Wind stability analysis of the rotating girder segment during construction of a rigid-frame bridge

3.1. Frictional moment before turning

To verify the stability of a rigid frame bridge during the preparation stage of rotational construction, the friction torque before rotation was first calculated. Based on domestic construction data of similar rotating bridges, the static friction coefficient of the spherical hinge obtained from balancing and weighing tests typically ranges from 0.018 to 0.1, lower than the design value of 0.1 [7]; the measured static friction coefficient for this bridge was determined to be 0.04. For the analysis of general stability, static friction coefficients of 0.04, 0.05, 0.06, 0.07, 0.08, and 0.09 were adopted. With a total superstructure weight of $G=$ 73500 kN, a spherical hinge radius of $R=$ 8 m , and a cantilever length of $L=$22 m , the friction torque before rotation was calculated as $M_{\mathrm{F}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}^{\mathrm{B}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{ }\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}$ = 23520 kN⋅m when ${\mu }_{Static}=$ 0.04, with results for other coefficients summarized in Table 1.

3.2. Stability calculation of the transverse bridge

During the preparation stage of bridge rotation construction, the sole main support between the superstructure and substructure is the spherical hinge system, making the transverse stability significantly affected by wind loads. Based on the wind speed, wind pressure, and geometric characteristics of the bridge, the design wind speed was set to 25 m/s, the terrain category was classified as B, the conversion factor $kc$ was taken as 1, and the air density was set to 1.25. The drag coefficient was obtained from Table 1 of the Wind Resistant Design Specification for Highway Bridges (JTG/T 3360-01-2018), and was herein taken as 1.4. Using the wind load calculation method specified in Eq. 1 of the aforementioned specification, the transverse wind-induced moment before rotation was calculated to be 14980 kN·m, and the transverse stability safety factor was subsequently determined using Eq. (1):

where, $M_H^{\mathrm{B}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{ }\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}$ represents the bending moment caused by wind loads on the transverse direction before rotation, while $M_{\mathrm{F}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{ }\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{ }}^{\mathrm{B}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{ }\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}$ represents the friction torque before rotation. Here, $M_f=F_{y}e=\mu Ne$, where $\mu$ is the static friction coefficient, $N$ is the reaction force at the support, and $e$ is the eccentricity (lever arm) from the point of action of the friction force to the center of rotation. The product is approximately 238 kN·m. Based on the above calculations, the safety stability factor for the transverse direction of the superstructure before rotation is obtained, as shown in Table 2.

As shown in Table 2, the transverse safety stability coefficient increases significantly with the static friction coefficient, indicating that the friction moment at the ball joint is crucial for transverse stability [8]. When the friction coefficient ranges from 0.04 to 0.09, the coefficient meets design requirements, demonstrating adequate stability against wind loads before rotation.

3.3. Longitudinal bridge direction stability calculation

Although the longitudinal direction of the superstructure of a rigid frame bridge is less affected by wind loads, it is necessary to consider the unbalanced weight between the two cantilever segments. Ten scenarios of unbalanced weight ($\delta$) ranging from 0, 1 %, 2 %, 3 %, 4 %, 5 %, 6 %, 7 %, 8 %, and 9 % were considered for calculating the longitudinal stability of the rotating structure. First, the overturning moment caused by the unbalanced weight was calculated as $M_0=W\delta L$, where $W$ is the total weight of the superstructure on one side of the rotation (here taken as $G=$73500 kN), $\delta$ is the unbalanced weight, and $L$ is the cantilever length (here taken as 22 m). The unbalanced moment $M_0$ generated by the unbalanced weight in the longitudinal direction of the superstructure is shown in Table 3.

Next, considering the safety stability factor of the longitudinal bridge, it is necessary to consider the wind load and unbalanced weight of the longitudinal bridge at the same time, and the safety stability factor of the longitudinal bridge direction is obtained from Eq. (2):

For the rigid frame bridge under site wind conditions, a friction coefficient of 0.04-0.09 and unbalanced weight within 0-9 ‰ ensure sufficient stability before rotation. Longitudinal and transverse safety factors increase with friction coefficient [9]. However, longitudinal stability decreases as unbalanced weight increases, showing its dominant influence on rotational stability [9]. Thus, strict control and detailed recording of concrete volume at cantilever ends are essential to limit unbalanced weight during construction.

4.1. Frictional moment during rotation

The stability of a rigid frame bridge during rotation is analyzed by calculating the frictional moment of the rotating system. For piers 6 and 7, the ratio of starting to sliding traction force is 0.56 and 0.53, respectively. Dynamic friction coefficients of 0.01-0.035 are used to evaluate superstructure stability. The total rotating mass, spherical radius $R=$ 8 m, and ball hinge friction moment (11760 kN·m) are considered. Friction moments under different coefficients are shown in Table 5. Results indicate that increasing the dynamic friction coefficient significantly enhances anti-overturning capacity, improving resistance to transverse wind loads during rotation.

To evaluate the stability of a rigid frame bridge during its rotation construction phase, the friction torque was first calculated. The friction coefficient for smooth surfaces is taken as 0.01, for rough surfaces as 0.02, and for very rough surfaces as 0.04. Based on this, six different scenarios were considered, ranging from smooth to rough contact surfaces, using dynamic friction coefficients of 0.01, 0.015, 0.02, 0.025, 0.03, and 0.035 to calculate the safety stability of the superstructure during rotation. For a particular rigid frame bridge with a total mass of $G=$7350 t and a spherical hinge radius of $R=$8 m, when the dynamic friction coefficient ${\mu }_{Dynamic}=$ 0.02, the friction torque of the spherical hinge $M_{\mathrm{F}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{ }\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}}^{\mathrm{D}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{ }\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}$ was calculated to be 11760 kN⋅m. Similarly, the friction torques for other dynamic friction coefficients are shown in Table 4. It can be concluded that an increase in the dynamic friction coefficient significantly enhances the anti-overturning capacity of the bridge during rotation, effectively mitigating the effects of lateral wind loads.

4.2. Stability calculation of the transverse direction

In the process of rotating construction, similarly, the bending moment generated by the transverse $M_H^{\mathrm{D}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{ }\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}=$2958 kN⋅m, bridge static wind load is calculated based on the wind load code, and the transverse bridge safety stability coefficient is calculated by Eq. (3):

The safety stability coefficient in the transverse direction is calculated according to the above formula, as shown in Table 5.

As shown in Table 6, the safety stability coefficient of the bridge rotating structure increases correspondingly as the dynamic friction coefficient rises from 0.01 to 0.035. Under all tested dynamic friction coefficients, the safety stability coefficients meet design requirements, demonstrating high reliability of the bridge during rotation. This indicates that the superstructure of the bridge maintains adequate safety stability under lateral wind loads during rotation construction [10].

4.3. Stability calculation of longitudinal bridge

Finally, considering the combined effects of wind loads and unbalanced moments on the longitudinal stability of the rotating bridge during construction, six scenarios of unbalanced weight ($\delta$) were selected: 0, 1 %, 2 %, 3 %, 4 %, and 5 %. Similar to Section 3.3, the overturning moment caused by the unbalanced weight was first considered. Using the formula $M_0=W\delta L$, the longitudinal unbalanced moments $M_0$ under the six scenarios of unbalanced weight were calculated, as shown in Table 6.

Next, considering the longitudinal safety stability coefficient ${\eta }_Z^{\mathrm{D}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{ }\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}$ under the combined effects of wind loads and unbalanced moments during bridge rotation, it was calculated using Eq. (4):

As shown in Table 7, the longitudinal safety stability coefficient during rotation is influenced by dynamic friction coefficient and unbalanced weight. Both coefficients increase with friction coefficient [11], while longitudinal stability decreases as unbalanced weight increases. Sufficient stability is achieved when friction coefficient > 0.015 or unbalanced weight ≤ 3 ‰. At 0.015 friction and 5 ‰ unbalance, the coefficient drops to 1.1, nearing the critical threshold – risking instability under dynamic effects.

Thus, unbalanced weight must be strictly controlled, ideally ≤ 4 ‰ via counterweights or construction adjustments. Increasing friction coefficient, especially by applying high-friction materials like modified epoxy resin on contact surfaces, significantly enhances stability. Controlling unbalance and enhancing friction are key to safe rotation.