The concrete slab is modeled by using a three-dimensional (3D) solid element SOLID65, which has the effects of tensile cracking and compression crushing. This type of element is defined by eight nodes and each node has 3°of freedom. The steel beam is simulated by using a SHELL43 element, each node of which has 6°of freedom of translation and rotation. This element has the characteristics of plasticity, creep, stress rigidness, large deformation, large strain, and element of life and death. The steel block at the loading position is developed by the SOLID45 element, with the characteristics of plasticity, creep, expansion, stress hardening, large deformation, large strain, and initial stress input. The number of nodes and degrees of freedom in SOLID45 are identical to those in SOLID65. The longitudinal reinforcing bars are simulated by a 3D truss element LINK8, each node of which has 3°of freedom. This type of element can bear axial tension and compression but not bending moment, with the characteristics of plasticity, creep, stress rigidification, and large deformation. A non-linear spring element COMBINE39 is used to simulate the longitudinal shear behavior of the stud connectors. The element has unidirectional characteristics with a nonlinear generalized force-deformation curve, and each node has 3°of freedom. At present, the element types used to simulate CFRP laminates include SOLID46 with layered structure characteristics and SHELL41 with membrane characteristics. In addition to the characteristics mentioned previously, SOLID46 allows more than 250 layers of different materials, with stress hardening and large deformation properties. SHELL41 is defined by four nodes, each of which has 3°of freedom. This type of element only has the plane inner membrane stiffness but no out-of-plane bending stiffness. It has the characteristics of stress rigidness, large deformation, non-linearity, and element of life and death. According to the contrast, the mechanical properties of SHELL41 are more consistent with the loading state of CFRP laminates in structural reinforcement. Thus, this element is used to simulate CFRP laminates in the modeling process of this study.

When the load acting on the composite beam is in the normal service stage, the slip between the steel and concrete is negligible compared with the slip between the steel beam and concrete slab (Nie et al., 2004). Therefore, it can be assumed that a good bond relationship exists between reinforcement and the concrete, and the coupling of node degrees of freedom is adopted in modeling. When CFRP laminates are introduced to strengthen the concrete slab of composite beams, high strength epoxy resin is usually used as a reliable adhesion agent, which can better realize the bond between the CFRP laminates and the concrete slab. Thus, it is also assumed that CFRP laminates and the concrete slab are perfectly bonded during the establishment process of the FEA model. The existing literature shows that the phenomenon of lifting will be produced between the steel beam and the concrete slab under the action of the load, and a certain tension will be generated in the studs. However, compared with the longitudinal slip, this phenomenon has little effect on the overall performance of composite beams. Therefore, the lateral and vertical degrees of freedom of the adjacent nodes at the interface between the top flange of the steel beam and the concrete slab are coupled to achieve the purpose of ignoring the lifting effect (Fan, 2003). The final FEA model is shown in Figure 3.

Based on the static test performed previously, an FEA model is established and verified in this section. It is worth explaining that the load-deflection curve of a structure generally includes a rising section and a descending section, and the latter is related to the ductility of the structure. However, the maximum deformation in the FEA is difficult to realize because of the low convergence and accuracy in the simulation calculation. Therefore, this study focuses on the failure state of the component corresponding to the peak load, and does not consider the maximum deformation of the descending section. The comparison result of calculated and tested values of the load-deflection curve is shown in Figure 7. The two calculation curves by the FEA model take into account two working conditions with slip and without slip effects between the concrete and the steel beam. When interface slip is considered, a spring element COMBINE39 is used to simulate the shear performance of the studs. When interface slip effect is not considered, the freedom degrees of nodes in the contact interface between the concrete and the top flange plate are coupled. As can be seen from the figure, the influence of interface slip effects can be ignored before the service load (50% F _u), because the finite element calculation results of the two cases coincide well. When the structure enters the elastic-plastic stage, the slip effect has a certain influence on the mechanical properties of the structure, mainly in the smaller flexural stiffness with considering the slip effect than that without considering. But the slip has little influence on the ultimate bearing capacity of composite beams. In addition, the bending stiffness obtained by the FEA model is higher than the experimental value, especially in the nonlinear stage. The assumption of a perfect bond between reinforcement and the concrete in the modeling process and the neglect of the weakening effect of shear connectors in concrete slabs are the main reasons for the difference in the results. But from the overall trend, the load-deflection curve calculated by the finite element is in good agreement with the experimental data, and the calculated results considering the interface slip effect are closer to the experimental values. The influence of the slip effect is considered in the FEA of the following contents.

The comparison of calculated and tested values of the test beam at the characteristic points are shown in Table 2, including the cracking state of the concrete slab, the normal limit state of the beam when the deflection reached the ultimate value (i.e., L/600), and the states of the compressive yield of the bottom profile, the tensile yield of longitudinal reinforcing bars, and the limit state of structure bearing capacity. L/600, f _sy, f _ry, and f _u in Table 2 represent deflections at the mid-span of each aforementioned stage, respectively. It can be seen from the table that the model calculation results of load or deflection at each characteristic point are quite consistent with the test values. Especially for the load, the error between the calculated and tested values is basically within 5%. It shows that the FEA model in this study can obtain an accurate inelastic mechanical behavior of a composite structure under negative moments.

In the FEA model, the strain of the bottom profile at the loading position is affected by the cushion block. Thus, the strain curves at the bottom of the web and the middle of the top steel profile are selected and compared with the experimental values. For the strain curve of the reinforcing bar, the main tension bar closest to the web of the steel beam at mid-span is selected as the research object. The comparison results of load-strain calculation curves and test values of the steel beam and the reinforcing bar are shown in Figure 8. It can be seen from the figure that the calculated strain of the steel beam is in good agreement with the experimental values. Because of the influence of concrete cracking, there is some deviation between the calculated and the experimental results after the structure enters the yield stage. But in general, the FEA model accurately simulates the loading process of the inverted composite beams. The results show that the reinforcing bars and steel beams in the negative moment region can work well together and give full play to their plastic deformation abilities.

Figure 9 shows the Mises equivalent stress cloud at the yield load of the bottom steel profile (701.8 kN). As can be seen from the figure, the maximum compressive stress of the steel beam occurs at a position about 20 cm away from the loading point, due to the existence of vertical stiffeners. At this moment, the maximum compressive strain reaches about 2,000 με, and then the steel plate enters the yield stage during the subsequent loading process. This phenomenon is consistent with the observation of the test.

As can be seen from Table 2, the finite element calculation result of the cracking load of concrete slabs is basically consistent with the test value, accounting for about 6.5% of the ultimate load. After reaching the cracking load of concrete, the ANSYS program will record the cracking situation of SOLID65 in each loading step and represent the cracking and fracturing behavior with circles and octagons, respectively. Figure 10 shows the crack propagation of a half-span concrete slab under different load grades. As can be seen from the figure, the initial cracks occur near the mid-span of the concrete slab, and then gradually propagate to the beam end with the increase of load grade. When the load reaches about 40% F _u, all transverse cracks appear basically, and several inclined cracks occurred near the support position which is consistent with the phenomenon observed in the test process.

To analyze the influence of the CFRP layout width on the mechanical properties of composite beams in negative moment regions, the layout width of CFRP laminates in the FEA model is equivalent to 100, 200, 300, 400, 500, and 600 mm respectively, under the premise of keeping the structural size of composite beams unchanged, as shown in Figure 11. Table 3 shows the cracking load, reinforcement yield load, and ultimate load of composite beams with different widths of CFRP laminates. The reinforcement ratio is chosen as 4.0% and 0.5%. It can be seen from the table that the mechanical properties of composite beams under negative moments with CFRP laminates are improved to a certain extent, especially when the reinforcement ratio is low. It is mainly manifested in the following aspects: the CFRP laminates can improve the crack resistance of the concrete slab in the negative moment region of composite beams, and the cracking load can be increased by about 5.0%; when the reinforcement ratio is high, the load at the yield stage of the reinforcing bar in the concrete slab with CFRP laminates is increased by less than 5% compared with the composite beam without CFRP laminates, and the growth rate reaches to 18.6% under a low reinforcement ratio; when the reinforcement ratio is high, the ultimate load of the composite beam with CFRP laminates is increased within 12% compared with the beam without CFRP, and the growth rate tends to be stable when the layout width exceeds 300 mm (Figure 12A); however, the growth rate at a low reinforcement ratio can reach up to 25.2%, and the ultimate load increases linearly with the increase of the layout width (Figure 12B).

When the concrete slab in the negative moment region of the composite beam is not bonded with CFRP laminates within the full width, an FEA model considering the layout position (Figure 13) is carried out to analyze the impact on the bearing performance of composite beams. In the FEA model, the total width of the CFRP layout is designed as 300 mm, and the reinforcement ratio of longitudinal reinforcement in the concrete slab is designed as 0.5%. The calculation results show that the load-deflection curves of the composite beams with the different layout positions of CFRP laminates coincide well, as shown in Figure 14A. When the total layout width of the CFRP laminates remains unchanged, the different layout positions have little effect on the bearing performance of the composite beam, with a cracking load, reinforcement yield load, and ultimate load of about 63, 490, and 830 kN, respectively. Figure 14B shows the curves of the maximum tensile stress of CFRP laminates varied with load. It can be seen that the maximum tensile stress of the CFRP laminates in the three layouts is basically the same before concrete cracking. However, the maximum tensile stress of the CFRP laminates arranged in three columns is smaller than that of the other two forms after the crack appears. Therefore, to reduce the stress of the CFRP laminates and control the crack width at the edge of the concrete slab, the CFRP laminates should be reasonably distributed along the width of the concrete slab in practical applications.

In practical engineering, the load-bearing performance of the structure is often guaranteed by increasing the number of CFRP adhesive layers. But this method tends to be conservative, resulting in a waste of materials. To study the influence of the layer number of CFRP laminates on the mechanical properties of composite beams under the action of negative moments, the numerical modeling and calculation analysis of composite beams with 1–5 layers of CFRP laminates on concrete slabs were carried out. The width of CFRP laminates is designed as 600 mm and it is arranged in three columns. Table 4 shows the cracking load, reinforcement yield load and ultimate load of composite beams with different layer numbers of CFRP laminates. The reinforcement ratio is chosen as 4.0% and 0.5%. It can be seen from the table that the cracking load of the concrete slab bonded with CFRP laminates can be increased by about 5.0%. But with the increase in the layer number, there is no significant improvement. When the reinforcement ratio is high, the load at the yield stage of the reinforcing bar in the concrete slab bonded with CFRP laminates is increased by less than 10% compared with the composite beam without CFRP laminates. However, the load-bearing performance of composite beams with more bonded layers is not significantly improved on the whole (Figure 15A). When the reinforcement ratio is low, the bearing performance of the composite beam with 1 ∼ 3 layers of CFRP laminates is greatly improved. The yield load of the reinforcing bar is increased by 26.5% and the ultimate load is increased by 30.3% when three layers of CFRP laminates are arranged. However, with the further increase in the number of adhesive layers, the load-bearing performance is not significantly improved (Figure 15B). In practical engineering, a reasonable number of layers should be arranged in combination with design requirements and theoretical calculations.

To study the influence of the longitudinal reinforcement ratio on the bearing capacity in negative moment regions of composite beams, the FEA model is developed with reinforcement ratios of 0.5%, 1.5%, 2.5%, 3.5%, and 4.5%, under the condition of keeping the structure and size of the composite beams unchanged. The layer number of CFRP laminates is designed as 1. The width of CFRP laminates is designed as 600 mm, and the layers are arranged in three columns. The curves of the ultimate bearing capacity of composite beams under negative moments with/without CFRP laminates varied with the reinforcement ratio are given in Figure 16. It can be seen from the figure that the longitudinal reinforcement ratio has a great influence on the bearing performance in the negative moment region of the composite beam. The ultimate load increases with the increase in the reinforcement ratio, and the growth rate reaches 15% ∼ 45%. Compared with the composite beam without CFRP laminates, the ultimate load of the composite beam bonded with CFRP laminates is increased by 10% ∼ 15%.

To study the effect of the shear connection degree on the bearing performance in negative moment regions of composite beams, the FEA model is developed with shear connection degrees of 40%, 60%, 80%, 100%, and 100%, under the condition of keeping the structure and size of composite beams unchanged. The longitudinal reinforcement ratio, the width of CFRP laminates, the layer number, and the column number of CFRP laminates are designed as 0.5%, 600 mm, 1, and 3, respectively. The curves of the ultimate bearing capacity of composite beams under negative moments with/without CFRP laminates varied with shear connection degrees are given in Figure 17. During the whole loading process of the FEA, it is found that when the shear connection degree is more than 40%, the maximum slip at the composite beam interface does not exceed 0.2 mm, which is less than the ultimate slip value of the stud connection (Fan, 2003). That is, no shear failure of the connectors occurs during structural loading. It can be seen from the analysis in Figure 17 that the ultimate load of the composite beam bonded with CFRP laminates is increased by about 15% compared with the beam model without CFRP laminates under the same loading conditions. However, the shear connection degree has little effect on the bearing capacity of the composite beam under negative moments. When the shear connection degree is 100% and above, the bearing capacity of the composite beam bonded with CFRP laminates in the negative moment region increases by about 5%, while the growth rate of the ultimate load of composite beams without CFRP laminates is less than 3%. This phenomenon is consistent with the research results in the existing literature (Loh et al., 2004), that is, the ultimate bearing capacity of composite beams in the negative moment region does not significantly decrease when the shear connection degree is reduced from 100% to 33%. However, considering the factors such as the fatigue performance of the connector and the complex force in the negative moment region, it is still recommended to design the studs based on the relevant specification (BS EN 1994-2, 2005; BS 5950, 1990), that is, the degree of shear connection should not be less than 100%.