33kN, respectively. The maximum strengths were measured as 1382.09kN, 1353.74kN, and 1346.95kN, respectively. The ductility values were 3.67, 3.69, and 3.51, respectively. Ultimately, it was observed that narrower shear connector spacing resulted in superior yield strength, maximum strength, and ductility.

Figure 4 illustrates the failure modes of the specimens. All specimens experienced the collapse of the upper slab, resulting in a decrease in the strength of the specimens. No slip of the lateral concrete was observed.

The theoretical calculation of the composite beam is determined based on the stress-strain distribution as shown in Figure 5. Using this stress-strain distribution, the compressive force $C$ acting on the concrete slab of the test specimen is determined as the minimum value among equations (1) to (3). Equation (3) involves the shear strength of the shear connector, denoted as $Q_n$, and in this study, an angled shear connector was used. Therefore, equation (4) for the shear strength of a channel-shaped shear connector was utilized to calculate $Q_n$[9].

$$C=A_s{F}_y$$

(1)

$$C=0.85f_{ck}{A}_c$$

(2)

$$C=\sum Q_n$$

(3)

$$Q_n=0.3\left(t_f+0.5t_w\right)L_a\sqrt{f_{ck}{E}_c}$$

(4)

Here, $A_s$ is the total cross-sectional area of steel section, $F_y$ is the yield strength of steel, $f_{ck}$ is the compressive strength of concrete, $A_c$ is the total cross-sectional area of concrete, $t_f$ is the flange thickness of the angle, $t_w$ is the web thickness of the angle, and $E_c$ is the elasticity modulus of the concrete.

Equation (5) calculates the design flexural strength in the positive moment region, taking into account the composite effect with the slab. After the manifestation of the composite action, it is assumed that no lateral-torsional buckling occurs due to the restraining effect.

$$M_n=C\left(d_1+d_2\right)+P_y\left(d_3-d_2\right)$$

(5)

Here, $d_1$ is the distance from the center of compression force $C$ in the concrete to the top of the steel member, $d_2$ is the distance from the center of compression force in the steel member to the top of the steel member(If there is no compression force on the steel member, $d_2$ is equal to 0), $P_y$ is the tensile strength of the steel member, and $d_3$ is the Distance from the center of action of $P_y$ on the steel member to the top of the steel member.

There is no available equation for the angle-shaped shear connector used in this study. Therefore, assuming a channel-shaped shear connector and calculating based on the existing equation for channel sections, the strength can be estimated to be approximately 802.84 kN, as shown in Table 4. Consequently, all specimens exhibit behavior corresponding to a composite ratio of 100%, resulting in a final strength of 1182.14 kN. However, since a reduction in strength was observed in the experimental results, it is desired to infer the strength of the angle-shaped shear connector through computational analysis.

Table 4. Theoretical Analysis Results of Specimens.

Table 4. Theoretical Analysis Results of Specimens.

No.	Specimen	$P_{max}$ (kN)	${\mathrm{P}}_{Theory}$ (kN)	$P_{max}/P_{Theory}$
1	CB-21	1382.09	1182.14	1.17
2	CB-13	1353.74		1.15
3	CB-9	1346.95		1.14

3. Finite Element Analysis

3.2. Concrete Analysis Model

The stress-strain behavior of concrete, encompassing both tension and compression, was represented using the Concrete Damaged Plasticity feature, which allows for the depiction of stress reduction after reaching the maximum stress. The input values used for this feature are presented in Table 4[17,18].

Table 4. Material properties of concrete.

Table 4. Material properties of concrete.

Concrete Compressive Behavior		Concrete Compression Damage
Yield Stress (MPa)	Inelastic Strain	Damage Parameter	Inelastic Strain
9.59	0	0	0
10.50	4.97E-05	0	4.97E-05
13.86	2.50E-04	0	2.50E-04
16.74	4.50E-04	0	4.50E-04
19.14	6.50E-04	0	6.50E-04
21.06	8.50E-04	0	8.50E-04
22.50	1.05E-03	0	1.05E-03
23.46	1.25E-03	0	1.25E-03
23.94	1.45E-03	0	1.45E-03
24.00	1.55E-03	0	1.55E-03
20.40	3.35E-03	0.2	3.35E-03
Concrete Tensile Behavior		Concrete Tension Damage
Yield Stress (MPa)	Cracking strain	Damage Parameter	Cracking strain
2.4	0	0	0
0.05	9.38E-04	0.97916667	9.38E-04

The Damage Parameter in concrete represents the coefficient that converts the elastic modulus of concrete when there is a decrease in strength after the elastic range. The calculation was carried out based on the abbreviated stress-strain behavior recommended by ABAQUS. The elastic modulus of concrete($E_c$) was calculated using the equation (6) provided by ACI 318, and a Poisson's ratio of 0.2 was used for concrete[19].

$$E_c=4700\sqrt{f_{ck}}$$

(6)

The stress-strain relationship of concrete was modeled using the formulation proposed by Hognestad, as shown in Figure 7. Although various concrete models accounting for strength degradation have been presented, the Hognestad model was selected for its computational speed and convergence characteristics[20,21]. The equations for the Hognestad model are given as follows: (7) and (8)[21].

$$f_c=f_{ck}\left[\frac{2{\epsilon }_c}{{\epsilon }_0}-{\left(\frac{{\epsilon }_c}{{\epsilon }_0}\right)}^2\right]\mathrm{for}{\epsilon }_0\le {\epsilon }_c\le 0$$

(7)

$$f_c=f_{ck}\left[1-0.15\frac{{\epsilon }_c-{\epsilon }_0}{{\epsilon }_{cu}-{\epsilon }_0}\right]\mathrm{for}{\epsilon }_{cu}\le {\epsilon }_c\le {\epsilon }_0$$

(8)

Here, $f_c$ is the stress in concrete, ${\epsilon }_c$ is the strain in concrete, $f_{ck}$ is the compressive strength of concrete, ${\epsilon }_{cu}$ is the ultimate compressive strain of concrete, and ${\epsilon }_0$ is the strain at maximum stress $f_{ck}$. In this analysis, ${\epsilon }_0$ is chosen as 0.002.

In the analysis of concrete beams, tension behavior of concrete has minimal influence. However, ABAQUS requires a tension softening curve for concrete. Therefore, a linear tension behavior was assumed for concrete. The tensile strength of concrete$(f_t)$ was taken as $f_t=0.1f_{ck}$, and the ultimate tensile strain$({\epsilon }_{tu})$ was selected as ${\epsilon }_{tu}=10f_t/E_c$.

In addition, the characteristic and damage variables for concrete compression and tension were generally set to the default values provided by ABAQUS. The dilation angle, which defines the post-peak behavior, was input as 38. The eccentricity, which is a measure of the ratio between compressive and tensile strengths of concrete, was set to 0.1.

The ratio of biaxial compression strength to uniaxial compression strength (ratio) can be calculated based on experimental results. However, in this study, the default value of 1.16 provided by ABAQUS was used. The parameter controls the yield surface of the concrete plasticity model and is known to have a value distribution based on previous research. In this study, the default value of 2/3 provided by ABAQUS was used[15].

The viscosity parameter can be adjusted to improve the convergence of the analysis by modifying the viscosity. However, in this analysis, the convergence was not significantly affected, so a value of 0 was used.

3.4. Verification and Parameter Analysis of the Analysis Model

Table 5 summarizes the analysis model, where the variables are represented by the number of shear connectors. The initial analysis and verification were conducted on the existing specimens CB-21, 13, and 9. The validated analysis method was then used to analyze the models with varying numbers of shear connectors. Additionally, in order to evaluate the strength in the absence of shear connectors, the analysis was also performed on models with zero shear connectors.

Table 5. Summary of variable analysis.

Table 5. Summary of variable analysis.

No.	Specimen	Spacing between stud (mm)	Number of studs (EA)
1	CB-21	400	21
2	CB-13	600	13
3	CB-9	1000	9
4	CB-7	1300	7
5	CB-5	1950	5
6	CB-3	3900	3
7	CB-0	-	0

Figure 8 illustrates the results of the analysis for the existing test specimens. The analysis was terminated at the point where overall concrete crushing occurred in the girder section. In the actual experimental results, due to the non-uniformity of concrete strength, there was almost no significant difference in strength according to the spacing of shear connectors. However, the computational analysis represented a uniform concrete strength, resulting in a slight decrease in strength as the spacing of shear connectors increased. The difference in maximum strength and ductility between the experiment and analysis is believed to be due to the non-uniformity of concrete. Considering that the initial stiffness and the actual failure mode are similar between the experiment and analysis, the analysis model is considered reliable. Figure 9 compares the failure modes of the experiment and analysis, where Figure 10(b) represents the degree of tensile damage in concrete. A value close to 1 indicates the occurrence of tensile failure in concrete. In the actual test specimen, tensile cracks occurred in the lower part, and the computational analysis is deemed to be similar to the actual experiment.

Figure 10 shows the load-displacement curves obtained from the variable analysis. It can be observed that as the number of shear connectors decreases, the maximum strength decreases. Up to 13 shear connectors, the behavior was similar to a composite action with a composite ratio of 100%. In the analysis results, no shear connector failure occurred, and the analysis was terminated based on concrete crushing. From 9 shear connectors or less, high stresses were observed in the shear connector region. Assuming a tensile strength of the weld material as 430 MPa, the analysis was terminated if the stress in the shear connector region exceeded this value.

In the actual experiments, even if concrete crushing occurred, the behavior of partial damage led to the ductile behavior of the composite beam. However, in the analysis, the strength reduction due to concrete crushing was not considered, resulting in no decrease in strength. Furthermore, in this study, the strength of the angle-type shear connectors was analyzed, and an analysis method based on this was proposed. Therefore, it was determined that the analysis method is reliable.

Table 6 compares the maximum strengths obtained from the experimental and analytical results. For the CB-21 specimen, the experimental strength was 1382.09 kN, and the analytical strength was 1359.28 kN. For the CB-13 specimen, the experimental strength was 1353.74 kN, and the analytical strength was 1335.49 kN. Lastly, for the CB-9 specimen, the experimental strength was 1346.95 kN, and the analytical strength was 1287.89 kN. The ratios between the experimental and analytical strengths were very close, with values of 1.02, 1.01, and 1.05, respectively.

Table 6. Results of analysis for existing specimens.

Table 6. Results of analysis for existing specimens.

No.	Specimen	$P_{max}$ (kN)	$P_{FEA}$ (kN)	$P_{max}/P_{FEA}$
1	CB-21	1382.09	1359.28	1.02
2	CB-13	1353.74	1335.49	1.01
3	CB-9	1346.95	1287.89	1.05

Based on the previous finite element analysis results, it was observed that the capacity of the composite beam varies with the number of shear connectors. This allows us to infer the strength of the angle-type shear connectors used in this study. In order for the composite beam to achieve its maximum composite action, it should be able to transmit all the compressive forces in the concrete, requiring a strength of approximately 4080 kN. Based on the experimental and analytical results, it can be concluded that up to 13 shear connectors exhibited complete composite behavior. Therefore, the individual strength of each shear connector can be calculated as 313.85 kN. Using this information, the resistance of the composite beam can be calculated using the equation provided earlier.

Table 7 compares the finite element analysis results with the inferred values of the shear connectors presented earlier. The ratios between the analysis results and the theoretical calculations range from 1.13 to 1.20, showing a consistent distribution. The higher strength observed in the analysis results can be attributed to a conservative evaluation of the maximum strength during the analysis. When the number of shear connectors is 0, the flexural strength of the steel and the bending strength of the concrete were calculated separately and combined, resulting in a ratio of 1.37. The larger deviation compared to other theoretical values can be attributed to the presence of minor composite behavior due to the friction between the concrete and the steel even in the absence of shear connectors.

Table 7. Results of variable analysis.

Table 7. Results of variable analysis.

No.	Specimen	$P_{FEA}$ (kN)	$P_{Thoery}$ (kN)	$P_{FEA}/P_{Theory}$
1	CB-21	1359.28	1182.14	1.15
2	CB-13	1335.49	1176.82	1.13
3	CB-9	1287.89	1100.12	1.17
4	CB-7	1231.49	1021.73	1.20
5	CB-5	1133.44	947.36	1.19
6	CB-3	1005.17	831.57	1.20
7	CB-0	952.81	706.25	1.37

Based on the previous experimental results, numerical analysis, and finite element analysis, the behavior of the composite beam using angle-shaped shear connectors was evaluated. The comparison between the experimental results and numerical analysis provided confidence in the reliability of the numerical analysis, allowing for the inference of the strength of the angle-shaped shear connectors. This inference enabled theoretical calculations, and it is deemed that the strength of the composite beam can be calculated based on this, taking safety into consideration during the design process.

4. Conclusions

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