Probabilistic Safety Factor Calculation of the Lateral Overturning Stability of a Single-Column Pier Curved Bridge under Asymmetric Eccentric Load

Abstract

At present, the calculation methods for the lateral overturning stability safety factor of a single-column pier curved bridge under asymmetric eccentric load in the highway bridge code adopt deterministic calculation models. This paper presents a transverse overturning stability safety factor study of single-column pier curved bridges under asymmetric eccentric load based on inverse reliability theory. This method, which both considers the influence of uncertainty factors among the structural parameters and satisfies the prescribed reliability level, was used to obtain safety factors through a target reliability index. Transverse overturning stability factors of single-column pier curved bridges under asymmetric eccentric load were estimated using the proposed method. A new method was proposed to calculate the design parameter of the safety factor of single-pier instability, and then an application was carried out. The results show that parameter uncertainty has a major effect on transverse overturning stability safety factors of single-column pier curved bridges under asymmetric eccentric load. Ignoring parameter uncertainty will produce an overestimation of the transverse overturning stability safety factors of single-column pier bridges. The vehicle load must be strictly controlled to avoid transverse overturning due to overloading. Compared to the deterministic model, the proposed method has the advantages of calculating the safety factors considering the randomness of parameters and establishing the relationship of the safety factor with the target reliability index.

1. Introduction

A single or similar single-column cast-in-situ continuous box girder structure is often used in urban overpass and trunk highway bridges in China. Single or similar single-column pier box girder bridges have the characteristics of light structure, reduced land occupation, improved substructure layout, increased vision, and beautiful bridge type [1,2,3]. However, many bridge overturning accidents have occurred during the use of this structural form (see

Table 1. A summary of overturning accidents of single-column pier bridges.

Time	Place	Structure Type	Cause of Accident
2007.10	Baotou, Inner Mongolia	Steel structure simply supported beam bridge	Heavy vehicle traffic (3 × 110 t)
2009.7	Tianjin	RC Continuous Beam Bridge	Heavy vehicle traffic (separate weight 147 t\142 t\140 t\54 t)
2010.11	Zhejiang Shangyu	RC	Heavy vehicle traffic (separate weight 125 t\125 t\110 t\30 t)
2011.2	Heilongjiang Harbin	Steel–concrete continuous beam bridge	Heavy vehicle traffic (The approved load of vehicles 1~4 was 102 t and the actual load was 395 t)

). For bridges where overturning accidents occur, the piers are mostly in the form of a single column or similar single-column piers, and the bearing spacing is small or a single support, resulting in a small anti-overturning moment of the bridge self-weight, while the overturning moment generated by external loads such as vehicle load increases, local scour on the bridge pier, earthquake, etc., increases the probability of overturning instability [4,5,6]. The danger of overturning is even greater when heavy-duty vehicles are mostly driven on one side. At present, China’s trucks are seriously overloaded, and some vehicles are overloaded with twice or even three times their appropriate load. At the same time, the existing highway bridge specifications in China do not make clear provisions on the lateral overturning stability of bridges, which leads to many bridge overturning accidents under the action of certain overloaded vehicles, although they meet the requirements of the design specifications.

At present, several researchers have investigated the lateral overturning stability of single-column pier curved bridges under asymmetric eccentric load. Yu studied the pier stability of an urban single-column pier viaduct using the finite element analysis software ABAQUS [7]. Wu et al. used the Midas/Civil finite element analysis program to construct the simulation models of 23 single-column pier continuous box girder bridges and analyzed the main factors affecting the stability of curved girder bridges [8]. Zhao et al. took a continuous beam bridge with a single-column pier at an expressway separation interchange as an example and analyzed the mechanical mechanism of transverse instability under overweight and asymmetric eccentric load [9]. Peng et al. proposed a practical calculation method of the anti-overturning bearing capacity, and a finite element model of the anti-overturning calculation of a single-column pier beam bridge was established by considering geometric nonlinearity and contact nonlinearity [10,11,12]. Jiang and Yang established a curved beam bridge model with a single-column pier by using Midas software based on the finite element theory and analysis method [13]. Wang et al. analyzed the transverse overall anti-overturning stability of a multispan single-column curved prestressed concrete bridge [14]. Zhou et al. discussed the selection of the overturning axis under the conditions of different support arrangements and different radii of curvature [15]. Zhuang studied the mechanical performance of the bearing and the interaction between the bearing and the box girder during the overturning of a box girder bridge [16]. Gao et al. established a calculation method for the transverse overturning stability of a beam considering rubber bearing deformation [17]. Lu et al. took a three-span continuous single-column pier curved beam bridge as the basic structure and calculated the anti-overturning stability safety factor [18]. Xu discussed the calculation formula of the overall anti-overturning stability of a box girder based on the assumption of overall rigidity of the box girder [19]. Sun et al. established a spatial finite element model using spatial beam elements and a beam lattice to analyze its overturning effect [20]. In conclusion, the general test method is to check the overturning stability safety of a structure through a deterministic model, without considering the influence of parameter randomness. Therefore, in fact, the overturning stability and safety of the structure is unknown [21,22].

Random analysis provides an effective analysis method for considering the random influence of parameters in the study of the lateral overturning stability and safety of single-column pier bridges. The current bridge design code has developed into a performance-based design concept; that is, the target reliability index of the structure is given in advance so that the safety of the structure can be ensured in its design [23,24]. In this way, it is necessary to correct the safety factor of overturning stability to ensure the predetermined reliability of the bridge transverse overturning stability. However, the current research methods for the overturning stability safety factor are not suitable for analyzing this problem. Therefore, on the basis of previous studies, this paper puts forward a research method for the overturning stability safety factor based on reliability back analysis theory and studies the transverse overturning stability safety factor of a single-column pier curved bridge under asymmetric eccentric load.

2. Reliability Back Analysis Theory

2.1. Principle

Der kiureghian A., Zhang Y., Li C. C. [25] defined the problem of structural reliability back analysis as follows:

$$‖u‖-{\beta }_{\mathrm{T}}=0$$

(1)

$$u+\frac{‖u‖}{‖{\nabla }_{u}G(u,\theta )‖}{\nabla }_{u}G(u,\theta )=0$$

(2)

$$G(u,\theta )=0$$

(3)

where $u$ is a vector of standard normal distribution space; ${\beta }_{\mathrm{T}}$ is the structural target reliability index; $G(u,\theta )$ is a function of the structure; ${\nabla }_u$ is the gradient operator; and $\theta$ is the design parameters to be determined.

Hong Li and Ricardo O. Foschi [26] proposed a direct algorithm to solve for these parameters. The basic idea of the algorithm is that ${\beta }_{\mathrm{T}}$ is known, and we solve for $\overline{\theta }$ (or the mean value of $\theta$), such the parameters to be solved meet $\min (u^{\mathrm{T}}u)={\beta }_{\mathrm{T}}^2$ and $G=G(u,\theta )=0$.

According to the theory of primary reliability analysis, the parameter vector $u$ satisfies the following formula at the design point:

$$u=[\frac{{({\nabla }_{u}G)}^{\mathrm{T}}u}{{({\nabla }_{u}G)}^{\mathrm{T}}{\nabla }_{u}G}]{\nabla }_{u}G$$

(4)

Thus, the target reliability index ${\beta }_{\mathrm{T}}$ of the structure is:

$${\beta }_{\mathrm{T}}=\frac{-{({\nabla }_{u}G)}^{\mathrm{T}}u}{{[{({\nabla }_{u}G)}^{\mathrm{T}}{\nabla }_{u}G]}^{1/2}}$$

(5)

The target reliability index ${\beta }_{\mathrm{T}}$ can be obtained by Taylor expansion at ${\beta }_{}^j$; the result is:

$${\beta }_{\mathrm{T}}={\beta }_{}^j+{\left.\frac{\partial \beta }{\partial K}\right|}_{K^j}(K_{}^{j+1}-K_{}^j)$$

(6)

where K^j+1 and K^j are the (j + 1)th and jth iteration values of the safety factor, respectively, which are the reliability index values ${\beta }_{}^j$ calculated for the jth iteration.

The iterative formula [27] of the safety factor can be obtained from Equation (6):

$$K_{}^{j+1}=K_{}^j+\frac{{\beta }_{\mathrm{T}}-{\beta }_{}^j}{{\left.\frac{\partial \beta }{\partial K}\right|}_{K^j}}$$

(7)

Equation (8) was selected as the convergence criterion of the reliability back analysis method adopted in this paper:

$$\left|K^{j+1}-K^j\right|\le \epsilon$$

(8)

In the formula, $\epsilon$ is taken as an appropriately small number; 0.0001 can be used in in specific calculations.

2.2. Steps and Procedure

The procedure for solving the lateral overturning stability safety factor of a single-column pier curved bridge under asymmetric eccentric load by using the above reliability back analysis method is as follows:

Step 1: The random variables and the initial value of the lateral overturning stability safety factor of the single-column pier curved bridge under asymmetric eccentric load are assumed, and the structural target reliability index ${\beta }_{\mathrm{T}}$ and convergence error $\epsilon$ are determined. The initial value of the random variable can be taken as the mean value of its probability distribution, and the initial value of the transverse overturning stability safety factor of the single-column pier curved bridge under asymmetric eccentric load can be calculated according to the deterministic model.

Step 2: Initialize the number of iterations j= 1, and calculate ${\beta }_{}^j$ and ${\left.\frac{\partial \beta }{\partial K}\right|}_{K^j}$.

Step 3: Print ${\beta }_{}^j$ and ${\left.\frac{\partial \beta }{\partial K}\right|}_{K^j}$ into (7), and then obtain the new K.

Step 4: Check whether it converges according to the convergence criterion of Equation (8). If it does not meet the convergence criterion, set j = j + 1 and go to Step 2. Otherwise, the calculation results are output.

3. Verification and Comparison of Methods

An example is given to verify the accuracy and effectiveness of the method proposed in this paper. Consider the following limit state equation [26].

$$G=\exp \left[-\eta \left(x_1+2x_2+3x_3\right)\right]-x_4+1.5$$

(9)

Here, the random variables $\left(x_1,x_2,x_3,x_4\right)$ obey the standard normal distribution and $\eta$ are independent of each other; these are the deterministic design variables (the safety factor calculated in this paper is the deterministic variable). The calculation results $\eta$ obtained by using the reliability back analysis method proposed in this paper are shown in

Table 2. Results Comparison of Inverse Reliability.

j	Reference [25]	Reference [26]	The Proposed Method	Error with Reference [26]
1	0.1000	0.150000	0.150000	0
2	0.4426	0.367813	0.200000	45.6%
3	0.4424	0.367722	0.320504	12.8%
4	0.4412	0.367428	0.356061	3.09%
5	0.3590	0.367288	0.366362	0.25%
6	0.3592	0.367218	0.367120	0.03%
7	0.3647	0.367183	0.367131	0.01%
8	0.3668
9	0.3670

.

It can be seen from the data in Table 2 that the calculation results of the reliability back analysis of the deterministic parameter problem are in good agreement with the literature results. In addition, by comparing the results of the methods from the literature and the proposed method, we found that the proposed method has a faster iterative convergence speed and the same iterative accuracy while also giving the target reliability index. This proves that the calculation results of reliability back analysis using the method proposed in this paper are accurate and reliable, and the calculation accuracy can meet engineering needs.

4. Reliability Model

When a heavy-duty vehicle runs on a single-pier curved bridge under asymmetric eccentric load, if the vehicle runs close to the outside, the single-pier curved bridge under asymmetric eccentric load is in danger of transverse overturning. Chapter 4.1.10 of the specifications [28] of “Code for design of highway reinforced concrete and pre-stressed concrete bridges and culverts” (JTG D62) stipulates that anti-overturning calculations of the superstructure should be carried out for medium- and small-span bridges with integral sections. The anti-overturning stability safety factor of the superstructure should meet the following requirement:

$$K=\frac{S_{\mathrm{bk}}}{S_{\mathrm{sk}}}\ge 2.5$$

(10)

Here, $K$ is the anti-overturning stability coefficient; $S_{\mathrm{sk}}$ is the standard value of vehicle load effects (including impact) to overturn the superstructure; and $S_{\mathrm{bk}}$ is the standard value combination of action effect to stabilize the superstructure.

For a curved bridge under asymmetric eccentric load, when all the supports of the mid-span pier are located on the inner side of the bearing line outside the abutment, the overturning axis is the bearing line outside the abutment; when all supports of the mid-span pier are located outside the connection between the supports outside the abutment, the overturning axis is taken as the connection between the support outside the abutment and the support of the mid-span pier. The anti-overturning safety factor of a box girder bridge is:

$$K=\frac{{\displaystyle \sum R_{\mathrm{Gi}}{x}_i}}{(1+\mu )(q_{\mathrm{k}}\Omega +P_{\mathrm{k}}e)}$$

(11)

where $R_{\mathrm{Gi}}$ is the support reaction force of each support in the completed bridge state; $x_i$ is the vertical distance from each support to the overturning axis; $\mu$ is the impact coefficient; $q_{\mathrm{k}}$ is the uniformly distributed load in lane load; $\Omega$ is the area enclosed by the overturning axis and transverse loading lane; $P_{\mathrm{k}}$ is the concentrated load in lane load; and $e$ is the maximum vertical distance from the transverse loading lane to the overturning axis.

According to the reliability back analysis theory, when evaluating the anti-overturning stability coefficient of a single-column pier curved bridge under asymmetric eccentric load, the overturning stability safety coefficient is taken as an unknown parameter. According to Equation (11), the calculation expression of the overturning stability safety coefficient is:

$$K=\frac{S_{\mathrm{bk}}}{S_{\mathrm{sk}}}$$

(12)

According to Equation (13), the function shown in Equation (14) can be established:

$$Z=S_{\mathrm{bk}}-K{S}_{\mathrm{sk}}$$

(13)

Combining Equations (12) and (14), a function of the anti-overturning stability of a straight bridge can be constructed as follows:

$$Z={\displaystyle \sum R_{\mathrm{Gi}}{x}_i}-K\left(1+\mu \right)(q_{\mathrm{k}}\Omega +P_{\mathrm{k}}e)$$

(14)

5. Case Analysis

We take a ramp bridge of an expressway in Sichuan as an example. The ramp bridge is located on a horizontal curve with a radius of 100 m and is a 3 * 25 m prestressed concrete continuous box girder bridge. The middle piers are all single-column piers, and the connecting ends are double-column piers with bent caps and double bearings (the transverse spacing of bearings is 3M). The arrangement of piers and bearings is shown in Figure 1. The large circle in the figure represents pier columns, the small circle represents bearings, and the double-column piers are all equipped with bent caps.

When calculating the lateral overturning stability of the single-column pier curved bridge under asymmetric eccentric load, a spatial model is established for calculation, and the prestress effect is considered. Under the standard combination condition, the values of all bearing reactions and other parameters outside the overturning axis are as follows:

$R_{G1}=2060 \mathrm{kN}$, $R_{G2}=2103 \mathrm{kN}$, $R_{G3}=350 \mathrm{kN}$, $R_{G4}=307 \mathrm{kN}$, $x_1=4.77 \mathrm{m}$, $x_2=4.77 \mathrm{m}$, $x_3=7.56 \mathrm{m}$, $x_4=7.56 \mathrm{m}$, $1+\mu =1.42$, $q_{\mathrm{k}}=10.5 \mathrm{kN}/\mathrm{m}$, $\Omega =106.2 {\mathrm{m}}^2$, $P_{\mathrm{k}}=260 \mathrm{kN}$, $e=3.04 \mathrm{m}$.

By substituting the above parameter values into equation (15), the structural overturning stability function can be obtained as

$$Z=\text{24,824.4}x(1)\cdot x(2)-1.42\cdot K\cdot x(3)\cdot [1115.1x(4)\cdot x(5)+790.4x(6)\cdot x(7)]$$

(15)

See

Table 3. Statistical properties of random variables.

Number	Random Variable	Probability Distribution Type	Average	Coefficient of Variation
x(1)	$R_{\mathrm{Gi}}/\mathrm{kN}$	Normal distribution	1.0148	0.0431
x(2)	$x_i/\mathrm{m}$	Normal distribution	0.9992	0.0061
x(3)	$1+\mu$	Extreme value type I Distribution	1.1957	0.0531
x(4)	$q_{\mathrm{k}}/(\mathrm{kN}/\mathrm{m})$	Extreme value type I Distribution	0.7995	0.0862
x(5)	$\Omega /{\mathrm{m}}^2$	Normal distribution	1.0059	0.0465
x(6)	$P_{\mathrm{k}}/\mathrm{kN}$	Extreme value type I Distribution	0.7995	0.0862
x(7)	$e/\mathrm{m}$	Normal distribution	0.9942	0.0218

for the statistical parameter values of each random variable [29].

5.1. Safety Factor Analysis Results

The calculation of the overturning stability safety factor of a single-column pier curved bridge under asymmetric eccentric load based on reliability back analysis theory solves for the overturning stability safety factor under the given target reliability index. Therefore, it is necessary to clarify the value of the target reliability index first. The target reliability index of components specified in the OHBDC highway bridge design code of Ontario, Canada and in the AASHTO code of the United States is 3.5. Therefore, unless otherwise specified, the target reliability index of the structure is 3.5, and the initial value of $K$ is 10.

In this example, the target reliability index of a curved bridge under asymmetric eccentric load is 3.5, and the calculated overturning stability safety factor is 6.555, which is greater than the specification limit of 2.5. The safety factor calculated based on the deterministic model is 9.174, the absolute error is 2.619, and the relative error is 39.95%. The overturning stability safety factor calculated by inverse reliability analysis is significantly less than that calculated by the deterministic model; this is because parameter uncertainty cannot be considered in the calculation of the deterministic model, which indicates that the parameter uncertainty has a great impact on the overturning stability safety factor. If the parameter uncertainty is ignored in the calculation, the overturning stability safety factor will be overestimated, which may lead to insufficient safety reserves of the structure.

5.2. Impact of Target Reliability Index

The calculation of the anti-overturning stability coefficient of a single-column pier curved bridge under asymmetric eccentric load based on reliability back analysis theory reverses the overturning stability safety coefficient under a certain reliability level. There is a corresponding relationship between the target reliability index and the overturning stability safety coefficient. Therefore, it is necessary to study the influence of the target reliability index on the overturning stability safety coefficient and the relationship between them. The value range of the structural target reliability index is 2.5~5.5, so as to analyze the impact of target reliability on the safety factor of overturning stability. The specific calculation results are shown in

Table 4. Influence of the Reliability Index on the Stability Safety Factor.

Target reliability index	2.5	3	3.5	4	4.5	5	5.5
Overturning safety factor	7.4952	7.0261	6.5549	6.0853	5.6234	5.1742	4.7424

.

It can be seen from the analysis in Table 4 that with increasing target reliability index, the safety factor of overturning stability shows a decreasing trend, indicating that with an increase in the target reliability index (as the failure probability gradually decreases), the actual required anti-overturning performance of the single-column pier curved bridge under asymmetric eccentric load gradually increases, and the safety reserve of the single-column pier curved bridge under asymmetric eccentric load gradually decreases. The anti-overturning stability coefficient calculated via inverse reliability analysis under each target reliability index is less than that calculated by the deterministic model, indicating that the parameter uncertainty has a great impact on the anti-overturning stability coefficient. Ignoring the parameter uncertainty will lead to an overestimation of the anti-overturning stability coefficient.

From the analysis of the above calculation results, it can be seen that the calculation of the overturning stability safety factor of a single-column pier curved bridge under asymmetric eccentric load based on reliability back analysis theory can not only consider the influence of parameter uncertainty, but also determine the overturning stability safety factor under different reliability levels and establish the relationship between the target reliability index and the overturning stability safety factor. By correcting the overturning stability safety factor to meet the preset reliability level, an evaluation system with dual control indexes of the target reliability index and anti-overturning stability coefficient is established.

5.3. Influence of Parameter Uncertainty on the Safety Factor

According to the analysis results of the overturning stability safety factor of a single-column pier curved bridge under asymmetric eccentric load using the deterministic model, a large overturning stability safety factor is obtained because the influence of parameter randomness is ignored. In order to analyze the influence of parameter randomness on the safety factor of overturning stability, we take the target reliability index of the structure as 3.5, increase the mean value of each random variable by −15% to +15%, and increase the coefficient of variation by 0.5~2 times [30,31,32]. The calculation results of the safety factor of overturning stability are shown in

Table 5. Effect of the mean values of random variables.

Mean Value	Safety Factor
	x(1)	x(2)	x(3)	x(4)	x(5)	x(6)	x(7)
	${\mathit{R}}_{\mathbf{G}\mathbf{i}}/\mathbf{k}\mathbf{N}$	${\mathit{x}}_{\mathit{i}}/\mathbf{m}$	$1+\mathit{\mu }$	${\mathit{q}}_{\mathbf{k}}/(\mathbf{k}\mathbf{N}/\mathbf{m})$	$\mathbf{\Omega }/{\mathbf{m}}^2$	${\mathit{P}}_{\mathbf{k}}/\mathbf{k}\mathbf{N}$	$\mathit{e}/\mathbf{m}$
−15%	5.5717	5.5717	7.7117	7.2387	7.2387	6.9273	6.9273
0	6.5549	6.5549	6.5549	6.5549	6.5549	6.5549	6.5549
+15%	7.5382	7.5382	5.6999	5.9801	5.9801	6.2113	6.2113

and

Table 6. Effect of the coefficients of variation of random variables.

Cov	Safety Factor
	x(1)	x(2)	x(3)	x(4)	x(5)	x(6)	x(7)
	${\mathit{R}}_{\mathbf{G}\mathbf{i}}/\mathbf{k}\mathbf{N}$	${\mathit{x}}_{\mathit{i}}/\mathbf{m}$	$1+\mathit{\mu }$	${\mathit{q}}_{\mathbf{k}}/(\mathbf{k}\mathbf{N}/\mathbf{m})$	$\mathbf{\Omega }/{\mathbf{m}}^2$	${\mathit{P}}_{\mathbf{k}}/\mathbf{k}\mathbf{N}$	$\mathit{e}/\mathbf{m}$
0.5	6.6752	6.5572	6.8591	6.7958	6.6048	6.5990	6.5586
1	6.5549	6.5549	6.5549	6.5549	6.5549	6.5549	6.5549
2	6.0278	6.5460	5.5291	5.5865	6.3739	6.1890	6.5397

and Figure 2 and Figure 3.

By analyzing the influence of the mean values of random variables on the safety factor, it can be seen that the safety factor increases with increasing mean value and decreases with increasing mean values of $\mu$, $q_{\mathrm{k}}$, $\Omega$, $P_{\mathrm{k}}$, and $e$. By analyzing the influence of the coefficients of variation of random variables on the safety factor, it can be seen that the safety factor decreases with increasing variability in $R_{\mathrm{Gi}}$, $x_i$, $\mu$, $q_{\mathrm{k}}$, $\Omega$, $P_{\mathrm{k}}$, and $e$. A comprehensive analysis of the impact of parameter uncertainty on the safety factor shows that the vehicle load has the greatest impact on the safety factor. In the actual operation process of the bridge, the vehicle load should be strictly controlled to prevent the bridge from overturning due to overload.

In conclusion, the randomness of parameters has an important impact on the quantitative description of safety factor analysis results. The parameter values of bridges in this paper refer to the existing specifications and engineering data of the same type, which have strong applicability and reliability.

5.4. Influence of Safety Factor on the Iterative Initial Value

In the process of reliability back analysis, the initial value $K^0$ of the transverse overturning stability safety factor of a single-column pier curved bridge under asymmetric eccentric load is arbitrarily selected, so it is necessary to analyze the influence of the initial value $K$ on the overturning stability safety factor. For initial $K$ values of 7, 8, 9, 10, 11, 12, and 13, the corresponding safety factor calculation results are shown in Figure 4 (the abscissa in the figure represents the number of iterations).

It can be seen from the analysis in Figure 4 that the selection of the initial value of the safety factor has no impact on the accuracy of the calculation results of the safety factor, only an impact on the convergence speed. This shows that the method proposed in this paper has high stability and can be applied to the calculation of the transverse overturning safety factor of a single-column pier curved bridge under asymmetric eccentric load.

6. Conclusions

In this paper, a research method to determine the lateral overturning stability safety factor of a single-column pier curved bridge under asymmetric eccentric load based on reliability back analysis was proposed; the existing calculation methods for the lateral overturning stability safety factor of a single-column pier curved bridge under asymmetric eccentric load in highway bridge codes use a deterministic calculation model. The calculated safety factor considers the influence of randomness of the structural parameters on the premise of meeting the target structural reliability index. A new method was proposed herein to calculate the design parameter of the safety factor of single-pier instability, and then an application was carried out. Firstly, the accuracy and effectiveness of this method were verified by an example, and then an example was analyzed; the following conclusions are drawn:

(1)

The randomness of parameters has a great impact on the safety factor of lateral overturning stability. For the structure evaluated via the deterministic method, randomness of the load will reduce the overturning stability safety.

(2)

The safety factor decreases with increasing structural target reliability index. The appropriate safety factor can be selected according to the target performance requirements.

(3)

During the actual operation of the bridge, the vehicle load should be strictly controlled to prevent the bridge from overturning due to overload.

(4)

The selection of the initial safety factor value has no effect on the accuracy of the analysis results, only an impact on the convergence speed of the results. The reliability back analysis method recommended in this paper has good applicability to determine the safety factor of overturning stability based on the target reliability index.

(5)

The proposed method has the advantages of calculating the safety factors considering the randomness of parameters and establishing the relationship of the safety factor with the target reliability index.

In this study, the lateral overturning stability of a single-column pier was treated as a rigid body problem. In fact, the elastic–plastic deformation of bearings has a certain impact on the overturning stability, and specific quantitative analysis of this needs to be carried out in further study.