Application of Bayesian Update Method in the Construction Control of Continuous Rigid Frame Bridge Girders with High Piers and Large Spans

Abstract

In the construction process of large-scale bridges, there are uncertainties and time-varying factors in the environment and construction loads. It is difficult to make accurate estimates of the theoretical calculation models of construction control in advance. In view of this situation, Bayesian dynamic updating method is introduced to re-estimate the predicted results of the theoretical model. When applying this method, first, the finite element calculation model is determined based on the response surface method, and its calculation results are used as prior information. Then, combined with the actual detection data during the construction process, the Bayesian update formula is derived based on the conjugate prior distribution to correct the theoretical prediction results of bridge construction monitoring. Finally, the actual stress detection data of the control section of high-pier and large-span continuous rigid frame bridges during the construction process illustrate the application process of Bayesian updating in improving the theoretical prediction model. Results indicate that the internal force of the bridge control section obtained by re-evaluating by Bayesian theory not only incorporates the priori information models but also actual monitors sample information during the construction process. The predicted results reflect the true deformation and stress state of the bridge during the bridge construction process and improve the precision of construction monitoring.

1. Introduction

With its advantages of convenient construction, reasonable force transmission and strong crossing ability, high-pier and long-span continuous rigid-frame bridges have become an important bridge type for crossing ravines and canyons in the southwestern mountainous regions of China [1,2]. The internal forces and deformations of the structure are the main control indicators during the cantilever construction process of continuous rigid-frame bridges, ensuring the reasonable completion of the bridge [3,4,5,6]. In existing bridge construction control, the results of finite element simulation analysis are widely used as the expected values for construction control [7,8]. However, due to various factors during the construction stage of bridges such as environmental conditions, material properties, geometric defects and physical parameter errors, the predicted results of finite element analysis for continuous rigid-frame bridges are frequently inconsistent with the actual construction state of the bridge [8,9,10]. To improve control accuracy, scholars both domestically and internationally have conducted effective research on bridge construction control theories and technologies. Zhang [11], Liu [12], and Liu [13] introduced grey theory into the assessment of bridge construction states. Ren [8] and Niu [14] proposed a finite element model updating method for bridges based on response surface methodology. Wang [15] and Bai [16] developed a deflection prediction model for continuous rigid-frame bridges based on the MEC-BP neural network. Sarwar [17], Xin [18], Chen [19], Cho [20], and Zhang [21] applied Kalman filtering estimation to predict the structural deformation in bridge construction. Innocenzi [22] established an improved bridge fine-scale finite element analysis model by using shell elements for all members in ANSYS. Nicoletti [23] calibrates the analytical model by building a digital twin model and continuously changing the structural input parameters of AM according to the current state of the bridge. These research results, which update the theoretical analysis model from the aspects of mathematical methods, refined finite element models, and digital twin models, have updated the theoretical analysis models and significantly improved the accuracy of bridge construction monitoring and prediction results. the numerous factors that can influence the calculation results of theoretical models for bridges such as environmental temperature, prestress loss, shrinkage, creep, and elastic modulus are characterized by their variability and time dependence. Although the above method can establish the finite element analysis model more accurately, the parameter correction of the theoretical calculation model cannot take into account all the influencing factors at the same time. Therefore, in order to predict the control indicators and parameters during bridge construction, it is necessary to find a method that can minimize the effects of these multi-factor uncertainties. Bayesian theory provides a method for an effective solution to this contradiction [24,25,26,27,28,29,30]. The basic idea [31,32] of Bayesian theory for parameter prediction is to emphasize the collection of prior information to form a prior distribution, make full use of existing sample observations, and then use Bayesian updates to combine prior information and actual sample observations to improve statistical forecast quality.

In the construction process of continuous rigid-frame bridges, the finite element analysis results of cross-sectional internal forces, despite potential deviations due to parameter variations or errors, can still reflect the patterns of real results. These calculation results can be used as prior information for predictions [33,34,35]. In this paper, Bayesian updating is applied to calculate and dynamically predict the internal forces and deformations of the control sections of high-pier and long-span continuous rigid-frame bridges based on the results of finite element model calculations and field monitoring data.

2. Determination of Internal Forces and Prior Information Model in Bridge Construction Sections

2.1. Preliminary Determination of a Priori Information of Internal Forces in Cross-Section

Continuous rigid-frame bridges with high piers and long spans are mainly cast with cantilever hanging baskets and self-erected section by section symmetrically. Due to the large spans, the cantilever construction causes obvious deflection. However, due to the high rigidity of this type of bridge, it is difficult to adjust the geometric alignment of the cantilever structure after the completion of construction. Therefore, it is essential to analyze the primary construction monitoring indicators and their influencing factors for the high-pier and long-span continuous rigid-frame bridge before and during construction and develop solutions in advance. Compared to the geometric monitoring of bridges, monitoring internal forces within the cross-section is more challenging and subject to a greater number of uncertain factors. Deviations in internal forces not only affect the geometric alignment of the structure but can also have severe implications for its safety, potentially leading to strength or instability failure in continuous rigid-frame bridges. This section primarily introduces the preliminary determination methods for the a priori information of internal forces in bridge construction.

Considering the characteristics of high-pier and long-span continuous rigid-frame bridges, the prior information of the section internal force in bridge construction can be determined through a forward analysis method and verified through a backward analysis method. When the forward analysis method determines the prior information of the control index, the finite displacement theory is mainly used for stage cycle analysis currently, and the calculation results are relatively accurate [36]. However, it is important to keep in mind the following five points to guarantee accurate results for the construction process analysis:

• The actual construction process and procedures should be strictly followed in the bridge structure forward analysis.
• The backward analysis based on the results of the actual construction should be used to determine the initial state of the structural analysis.
• The initial state of each stage analysis should be based on the results of the previous construction stage.
• The forward analysis procedure should gradually take the time lag effects such as concrete shrinkage, creep, prestress, and temperature, into account.
• The finite element model used in the forward analysis needs to be accurate enough to accurately represent the current state of the continuous rigid-frame bridge.

2.2. Modification of a Priori Information Model for Bridge Construction Cross-Sectional Internal Forces

In order to accurately reflect the structural characteristics of the high-pier and long-span continuous rigid frame bridge, the finite element model established based on the design drawings must be modified to take into account the effects of construction errors, model simplification, and parameter changes. The model modification can use the finite element model modification method based on the response surface methodology [37,38,39], whose basic principle is to select appropriate structural parameters and responses within a reasonable range of structural parameters, and fit response surface models between structural parameters and structural responses by calculating the response at sample points by varying parameter values within the design space. Taking the measured response as the target, the response surface model is used to replace the initial finite element model for parameter optimization, and finally the final correction value and correction model of the parameters are obtained.

Figure 1 depicts the specific process of the finite element model modification.

The calculation results of cross-sectional internal forces obtained by the modified finite element analysis model can serve as the prior information for controlling the internal forces at various sections during bridge construction.

Figure 2 depicts the method for determining the prior information of the internal forces of bridge sections.

3. Correction and Prediction of Construction Monitoring Results for Continuous Rigid Frame Bridges

According to the central limit theorem [40,41], if variables follow similar distributions and have weak correlations, their means will approximately follow a normal distribution. There are many factors that affect the internal forces characteristics of continuous rigid frame bridges during construction such as environmental temperature, elastic modulus of concrete materials, shrinkage and creep, section size, construction loads, and concrete compactness. These factors are independent of each other and their detection results follow an approximately normal distribution during the bridge construction. Therefore, the random distribution model of predicted results for the internal forces of the bridge section can be considered to follow a normal distribution. The statistical parameters (variance σ² and mean value μ) of on-site test samples for the section internal forces of the rigid frame bridge are not constant during different construction stages. Additionally, the variance of prior samples cannot be determined when dynamically updating and predicting detection data for the first time. Consequently, when using Bayesian theory to estimate the section of internal forces of the rigid frame bridge during the construction process (i.e., correcting the theoretical calculation results for both), it is necessary to take into account two working conditions. The first condition involves predicting the posterior distribution of the mean when both the sample variance and mean are unknown, while the second one involves predicting the posterior distribution of the mean when the variance is known.

3.1. Bayesian Estimation When Population Sample Mean and Variance Are Unknown

The detection samples of section stress during the bridge construction process are denoted as x₁, x₂, x₃, …, x_n. It is assumed that they are drawn from the population sample N (θ, σ²), and both θ and σ² are unknown. The prior distributions for θ and σ² can be taken as normal-inverse-gamma prior [42], i.e., θ|σ²~N (μ, σ²/k), σ²~Γ⁻¹(r/2, λ/2), where μ, k, r, λ are known hyperparameters. Equation (1) represents the likelihood function of (θ, σ²).

$$l\left(\theta ,{\sigma }^2|x\right)\propto {\sigma }^{-n}\exp \left\{-\frac{1}{2{\sigma }^2}\left[{\displaystyle \sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}+n{(\overline{x}-\theta )}^2\right]\right\}$$

(1)

The joint density of the normal-inverse-gamma prior distribution for (θ, σ²) is shown in Equation (2) [43]:

$$\begin{gathered} \pi \left(\theta ,{\sigma }^2\right)={\pi }_1\left(\theta |{\sigma }^2\right)\cdot {\pi }_2\left(\theta |{\sigma }^2\right) \\ \propto {\left({\sigma }^2\right)}^{-\left[\left(r+1\right)/2+1\right]}\exp \left\{-1/\left(2{\sigma }^2\right)\cdot \left[k{\left(\theta -\mu \right)}^2+\lambda \right]\right\} \end{gathered}$$

(2)

According to Equations (1) and (2), the joint posterior density of parameters θ and σ² is shown in Equation (3):

$$\begin{gathered} \pi \left(\theta ,{\sigma }^2|x\right)\propto l\left(\theta ,{\sigma }^2|x\right)\cdot \pi \left(\theta ,{\sigma }^2\right) \\ \propto {\left({\sigma }^2\right)}^{-\frac{r+n+1}{2}+1}\exp \left\{-\frac{1}{2{\sigma }^2}\cdot \left[\left(n-1\right)s^2+\lambda +H\right]\right\} \end{gathered}$$

(3)

where

$$\begin{gathered} H=n{\left(\overline{x}-\theta \right)}^2+k{\left(\theta -\mu \right)}^2 \\ =\left(n+k\right){\left(\theta -\frac{n\overline{x}+k\mu }{n+k}\right)}^2+\frac{nk}{n+k}{\left(\overline{x}-\mu \right)}^2 \end{gathered}$$

If v_n = n + r, k_n = n + k, Equation (3) could be obtained by transformation:

$$\pi \left(\theta ,{\sigma }^2|x\right)\propto {\left({\sigma }^2\right)}^{-\frac{{\nu }_n+1}{2}+1}\cdot e^{-\frac{1}{2{\sigma }^2}\left[({\nu }_n{\sigma }_n^2+k_n{\left(\theta -\mu \left(\overline{x}\right)\right)}^2\right]}$$

(4)

where

$$\begin{gathered} \mu \left(\overline{x}\right)=\frac{n\overline{x}+k\mu }{n+k} \\ {\nu }_n{\sigma }_n^2=\left(n-1\right)s^2+\lambda +\frac{nk}{n+k}{\left(\overline{x}-\mu \right)}^2 \end{gathered}$$

(5)

According to Equations (4) and (5), when both the mean and variance of the population sample are unknown, Equation (5) can be used to predict the internal forces during the construction process of the continuous rigid-frame bridge. In Equation (5), μ($\overline{x}$) represents the predicted or adjusted result of the internal forces of the bridge during the construction process at time t, which μ can be understood as the theoretical solution obtained by numerical analysis or theoretical calculation at time t, $\overline{x}$ is the mean value of the on-site detection sample, k is the prior degree of freedom of θ, which is determined by the confidence level of the prior information and can be set to the sample size of the first detection data, and n is the actual number of detection samples, which can be set to the sample size at time t or the number of samples in the second detection.

3.2. Bayesian Estimation When Population Sample Variance Is Known

The population samples of section stress are denoted as X~(θ, σ²), where σ² is known, while θ is unknown. The prior distribution π(θ) of θ obeys N(μ, τ²), where μ and τ² are determined through statistical analysis of the actual monitoring results of cross-sectional stress at time t (second monitoring).

Therefore, for the posterior distribution π(θ|x) of θ,

$$\begin{gathered} \pi \left(\theta |x\right)\propto f\left(x|\theta \right)\pi \left(\theta \right) \\ \propto \exp \left\{-\frac{1}{2}\left[\frac{{\left(x-\theta \right)}^2}{{\sigma }^2}+\frac{{\left(\theta -\mu \right)}^2}{{\tau }^2}\right]\right\} \end{gathered}$$

(6)

If ρ = 1/τ² + 1/σ²,

$$\pi \left(\theta |x\right)\propto \exp \left\{-\frac{\rho }{2}{\left[\theta -\frac{1}{\rho }\left(\frac{\mu }{{\tau }^2}+\frac{x}{{\sigma }^2}\right)\right]}^2\right\}$$

(7)

$$\mu \left(x\right)=\frac{1}{\rho }(\frac{\mu }{{\tau }^2}+\frac{x}{{\sigma }^2})=\frac{{\sigma }^2}{{\sigma }^2+{\tau }^2}\mu +\frac{{\tau }^2}{{\sigma }^2+{\tau }^2}x$$

(8)

It is evident that, when σ² is known, given mean μ and variance τ² of the prior information, the accurate values of the internal forces in bridge construction can be predicted using Equation (8), where x is obtained through numerical calculation or theoretical solution.

Additionally, if the third inspection sample x₁, …, x_n at time t + 1 follows the normal distribution N(θ, σ²), the joint density of x = (x₁, …, x_n) is represented by Equation (9):

$$f\left(x|\theta \right)={\left(2\pi {\sigma }^2\right)}^{-n/2}\exp \left\{-\frac{1}{2{\sigma }^2}{\displaystyle \sum _{i=1}^n{\left(x_i-\theta \right)}^2}\right\}$$

(9)

In Equation (7), when x is replaced by $\overline{x}$ ($\overline{x}$ is the mean value of samples x₁, …, x_n, i.e., the mean value of on-site detection samples), and σ² is replaced by σ²/n, the posterior distribution π(θ|x) of θ can be obtained as follows:

$$\pi \left(\theta |x\right)=\frac{1}{\sqrt{2\pi }{\eta }_n}\exp \left\{-\frac{1}{2{\eta }_n^2}{\left[\theta -{\mu }_n\left(\overline{x}\right)\right]}^2\right\}$$

(10)

where

$$\begin{gathered} {\mu }_n\left(\overline{x}\right)=\frac{{\sigma }^2/n}{{\sigma }^2/n+{\tau }^2}\mu +\frac{{\tau }^2}{{\sigma }^2/n+{\tau }^2}\overline{x} \\ {\eta }_n^2=\frac{{\sigma }^2{\tau }^2}{n{\tau }^2+{\sigma }^2} \end{gathered}$$

(11)

Based on Equations (10) and (11), it is possible to predict the internal forces of a continuously rigid frame bridge during the construction process, given the variance σ² of the population sample, using Equation (11). In Equation (11), μ_n($\overline{x}$) represents the predicted or adjusted results of internal forces at time t during the construction process of the continuously rigid frame bridge. σ² can be obtained through statistical analysis of the initial inspection sample. τ² and n represent the variance and sample size of the second inspection sample, respectively. μ is the theoretical solution obtained by theoretical calculation or numerical analysis at time t, and $\overline{x}$ is the mean value of the second inspection sample.

3.3. Bayesian Update of Construction Monitoring Indexes for Continuous Rigid Frame Bridges

From the perspective of Bayesian prediction theory, Equation (5) provides the revised value or prediction value of the mean μ(x) when the population sample mean and variance (θ and σ²) cannot be accurately determined, based on the prior sample mean μ or theoretical calculated value. Equation (11) provides the mean value μ(x) based on the prior sample mean μ (or theoretical calculated value) and the information of the detection sample (sample size n and sample mean $\overline{x}$), when the population sample variance σ² is known and the mean value θ is unknown.

By analyzing Equations (5) and (11), it can be concluded that in the process of bridge construction monitoring, the predicted values of the internal forces of the section can be represented as weighted results of the detection sample mean and theoretical calculated results. When the variance and mean are unknown, the weighting factors for the detection sample mean and theoretical calculated result are, respectively, the following: k/(n + k) and n/(n + k). When the variance is known, the weighting factors for the detection sample mean and theoretical calculated result are, respectively, the following: σ²/n/(σ²/n + τ²) and τ²/(σ²/n + τ²). To standardize the expression, if λ₁ = n/(n + k) or λ₂ = τ²/(σ²/n + τ²), Equation (5) can be transformed into the following: μ($\overline{x}$) = λ₁$\overline{x}$ + (1 − λ₁)μ = μ + λ₁($\overline{x}$ − μ), Equation (11) can be transformed into: μ_n($\overline{x}$) = (1 − λ₂)μ + λ₂$\overline{x}$ = μ + λ₂($\overline{x}$ − μ). Hence, the Bayesian update model at a future time can be expressed as the following:

$${\mu }_i\left(\overline{x}\right)=\mu +{\lambda }_i\left({\overline{x}}_0-{\mu }_0\right)$$

(12)

where i = 1 or 2, and they represent the two working conditions of Equations (5) and (11), respectively. In these equations, μ_i($\overline{x}$) represents the predicted value of the section stress of the section of the high-pier and long-span continuous rigid structure bridge in a certain construction stage. ${\overline{x}}_0$ represents the mean of the detected sample of the section stress at time t₀. μ₀ represents the predicted value of the prior model of internal force at time t₀. μ represents the predicted results of the section stress at time t obtained through finite element analysis. When using Equation (12) to modify the monitoring results during the construction process, the key lies in determining the weighting coefficients λ_i. For the condition of unknown variance, according to the existing statistical results, the value of k can be the number of samples required to satisfy a certain confidence level condition, while n can be selected corresponding to the number of samples for the sample mean $\overline{x}$. Additionally, based on statistical analysis of multiple previous monitoring results, the existing sample variance can be approximated as the population sample variance, allowing the use of λ₂ for Bayesian prediction. In this case, σ² can be calculated using Equation (13), and n and τ² represent the sample size and variance, respectively, corresponding to the monitoring samples.

$${\sigma }^2={\widehat{\sigma }}^2=\frac{1}{n}{\displaystyle \sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}$$

(13)

Figure 3 illustrates the updating process of concrete carbonation depth based on Bayesian method.

4. Analysis of a Calculation Example

The new prediction model, which is based on Bayesian theory, takes into account the calculation results of the original theoretical model as well as the results of the on-site monitoring data collected throughout the construction process. To illustrate the process of using Bayesian theory to predict the theoretical data of construction monitoring indicators for high-pier and large-span continuous rigid-frame bridges and examine its effectiveness, this paper introduces and analyzes the construction monitoring results of two stress control sections of the Qingshuijiang high-pier and large-span continuous rigid-frame bridge on the Kaixi Ring Expressway in Guizhou Province.

Project Overview

The Qingshuijiang Bridge is located in the northern section of the Kaili Ring Expressway in Guizhou Province. The main span of the bridge is a prestressed concrete continuous rigid frame bridge with a length of (81 + 150 + 150 + 81) m. The route is divided into sections, and the left and right sections are structurally independent. The height of the main pier is 112 m, and the box beam adopts a variable cross-section single-box flat section. Considering that the main factors affecting the stress state of a large-span prestressed concrete continuous rigid-frame bridge are the sectional stress of the piers and main beams, this paper takes the stress test results of the 5# pier Section 2-2 (the cantilever root section of the main beam) and the 6# pier Section 4-4 (the main beam L/4 section) after pouring and tensioning in each section of the top plate as examples for analysis, and the application of Bayesian updating in the dynamic prediction of construction control indicators of high-pier and large-span continuous rigid-frame bridges is explained in detail. Figure 4 depicts the longitudinal layout of the main beam stress (strain) test section.

The numbering of the cantilever construction beam sections for the main beam is shown in Figure 5.

During the construction monitoring process of this project, two Bayesian updates were conducted for stress prediction of the 5# pier Section 2-2 and 6# pier Section 4-4. The first Bayesian update for the 5# pier Section 2-2 was conducted after the completion of casting and tensioning block 1, and the second update was conducted after the completion of block 6. The first Bayesian update for the 6# pier Section 4-4 was conducted after the completion of casting and tensioning block 9, and the second update was conducted after the completion of block 12. Figure 6 depicts the stress monitoring points for each section. The stress values of the top plates at control Sections 2-2 and 4-4 during cantilever construction of the main girder are shown in

Table 1. The measured value of the roof stress of the main beam cantilever construction control Section 2-2/4-4.

Main Beam Cantilever in the Construction Stage	Left 5# Pier Control Section 2-2		Left 6# Pier Control Section 4-4
	Stress after Pouring (MPa)	Stress after Tensioning (MPa)	Stress after Pouring (MPa)	Stress after Tensioning (MPa)
1#	−0.40825	−0.76325
2#	−0.33725	−1.3845
3#	−2.28975	−3.692
4#	−3.56775	−5.041
5#	−4.26	−6.51425
6#	−4.828	−6.51425
7#	−6.47875	−7.171
8#	−6.44325	−7.93425
9#	−7.93425	−8.53775	−1.65075	−2.39625
10#	−7.739	−9.28325	−1.97025	−3.28375
11#	−8.67975	−10.1885	−3.124	−5.2895
12#	−9.35425	−10.63225	−4.544	−6.3545
13#	−9.46075	−10.20625	−5.254	−7.40175
14#	−9.6205	−10.437	−6.39	−8.2005
15#	−9.46075	−10.66775	−7.08225	−8.53775
16#	−9.7625	−10.721	−7.526	−8.875
17#	−9.92225	−10.4725	−7.597	−8.165

, and the data in the table represent the average measured values of the two stress measurement points on the top plate.

The Bayesian estimate weight coefficients at various stages can be obtained by thoroughly analyzing all the stress monitoring data of Sections 2-2 and 4-4 under various construction conditions, as shown in

Table 2. Bayesian update weights of control section stress in different construction stages.

Section	Construction Stage	Bayesian Updating Phase	Weight Coefficient
Section 2-2	Concrete placement	The first update	0.3497
		The second update	0.3923
	Prestressed tension	The first update	0.3223
		The second update	0.3423
Section 4-4	Concrete placement	The first update	0.4444
		The second update	0.4589
	Prestressed tension	The first update	0.4317
		The second update	0.4320

.

Figure 7, Figure 8, Figure 9 and Figure 10 show the predicted curves of the theoretical calculation results for Sections 2-2 and 4-4 after Bayesian updating. They are compared to the theoretical results and actual test values. The predicted curve for the stress values at Section 2-2 (4-4) after the first Bayesian update was determined by modifying the model based on the detected data after pouring and tensioning of block 1# (9#). The predicted curve of the second Bayesian update was determined after model correction based on the detection data obtained after pouring and tensioning of block 6# (12#).

From Figure 7, Figure 8, Figure 9 and Figure 10, conclusions could be observed as follows: (1) There is a large discrepancy between the control section stress obtained directly from the initial finite element calculation model and the actual measured stress during the construction process, especially for Section 2-2 (main beam root section). The error in stress increases as the construction segment progresses, and the maximum error reaches 16.25% (after pouring) and 15.87% (after prestressing). (2) The predicted curve after Bayesian updating is in good agreement with the development law of the theoretical analysis prediction curve, and the prediction result becomes closer to the measured value after each Bayesian model update, which can better reflect the stress state of the control section. (3) The model’s accuracy of prediction can be significantly increased following each Bayesian update. For example, for Section 2-2, the error between the measured value and theoretical value of section stress monitored after concrete prestressing was reduced from 15.87% to 12.21% and 7.75% under the conditions of no Bayesian update and once and double Bayesian update, respectively. For Section 4-4, the error was reduced from 10.47% to 6.80% and 4.25%, respectively. The above analysis results show that the Bayesian update that integrates theoretical models and actual measured data has a good prediction effect. It is necessary to introduce Bayesian updating to re-evaluate the calculation results of conventional theoretical models in light of the construction environment of high-pier and long-span continuous rigid frame bridges and the numerous uncertain factors that affect the bridge stress state.

5. Conclusions

In order to reduce the theoretical calculation error of the control indicators for construction monitoring of continuous rigid-frame bridges with high piers and long spans, research on parameter re-estimation based on Bayesian theory was carried out.

• A Bayesian update method is established under the condition that the prior information sample mean and variance are unknown, as well as that the prior information sample variance is known but the mean value is unknown. The prediction formula for the sample mean value can be adopted as Equations (5) and (11), respectively. Bayesian update can quickly reduce the error between theoretical prediction results and measured values, and better prediction results can be achieved after fewer times of Bayesian updates.
• The estimation of bridge construction monitoring and control indicators based on Bayesian updating simultaneously integrates prior information and detection information obtained during the actual construction process. The prediction curve is consistent with the theoretical calculation curve, and the prediction results can more accurately reflect the stress and linear state during the bridge construction process.
• The finite element calculation model corrected based on the response surface method has better prior information, higher prediction effect and efficiency, and requires fewer updates to achieve higher prediction accuracy.
• The results of the finite element analysis of cross-sectional stresses at various sections during bridge construction follow a similar pattern to the measured values after two rounds of Bayesian updates. The prediction accuracy and efficiency are significantly increased as a result of the error being cut in half. These research results can serve as a theoretical basis for the construction control of continuous rigid-frame bridges.