4.1. Internal Force Redistribution at the Middle Support during the Fatigue Progress
Previous studies have shown that the formation of the plastic hinge will cause obvious internal force redistribution at the middle support of continuous composite box beams [
20,
21,
22,
23,
24,
25,
26,
27].
Table 6 shows the measured force (not including self-weight) at the middle support under the target number of loading cycles, with an applied static load of 340 kN. The test results show that the middle support force decreases with the increase in the number of fatigue load cycles, while the side support force increases with the increase in the number of fatigue load cycles. When approaching fatigue failure, the ratio of side support force to middle support force rapidly increases. According to the test results, this phenomenon can be attributed to the change in flexural stiffness and the effect of slippage. In the early stages of fatigue loading, the flexural stiffness of the beam is not significantly affected, and the load is distributed evenly among the supports. As the number of fatigue cycles increases, the cumulative fatigue damage results in the formation of the plastic hinge in the negative moment zone, leading to internal force redistribution in the beams.
The internal force redistribution at the middle support can be expressed by Equation (3) [
27]:
where
βexp is the experimental moment modification coefficient at the middle support,
M is the elastic calculated moment (
M = 3
FL/32),
M′ is the measured moment,
F is the applied load,
Fm is the measured force at the middle support, and
l is the length of one span.
The moment modification coefficient
βexp at the middle support of FSCB-1 to FSCB-8 versus the fatigue loading process is shown in
Figure 13. The moment modification coefficient increases with the increase in the number of fatigue load cycles, and the curves show a quadratic function pattern. When
Nc/Ni = 0, the difference in the moment modification coefficient between FSCB-1 and FSCB-8 is less than 5%. When 0 <
Nc/Ni < 0.5, the moment modification coefficient increases slowly. At this stage, the moment modification coefficients of FSCB-1 to FSCB-8 increase by approximately 11%, 13%, 18%, 10%, 11%, 31%, 9%, and 22%, respectively. When 0.5 <
Nc/Ni < 1, the moment modification coefficients increase faster, and the moment modification coefficients of FSCB-1 to FSCB-8 increase by approximately 27%, 34%, 28%, 16%, 34%, 23%, 19%, and 22%, respectively.
Figure 13 indicates that the growth of the internal force redistribution coincides with the stiffness degradation curve, suggesting that plastic hinges fully develop towards the end of the fatigue loading process. According to the findings in [
24], adding reinforcement can effectively restrict concrete cracking in the negative moment zone, reducing the stiffness degradation and internal force redistribution in the entire beam. These results are consistent with the findings for FSCB-4 and FSCB-5 presented in this study. Therefore, stiffness degradation is a critical parameter for calculating steel–concrete continuous composite box beams. As it can be difficult to determine support forces from the existing literature, this study serves as a valuable reference for investigating internal force redistribution in continuous composite box beams throughout the entire fatigue process.
To take the effect of cumulative fatigue damage into account, a quadratic function model is proposed to express the law of the test data, which can be simplified as shown in Equation (5):
where
a and
b are the fatigue effect coefficients,
βn is the moment modification coefficient when
Nc/Ni =
n,
βu is the moment modification coefficient when
Nc/Ni = 1, and
βs is the moment modification coefficient when
Nc/Ni = 0.
By substituting the relevant data in this study into Equation (4), values of
a = 0.4 and
b = 0.6 were obtained. The fitting curves are shown in
Figure 14. According to the test data, the determination coefficients (R
2) of all models are above 0.86, as shown in
Table 7. This indicates that although the moment modification coefficient is influenced by various factors—such as the amplitude of the fatigue load, reinforcement ratio, stirrup ratio, and stud layout—Equation (5) provides a reasonably accurate fit for all specimens with different parameters.
4.2. Internal Force Redistribution at the Middle Support When Approaching Fatigue Failure
Figure 15 shows the moment modification coefficient–load curves for FSCB-1 to FSCB-8 when approaching fatigue failure. The curves indicate that when approaching fatigue failure, a significant moment modification occurred under small loads, but the change in the moment modification coefficient was small with increasing static load. For example, the moment modification coefficient for FSCB-1 was 39% at a load of 10 kN and increased by only 2% to 41% at a load of 340 kN. These results suggest that the plastic hinge in the negative moment zone had fully rotated towards the end of the fatigue loading process. Therefore, for steel–concrete continuous composite box beams approaching fatigue failure, accumulated fatigue damage is the primary factor causing internal force redistribution, rather than the static load.
The difference in the ultimate moment modification coefficient from FSCB-1 to FSCB-8 is shown in
Figure 16. The moment modification coefficient for FSCB-1 to FSCB-8 under 340 kN varied from 27% to 54%. Based on the comparison of each parameter, the following conclusions can be drawn: (1)
Figure 16a shows that when the upper and lower load limits increased from 160 kN to 200 kN and from 340 kN to 380 kN, respectively, the moment modification coefficient increased by 9%; when only the upper limit increased from 160 kN to 200 kN, the moment modification coefficient increased by 6%. (2)
Figure 16b indicates that when the reinforcement ratio increased from 3.38% to 4.40% or decreased from 3.38% to 2.37%, the moment modification coefficient decreased by 14% or increased by 9%, respectively; when the stirrup ratio increased from 0.54% to 0.81% or decreased from 0.54% to 0.27%, the moment modification coefficient decreased by 13% or increased by 13%, respectively. (3)
Figure 16c shows that reducing the shear connection degree from 100% to 76% increased the moment modification coefficient by 5%. This indicates that the moment modification coefficient caused by fatigue load decreases with increasing stirrup ratio, reinforcement ratio, and shear connection degree, and increases with increasing load amplitude and load limit. Among the three comparison groups, the change in the moment modification coefficient with different reinforcement ratios and stirrup ratios is significant (13%), implying that the reinforcement ratio and stirrup ratio are important factors that control the moment modification coefficient under fatigue load.
Based on the plastic hinge rotation theory, Sun [
27] obtained the following equations for calculating the moment modification coefficient:
where
θ is the angle displacement caused by the modified moment Δ
M,
l is the length of one span,
m is the ratio of the length of the negative moment zone to the length of one span,
n is the ratio of the stiffness at the positive moment to the stiffness at the negative moment,
B is the stiffness at the positive moment,
θu is the ultimate plastic angle displacement of the plastic hinge at the middle support,
φu is the ultimate curvature when the bottom of the steel box yields,
φy is the curvature when the reinforcement yields,
lp is the length of the negative moment zone,
Mu is the ultimate bending moment at the negative moment, and
βu,cal is the calculated moment modification coefficient.
According to Equations (6)–(9), the moment modification coefficients for FSCB-0 to FSCB-8 and the specimens in references [
24,
26] were calculated as they approached the ultimate state. The
βu,cal and
βu,exp of each specimen are shown in
Table 8. It can be seen from the results that the
βu,cal values of the specimens under static loads are in good agreement with the
βu,exp values. However, for the specimens under fatigue loads, the calculated values are in good agreement with the experimental values for FSCB-1, FSCB-4, FSCB-5, and FSCB-8, while there is some deviation for FSCB-2, FSCB-3, FSCB-6, and FSCB-7. This discrepancy is due to the fact that the stiffness calculation takes into account the influence of the reinforcement ratio and the shear connection degree, but neglects the effects of the fatigue amplitude and stirrup ratio on the stiffness of the negative bending moment zone. In addition, most
βu,cal values under fatigue failure are lower than the
βu,exp values for the specimens under fatigue loads. This is because after a period of fatigue loading, the neutral axis moved downward, so the
Mu of specimens was overestimated and the
θu was underestimated. Hence, it is necessary to take the load amplitude and stirrup ratio into consideration in the calculation of fatigue moment modification.
Overall, the βu,cal of FSCB-1 to FSCB-8 was within 20% of the βu,exp. This indicates that Equations (6)–(9) can be applied to steel–concrete composite beams under fatigue loads, but it is necessary to consider the effects of the load amplitude and stirrup ratio on the stiffness and length of the negative moment zone to improve the calculation accuracy.