3.1. Analysis of Influence Parameters of Curvature Change
Curved beam cable-stayed bridges are more complexly affected by wind loads due to their characteristics. Previous studies [
13,
29] have shown that an unbalanced structural response is generated due to the change of curvature, the different rigidity of the inner and outer beams, and the incoming-flow wind loads acting on the curved beam bridge at a specific wind attack angle. As a result, an unbearable tensile force on the cable is produced, and a fortuity of cable break is caused. Therefore, this Section will study the influence of its changes on the dynamic response of the broken cable of the curved beam stayed bridge by wind loads from five different curvatures
k of 0.0000, 0.0024, 0.0048, 0.0072, and 0.0097.
The curvature of the five finite element models adopted in this section is based on the 1 to 5 cases of the CBUSB literature studied by Zhao et al. [
1]. The finite element models under the 5 cases are shown in
Figure 3. In the process of the curvature change, the cross-sectional shape of the bridge and the positions of the fixed supports at both ends are not changed. The position of the hanging point of the cable and the main beam of the bridge also remains unchanged, and the projection of the cable in the
X-
Y plane is kept on the same line as the cantilever beam. The
Z-axis coordinates of the fixed hinge support at the suspension point of the cable remain unchanged, as shown in
Figure 1. C15# is selected as the breaking target cable, and the break simulation of the cable under 5 cases is carried out.
Figure 4a shows the
of C14# under three states of cable breakage and five curvature cases, where
is the biggest value of the cable force and subscript 1, 2, and 3 denote the moments
t <
t0,
t0 ≤
t ≤
t1, and
t >
t1, respectively. When curvature increases, the
throughout the cable breakage increases (as shown in
Figure 4a). As the curvature increases, the force on the remaining cable of state 3 increases, indicating that remaining cables are more likely to approach the allowable value of the maximum cable strength stress to cause cable fatigue failure. As shown in
Figure 4a,
is increased by the impact loads generated in state 2. With the impact loads tending to 0, the internal force of the bridge structure is redistributed, so
is higher than
.
Figure 4b shows the cable force spectrum of C14# throughout the cable breakage in five curvature cases. In
Figure 4b,
f is the frequency and
ST is the cable force power spectral density. The peak frequencies in
ST include the bridge natural frequencies of 0.595 Hz, 0.994 Hz, 1.271 Hz, 1.304 Hz, and 1.310 Hz and many peak frequencies that are quite different from the natural frequency, such as 0.820 Hz, 1.466 Hz, and 2.013 Hz. The structural vibration modes of the bridge and cable corresponding to these peak frequencies contribute significantly to the CBUSB wind vibration, which shows that the design of cable-stayed bridges should consider the adverse influence of the strong nonlinear vibration of the cable on CBSUBs after the cable breakage.
The peak along-wind displacement
throughout the cable breakage at the connection point under the 5 curvature cases when the target cable C15# (
X = 56.726) breaks are shown in
Figure 5a, where
U is the along-wind displacement and
X is the
X-axis coordinate value (coordinate system as shown in
Figure 1) of the bridge and cable connection point. As the curvature increases, the peak
decreases, indicating that increasing the curvature can make the lateral stiffness of the bridge larger and reduce the
caused by impact loads. However, the inhibition of
U by increasing curvature is nonlinear, and the inhibition will level off when a certain value is reached. The
of case 3 throughout the cable breakage is shown in
Figure 5b. The impact loads generated by the cable breakage increases
from wind-induced vibration. The closer the location of the broken cable, the greater the influence of impact loads generation on
. The along-wind displacement spectrum of case 5 within 3 states is shown in
Figure 4c. The first-order frequency of states 1, 2, and 3 is 0.60 Hz, 1.18 Hz, and 1.20 Hz, respectively, which indicates that the along-wind displacement spectrum of case 5 throughout the cable breakage can give a good inclusion of the along-wind displacement at different cable breakage states.
Combined with the analysis of
Figure 5a,c, after using the band-pass filter method [
30], it is obtained that the ratio of the resonance response to the fluctuating response in cases 1 and 5 is 44% and 28%, respectively. As the curvature increases, this ratio decreases, indicating that the curvature increases, the
decreases (indirectly reflecting the decrease of the stress amplitude of the structure) and the ratio of the resonance response decreases (indicating the decrease in the number of dynamic cycles of the structure), which can reduce the possibility of fatigue damage.
The first-order frequency of the CBUSB increases with increasing curvature, demonstrating that the increase of curvature moves the excitation frequency of the resonance response far away from the wind dominant frequency (0.006–0.037 Hz); thus, the resonance energy excited by wind loads decreases. Considering the influence law of the CBUSB along-wind displacement response and cable tension response, when 0.0048 ≤ k ≤ 0.0052, the influence of broken cables on the wind-induced vibration response of bridges is the least significant, which can ensure the safety of CBUSBs.
3.2. Analysis of the Cable Arrangement Case
As the cable arrangement cases changes, the mechanical and dynamic characteristics of the CBUSB also change, thereby affecting the structural dynamic response after cable breakage. This Section will study the influence of different cable arrangement cases on the dynamic response of the CBUSB through the following four cable arrangement cases.
Figure 6a,b show the arrangement of cables on the side close to or far from the center of curvature;
Figure 6c,d show that the position of cable breakage is close to and far from the middle side of the curvature, respectively, under the arrangement of cables on both sides. C15#, as the breaking target cable, is used to simulate the breaking process of the cable by the dynamic response method, and numerical analysis is carried out for each case.
Figure 7a shows the
of the analysis cable C14# throughout the cable breakage under different cable arrangement cases. The impact loads applied to the unilateral arrangement of the cable throughout the breakage are larger than that of the bilateral arrangement of the cable (as shown in
Figure 7a). Compared with the 4 cases, the influence of the bilateral arrangement of cables on the remaining cable force in state 3 is minimal. The maximum
of case 4 is 1.6 × 10
5 kN, and the cable force is only increased by 6.4%. When cables are arranged unilaterally on the side away from the curvature center, it leads to a more pronounced redistribution of the wind-induced vibration internal force. The
ST throughout the cable breakage under four cases is shown in
Figure 7b. The resonance responses of cases 2 and 4 account for 54% and 32% of the fluctuating response, respectively. The analysis shows that when the cable arrangement is unilateral, the structure is more affected by the impact load caused by the cable breakage, and resonance damage easily occurs after the cable breakage.
When the target cable C15# breaks, the
and its power spectrum
SU under 4 cable arrangement cases are shown in
Figure 7c,d, respectively. As shown in
Figure 7c,d, the mid-span point
under case 1 and case 2 is 22.55 mm and 27.76 mm, respectively, which indicates that the static wind action of case 1 is less than that of case 2. The first-order frequency of case 1 and case 2 is 1.36 Hz and 1.31 Hz, respectively, which indicates that case 1 is far from the wind dominant frequency (0.006–0.037 Hz) and has less resonance response than case 2. Additionally, the double-sided arrangement of the cable reduces the energy generated by the structural resonance and the influence on the dynamic response of the CBUSB in state 2. The first-order frequencies of case 3 and case 4 are 1.69 Hz and 1.52 Hz, respectively, which shows that it is less prone for case 3 to resonate due to excitation by the wind. Through peak value statistical analysis, the case 3 and case 4 ration of mean displacement response to the total response accounts for 98.37% and 98.49%, respectively, which indicates that the CBUSBS is quasi-static after cable breakage when the cables are arranged on both sides. From the perspective of reducing the damage of CBUSBs caused by cable breakage and wind, cables should be arranged on both sides of the curved beam and the cables close to the curvature center should be designed more strictly. If the unilateral arrangement case is adopted, the cables should be arranged at the side close to the center of curvature.