Bridge overturning is one of the worst antiresilience scenarios, because it may result in catastrophic collapse. After the bridge overturned, looking at the resilience triangle in Fig. 1, the angle α representing the speed of bridge recovery is usually quite small, which means that traffic is immediately interrupted and takes a long time to reconstruction.
The bridge types of most likely overturning are as follows: bridge with incorrectly calculated antioverturning factor, curved bridge with too small curve radius, and bridge with unreasonable bearing settings.
Up to date, the evaluation criteria for bridge overturning is controversial, and the antioverturning design method is to be developed. The indepth study of antioverturning design including antioverturning factor, geometry of curved bridges, disposition of bearings, etc.
5.1 Calculation methods of antioverturning factor
The calculation of the antioverturning factor is an important aspect of bridge antioverturning design. However, discrepancies in understanding among designers may lead to disparities in calculation results, which may result in inadequate structural robustness. Therefore, it is necessary to first discuss calculation methods of structural antioverturning factor.
At present, the calculation methods of the antioverturning factor are mainly divided into two types. The first one is calculated in terms of the deformable body, that is, the overturning problem of the beam is not only a rigid body contact rotation problem, but also contains multiple nonlinear behaviours including volume nonlinearity, material nonlinearity and contact nonlinearity. The second one is calculated in terms of the rigid body, that is, the influence of the stress state change and material deformation of the beam is ignored to facilitate the engineering calculation. In addition, there are two methods of dividing dead load to calculate the antioverturning factor according to the rigid body: 1) the unified barycentre; 2) the axis of overturning. The two antioverturning factors are calculated as follows:

(1)
Structure seen as a deformable body
There are two definite characteristic states in the overturning process: in the state 1, the unidirectional support of the box girder begins to release from the compression state; in the state 2, the torsion support of the box girder fails completely. The typical failure state is shown in Fig. 16.
The antioverturning factor is calculated as follows:
$${K}_{qf}=\frac{\sum {S}_{\textrm{bk},i}}{\sum {S}_{\textrm{sk},i}}=\frac{\sum {R}_{\textrm{Gk},i}{l}_i}{\sum {R}_{\textrm{Qk},i}{l}_i}$$
(11)
where ∑S_{bk, i} is the design value of the effect stabilizing the superstructure; ∑S_{sk, i} is the design value of the effect destabilizing the superstructure; l_{i} is the centre distance between the failure support and effective support on the i # pier; R_{Gki} is the support reaction force of the ineffective support on the i # pier under permanent actions; R_{Qki} is the support reaction force of the ineffective support on the i # pier under variable actions.
The above method abandons the concept of overturning axis and adopts the concept of overturning limit state. However, it needs to establish a fine finite element model and the calculation is more complicated.

(2)
Structure seen as a rigid body
$${K}_{qf}=\frac{\sum {S}_{\textrm{bk},i}}{\sum {S}_{\textrm{sk},i}}=\frac{\sum {M}_{\textrm{K}}}{\sum {M}_{\textrm{Q}}}$$
(12)
where ∑M_{K} is the antioverturning moment; ∑M_{Q} is the overturning moment.
The following will compare the two methods to calculate the antioverturning factor.
$${K}_1=\frac{\sum {M}_{\textrm{K}}}{\sum {M}_{\textrm{Q}}}=\frac{\textrm{W}\times {\textrm{L}}_{\textrm{W}}}{\textrm{P}\times {\textrm{L}}_{\textrm{P}}}$$
(13)

Algorithm 2: the section is separated by the overturning line at the support. The dead load on the side of the live load generates the overturning moment, and the antioverturning moment is generated on the other side. The antioverturning factor expression is shown in the following formula, and the calculation diagram is shown in Fig. 18.
$${K}_2=\frac{\sum {M}_{\textrm{K}}}{\sum {M}_{\textrm{Q}}}=\frac{{\textrm{W}}_{\textrm{K}}\times {\textrm{L}}_{\textrm{WK}}}{\textrm{P}\times {\textrm{L}}_{\textrm{P}}+{\textrm{W}}_{\textrm{Q}}\times {\textrm{L}}_{\textrm{WQ}}}$$
(14)
In order to make a more obvious comparison between the two algorithms, the following takes a 35 m simply supported linear bridge as an example. The bridge is 38 m wide and the beam is 2.4 m high. The section size is shown in Fig. 19. The value of vehicle load is determined according to JTG D602015 (namely General Specifications for Design of Highway Bridges and Culverts), and the calculation results of the two algorithms are shown in Fig. 20.
As can be seen from Fig. 19, the factors obtained by the two algorithms for the same section are quite different. Compared with algorithm 1, the antioverturning factor obtained by algorithm 2 decreased significantly. For the bridge, the support spacing worth paying attention to is 3.5 m ~ 9.0 m. In this range, algorithm 1 meets the requirements of the specification, and the other does not meet the requirements. However, both algorithms are used in actual bridge design.
5.2 Antioverturning stability of curved bridges
Different from the linear bridge, the stress state of the curved bridge is more complicated. Due to the influence of curvature, the barycentre of the beam is inclined to the outer side of the centre line, which can generate torque and result in inconsistent side support reaction, or even support separation. In addition, some other factors, such as temperature of beams, superelevation, cross slope, automobile centrifugal force etc., make the design of the curve beam more complex; more attention should be paid to the resilience design of the curved bridge. The barycentre position of the curved beam under different curvature conditions is shown in Fig. 21. In this figure, R is the outer diameter of the curved beam; r is the inner diameter of the curved beam; B is the beam width; b is the support spacing; θ is half of the central angle; X_{c} is the distance between the barycentre and the centre; X_{qf} is the distance between the overturning line and the centre; R_{e} is the distance between the outer support and the centre.
To simplify the calculation, the beam is regarded as a homogeneous plate. When the barycentre falls on the overturning line, the calculation formulas of X_{c} and X_{qf} are as follows.
$${\textrm{X}}_{\textrm{c}}=\frac{2\left({R}^3{r}^3\right)\sin \theta }{3\left({R}^2{r}^2\right)\theta }$$
(15)
$${\textrm{X}}_{\textrm{qf}}={R}_{\textrm{e}}\cos \theta$$
(16)
If vehicle load is taken into account, the calculation formula of antioverturning coefficient is shown in the following formula.
$$K=\frac{W\left({R}_{\textrm{e}}\cos \theta {X}_{\textrm{c}}\right)}{Q\left({X}_{\textrm{qf}}{R}_{\textrm{e}}\cos \theta \right)}\le {K}_{\textrm{qf}}$$
(17)
Taking a simply supported curve beam as an example, the basic information: the width is 12 m, support spacing is 4 m, when the barycentre falls on the overturning line, the relationship between r and θ is shown in Fig. 22.
As can be seen from the figure above, the area on the upper side of the curve is X_{c} > X_{qf}, while the area on the lower side is X_{c} < X_{qf}. In the actual design, the relationship between r and θ should be ensured to be located on the lower side of the curve. For the bridge, particular attention should be paid to the area between the two dotted lines, that is, 35 m ~ 50 m is the range of the commonly designed span, which is prone to occur designer’s design mistakes that the beam barycentre is outside the overturning line.
In the actual situation, the curve beam is also faced with other adverse factors, such as singlecolumn piers, small spacing support, overload, puncture, support disease, lateral wind load, vehicle impact, and earthquake. Therefore, under design principles for bridge resilience, the reasonable design range should be formed to meet the requirements of robustness and redundancy.
5.3 Reasonable disposition of supports
Antioverturning design mainly focuses attention on the beam itself, and ensures the supports not separation. However, in the support design, the design value of the support reaction is calculated according to the standard value of vertical load, so support collapse may occur before overturning coefficient of the beam reaches 2.5, resulting in the beam overturn. Therefore, the support design should be unified with the antioverturning design of the beam, but not design them separately.
5.4 Antioverturning countermeasures
In bridge design, the designer should consider the factors affecting the bridge antioverturning comprehensively. The correspondence between structural antioverturning and resilience design criteria is shown in Fig. 23.
In practice, some retrofit options against overturning are shown in Fig. 24, including adding steel tie plate, adding cap beam, adding steel pipe column, and enlarging bridge pier.