The Randomness Analysis of Shrinkage and Creep Mechanical Behavior in Continuous Rigid-Frame Bridges with Ultra-High Piers

Abstract

To quantify the influence of shrinkage and creep randomness on the long-term deformation and closure pushing force of ultra-high pier continuous rigid-frame bridges, this study proposes a probabilistic analysis framework. Based on the CEB-FIP (1990) model, seven random factors were introduced to account for model and parameter uncertainties. The Response Surface Method (RSM) was employed to establish an explicit relationship between structural responses and random variables, followed by Monte Carlo simulation for probabilistic analysis, using the Lugou River Grand Bridge as a case study. Results show that the quadratic response surface fits the finite element results accurately (R² > 0.9). The probability distribution of pier-top displacement approximately follows a log-normal distribution, while mid-span deflection and pier-base moment are approximately normally distributed. The pushing force when considering randomness is 1.16 times the deterministic value, a difference that cannot be ignored. The proposed method provides a probabilistic basis for determining the pushing force, overcoming the limitations of traditional deterministic analysis.

1. Introduction

With the continuous expansion of transportation infrastructure into complex mountainous areas, long-span continuous rigid-frame bridges have been widely adopted in modern bridge engineering due to their excellent spanning capacity and structural robustness [1]. To accommodate challenging topographical conditions, the pier heights of such bridges have steadily increased, giving rise to the distinct structural characteristics of high piers and long spans. However, the resulting increase in structural flexibility also makes these bridges more sensitive to time-dependent effects of concrete. The frequently observed progressive deflection of the main girder during long-term service has become a critical issue threatening the safety and service life of these structures. To improve the internal force distribution and geometric alignment of the completed bridge, applying a pushing force during the closure phase is recognized as an essential structural control measure [2,3,4]. Nevertheless, the appropriateness of the pushing force magnitude heavily depends on the accurate prediction of long-term structural deformation, especially the effects of concrete shrinkage and creep.

For long-term deformation prediction, current engineering practice mainly relies on deterministic analysis methods that employ the statistical mean values derived from experiments. The shrinkage and creep models recommended by various design codes (e.g., ACI209(92) [5], CEB-FIP1990 [6], GL2000 [7], AASHTO [8], B4 [9], and FIB2010 [10]) fall into this category of deterministic models. However, as pointed out by Bazant [11], the influencing factors for concrete shrinkage and creep are numerous, and their evolution exhibits both time dependency and randomness, demonstrating high complexity. Diamantidis et al. [12] further highlighted, through comparative studies, significant discrepancies between deterministic predictions of concrete creep effects and measured data. They systematically attributed these discrepancies to model uncertainty, parameter variability, and the inherent randomness of microscopic physical mechanisms. These intrinsic stochastic factors prevent deterministic models from accurately reflecting the true time-dependent structural behavior [13], thereby significantly impacting the rational determination of the closure pushing force. In long-span continuous rigid-frame bridges with high piers, the increased span length and the use of flexible piers further amplify the structural sensitivity to shrinkage and creep, substantially increasing the difficulty of long-term deformation prediction. Existing research indicates that the deviation between deterministically calculated mid-span deflection values and field measurements can exceed 30%, revealing a non-negligible gap between theoretical models and actual structural response [14]. Consequently, relying solely on qualitative or deterministic analysis is insufficient to ensure the safety and serviceability of such bridges during construction and operation, making it imperative to account for the randomness inherent in long-term deformation in any predictive framework [15].

To overcome the limitations of deterministic analysis, probabilistic methods that account for both model and parameter uncertainties have emerged as an inevitable trend. To address model uncertainty, Criel et al. [16] proposed a weighting method that combines predictions from multiple classical creep models using optimal weights, aiming to leverage the strengths of individual models and enhance the consistency and reliability of long-term predictions. Regarding parameter uncertainty, Jia et al. [17] employed the Hamiltonian Markov Chain Monte Carlo (HMCMC) algorithm to perform Bayesian updating of the joint probability distribution of state variables, including creep, structural rigidity, and prestress loss, based on deflection monitoring data from a prestressed concrete bridge. This approach significantly reduced associated uncertainties and enabled a probability-based condition assessment. Similarly, Tonelli et al. [18] utilized the Markov Chain Monte Carlo (MCMC) method integrated with monitoring data from an actual bridge in Italy to update structural parameters based on the Bazant B3 model, effectively reducing prediction uncertainty and quantitatively enhancing the reliability assessment of the in-service bridge.

In recent years, surrogate modeling, machine learning, and Bayesian updating have advanced probabilistic reliability analysis of engineering structures. Surrogate models such as Kriging, neural networks, and response surface methods replace costly finite element simulations, improving uncertainty propagation efficiency. Zemed et al. [19] combined Kriging with support vector regression for time-dependent reliability and sensitivity analysis of RC bridges under creep, shrinkage, and traffic loads. Machine learning excels at capturing nonlinear mappings between random inputs and structural responses. Using LSTM and PCA, a surrogate procedure was proposed to predict the long-term behavior of large-scale PC bridges based on five critical variables [20]. Bayesian updating with monitoring data and stochastic sampling enables quantitative calibration of model parameters and reduction in multi-source uncertainty. Yan et al. [21] established a Bayesian framework to update the rigidity ratio of aging RC hollow slab bridges using deflection measurements, with a Kriging surrogate model accelerating MCMC sampling. A relevant study by Amini Pishro et al. [22] applied physics-informed machine learning to structural dynamics and seismic analysis. Nevertheless, most existing advanced surrogate models and machine learning algorithms require complex training or large sample databases, limiting their practical application to probabilistic analysis of long-span rigid-frame bridges [23].

However, although the aforementioned Bayesian methods that are based on high-fidelity models and sophisticated sampling algorithms offer high accuracy, their rigorous numerical simulation processes require extensive iterative computations, resulting in high computational costs. This hinders their direct application to probabilistic analysis and design optimization of large-scale bridge structures. To strike a balance between analytical thoroughness and engineering efficiency, this study proposes an RSM-MCS framework tailored for high-pier continuous rigid-frame bridges. It requires only 15 finite element runs, quantifies seven key sources of randomness (creep, shrinkage, strength, modulus, humidity, loading age, and self-weight), and provides a probabilistic pushing force for closure control. This paper introduces the Response Surface Method (RSM) as an efficient surrogate modeling strategy. Using the Lugou River Grand Bridge as a case study, a Midas finite element model is established to compute displacement responses induced by various factors. The study adopts the CEB-FIP (1990) model for shrinkage and creep, while considering the randomness associated with the model itself and relevant material parameters. On this basis, the RSM [24] is employed to construct an explicit functional relationship between displacements and the random variables. Combined with Monte Carlo sampling, this framework is used to systematically investigate the influence of shrinkage and creep randomness on the selection of the closure pushing force, thereby providing a technically sound and feasible approach for long-term deformation prediction that incorporates stochastic effects.

2. Random Analysis Method for Shrinkage and Creep

2.1. Probabilistic Modeling of Creep Random Variables

Current research categorizes the factors influencing concrete shrinkage and creep into internal (e.g., cement type, mix ratio, concrete strength) and external (e.g., environmental conditions, stress state, load duration). The inherent complexity and variability of these factors impart significant randomness to concrete shrinkage and creep. Therefore, this study adopts a probabilistic approach based on the CEB-FIP90 model. A total of seven random factors are introduced to quantitatively represent the primary uncertainties. Specifically, these factors encompass two distinct types: two account for the inherent model uncertainty of the prediction models themselves, while the remaining five represent the parameter randomness associated with material properties, environmental conditions, and loading conditions.

Regarding model uncertainty, to correct for the inherent bias in the prediction models for shrinkage and creep, which is independent of material and environmental factors [25], two global random factors, α₁ and α₂, are introduced. These factors are multiplied by the creep model and the shrinkage model, respectively, thereby incorporating this overall uncertainty into the long-term strain calculation. The core expression is given by

$$\epsilon (t,t_0)={\alpha }_1\sigma J(t,t_0)+{\alpha }_2{\epsilon }_{\mathrm{sh}}(t)={\alpha }_1\sigma \left({\displaystyle \frac{1}{E_c(t,t_0)}}+{\displaystyle \frac{\phi (t,t_0)}{E_c}}\right)+{\alpha }_2{\epsilon }_{\mathrm{sh}}(t)$$

(1)

In this equation,${ \alpha }_1$ represents the random factor for the creep model, and ${\alpha }_2$ represents the random factor for the shrinkage model; $t$ and $t_0$ represent the calculation age and the loading age, respectively; $J(t,t_0)$ represents the creep coefficient; $E_c({t,t}_0)$ represents the elastic modulus at the loading age; $E_c$ represents the elastic modulus at 28 days.

In terms of parameter randomness, five factors (α₃ to α₇) are incorporated to capture the stochastic nature of key inputs. Generally speaking, there exists a stochastic correlation between the compressive strength of concrete and its elastic modulus. The CEB-FIP90 shrinkage and creep model provides a deterministic relationship between the two, as shown in the following formula:

$$E_c=2.15\times 10^4{\left[{\displaystyle \frac{f_{cm}}{10}}\right]}^{{\displaystyle \frac{1}{3}}}(\mathrm{MPa})$$

(2)

In this study, a stochastic factor ${\alpha }_3$ is introduced to represent the mean compressive strength, and by multiplying this factor with ${\alpha }_4$ using the deterministic relationship formula, the stochastic correlation between concrete compressive strength and elastic modulus is expressed:

$$E_c(t_0)=E_c\sqrt{\mathit{exp}\left\{s\left(1-\sqrt{{\displaystyle \frac{28}{t_0}}}\right)\right\}}$$

(3)

$$E_c={\alpha }_4\left\{2.15\times 10^4{\left[{\displaystyle \frac{f_{cm}}{10}}\right]}^{{\displaystyle \frac{1}{3}}}\right\}(\mathrm{MPa})$$

(4)

This formula is adopted to describe the time-varying variation law of concrete elastic modulus with the initial loading age. Ec is the reference elastic modulus of concrete; s is the material fitting parameter of concrete.

$\phi (t,t_0)$ represents the creep coefficient, defined as follows:

$$\phi (t,t_0)={\varphi }_0{\beta }_c(t-t_0)$$

(5)

where ${\varphi }_0$ is the nominal creep coefficient, ${\beta }_c$ is the time coefficient of the creep process, t is the time of creep calculation, and $t_0$ is the loading age.

$${\varphi }_0={\varphi }_{RH}\beta (f_{cm})\beta (t_0)=\left[1+{\displaystyle \frac{1-H}{0.46(h/100{)}^{1/3}}}\right](5.3/\sqrt{f_{cm}/f_{cm0}}){\displaystyle \frac{1}{0.1+(t_0/t)}}$$

(6)

The creep development equation is expressed as

$${\varphi }_c(t-t_0)={\left[{\displaystyle \frac{(t-t_0)/t_0^{\prime }}{{\beta }_H+(t-t_0)/t_0^{\prime }}}\right]}^{0.3}$$

(7)

where h is the theoretical thickness of the component, with $h_0$ taken as 100 mm; $f_{cm}$ (28) is the average compressive strength of concrete at 28 days as specified by CEB-FIP 1990, $f_{cm}$ = $f_{ck}$ + 8 MPa, where $f_{ck}$ is the characteristic compressive strength of a cube; $f_{cm0}$ is taken as 10 MPa; H is the relative humidity (%) of the environment; and $t_0^{\prime }$ is taken as 1 day in the following equation:

$${\beta }_H=150\left[1+(1.2{\displaystyle \frac{RH}{R{H}_0}}{)}^{18}\right]{\displaystyle \frac{h}{h_0}}+250\le 1500$$

(8)

In this study, the original H and $t_0$ are multiplied by stochastic factors ${\alpha }_5$ and ${\alpha }_6$ to describe the randomness of environmental humidity and loading age, respectively. Additionally, the randomness of self-weight load is considered, with a stochastic factor of ${\alpha }_7$. The statistical parameters of each stochastic variable are shown in

Table 1. Statistical characteristics of random parameters.

Description	Symbol	Distribution Type	Mean	Coefficient of Variation
Randomness in Creep Model [6]	α1	Normal Distribution	1.0	0.35
Randomness in Shrinkage Model [6]	α2	Normal Distribution	1.0	0.46
Randomness in Compressive Strength [25]	α3	Normal Distribution	1.0	0.1
Randomness in Elastic Modulus [26]	α4	Normal Distribution	1.0	0.2
Randomness in Environmental Humidity [27]	α5	Normal Distribution	1.0	0.16
Randomness in Loading Age [26]	α6	Uniform Distribution	1.0	0.11
Randomness in Self-weight Load [26]	α7	Normal Distribution	1.0	0.05

.

The CEB-FIP 1990 model defines the shrinkage development equation as

$${\epsilon }_{cs}(t,t_s)={\epsilon }_{cs0}{\beta }_s(t-t_s)$$

(9)

In the following equation:

$${\epsilon }_{cs0}={\epsilon }_s(f_{cm}){\beta }_H,{\epsilon }_s(f_{cm})=\left[160+10{\beta }_{sc}\left(9-{\displaystyle \frac{f_{cm}}{10}}\right)\right]\times 10^{-6}$$

(10)

Considering the above random factors, the long-term random equation for concrete is expressed as follows [25]:

$$\begin{array}{ll}\epsilon \left(t,t_0\right)=\sigma \left(t_0\right)·& {\alpha }_1·{\displaystyle \frac{1}{{\alpha }_5{E}_c\left({\alpha }_4{f}_{cm},t_0\right)}}+\sigma \left(t_0\right)·{\alpha }_1·{\displaystyle \frac{\varphi \left(t,t_0,{\alpha }_4{f}_{cm},{\alpha }_3,h\right)}{{\alpha }_5{E}_c\left({\alpha }_4{f}_{cm}\right)}}\\ & +{\int }_{t_0}^t{\alpha }_1\left[{\displaystyle \frac{1}{{\alpha }_5{E}_c\left({\alpha }_4{f}_{cm},t_0\right)}}+{\displaystyle \frac{\varphi \left(t,t_0,{\alpha }_4{f}_{cm},{\alpha }_3,h\right)}{{\alpha }_5{E}_c\left({\alpha }_4{f}_{cm}\right)}}\right]d\sigma \left(t_0\right)+{\alpha }_2{\epsilon }_{sh}(t,{\alpha }_4{f}_{cm},{\alpha }_3,h)\end{array}$$

(11)

2.2. Response Surface Fitting Technique

The probabilistic models for the seven random variables (α₁ to α₇), as defined in Section 2.1, define the input domain for the analysis. To efficiently propagate these uncertainties through the complex finite element model and obtain the probability distribution of structural responses (e.g., pushing force), a surrogate model is employed. This study utilizes the Response Surface Method (RSM) for this purpose. RSM approximates the implicit relationship between the input random variables and the output structural response by fitting an explicit polynomial function to a limited number of deterministic finite element analyses. Within a local range, RSM accurately approximates functional relationships using a reduced number of sample points, and complex response relationships are expressed in a simple algebraic form. This approach circumvents the enormous computational cost of performing Monte Carlo simulation directly on the finite element (FE) model, making the probabilistic analysis of concrete shrinkage and creep effects computationally feasible [25].

The basic idea is to select a polynomial form that approximates the implicit actual response function, followed by determining the undetermined parameters in the approximate function through a series of experimental points. By reasonably selecting experimental points and iteration strategies, the approximate response function can be ensured to converge to the true implicit response function. Response Surface Methodology (RSM) is highly operable and can be combined with finite element analysis for the optimization analysis of complex structures.

Currently, the most commonly used response surface forms are linear polynomials and full/partial quadratic polynomials. The quadratic response surface, which includes second-degree terms, can reflect the nonlinearity of the implicit response function to a certain extent. If the order of the true response function is not very high, the quadratic response surface can indeed produce satisfactory results. However, if the order of the implicit response function is much higher than quadratic, using only quadratic terms to reflect the high nonlinearity of the true response function may result in low accuracy, and in some cases, even erroneous results.

2.3. MCS Random Analysis Method

The Monte Carlo Simulation (MCS) method, also known as the random sampling or statistical experimental simulation method, is a general term for computational methods that involve drawing a large number of random samples from a population. The Monte Carlo method is based on specific data generation processes and sample sizes, using simulation to approximate the sampling distribution of an estimator. It can realistically simulate actual physical processes, making it highly applicable for solving real-world problems and yielding highly satisfactory results. The basic computational principle is shown in Figure 1.

2.4. Random Analysis Method for Creep Response Based on Response Surface and MCS

2.4.1. Response Surface Method Based on Monte Carlo Sampling

Although the direct Monte Carlo method can overcome the difficulties associated with solving implicit functions, it imposes a significant computational burden, which often makes it unsuitable for the random analysis of shrinkage and creep effects in large-scale structures. To overcome these shortcomings, this study adopts the Monte Carlo method based on the Response Surface Method (RSM) [24]. In this analysis, the long-term deformation of the structure was first obtained using a Midas finite element model. Subsequently, the response surface function of the structure was established. Random sampling was then performed on the random variables, and the obtained samples were input into the constructed response surface function. Statistical analysis of the results yields the distribution characteristics of the long-term deformation of the structure. The response surface method generally adopts a simple quadratic polynomial form. Depending on the complexity of the problem and computational cost considerations, the quadratic polynomial can take two forms: with or without cross-terms. To ensure both computational accuracy and efficiency, this study constructs a response surface equation using a quadratic polynomial without cross terms:

$$F(X)=a+\sum _{j=1}^n{b}_j{X}_j+\sum _{j=1}^b{c}_j{X}_j^2$$

(12)

where X_j values are the random variable factors; n is the number of random variables; and a, b_j, and c_j are 2n + 1 undetermined coefficients, which need to be determined by establishing 2n + 1 equations through 2n + 1 finite element analyses. The specific process is illustrated in the flowchart.

2.4.2. Random Factor Sampling Points

The first step in solving the response surface coefficients is parameter design. This step is a critical task, primarily involving the selection of appropriate sampling points within the parameter space. These sampling points are used to construct a response surface that approximates the actual structural response within the effective region. Based on the distribution types and parameter values of the random variables, the Central Composite Design method is used to design the response surface sampling, which yields the sampling points of random factors as shown in

Table 2. Sample of random factors.

Case	a1	a2	a3	a4	a5	a6	a7
1	1.0	1.0	1.0	1.0	1.0	1.0	1.0
2	0.65	1.0	1.0	1.0	1.0	1.0	1.0
3	1.35	1.0	1.0	1.0	1.0	1.0	1.0
4	1.0	0.54	1.0	1.0	1.0	1.0	1.0
5	1.0	1.46	1.0	1.0	1.0	1.0	1.0
6	1.0	1.0	0.9	1.0	1.0	1.0	1.0
7	1.0	1.0	1.1	1.0	1.0	1.0	1.0
8	1.0	1.0	1.0	0.8	1.0	1.0	1.0
9	1.0	1.0	1.0	1.2	1.0	1.0	1.0
10	1.0	1.0	1.0	1.0	0.84	1.0	1.0
11	1.0	1.0	1.0	1.0	1.16	1.0	1.0
12	1.0	1.0	1.0	1.0	1.0	0.89	1.0
13	1.0	1.0	1.0	1.0	1.0	1.11	1.0
14	1.0	1.0	1.0	1.0	1.0	1.0	0.95
15	1.0	1.0	1.0	1.0	1.0	1.0	1.05

. After fitting the response surface, Latin Hypercube Sampling is used to extract samples for verification of the response surface error.

2.4.3. Sensitivity of Random Variables

Sensitivity refers to the gradient of the objective response function $F(X)$ with respect to the random variables $X_i$. Sensitivity analysis is typically an indispensable component of structural response analysis, given that different random variables exert heterogeneous contributions to the structural response values. Specifically, sensitivity analysis facilitates the quantification of the importance of each random variable involved [28].

Typically, sensitivity ${S^{\prime }}_{X_i}$ obtained after deriving the response surface by taking the first derivative of the response surface equation F(X) with respect to each random variable. In this study, the discreteness of the random variables is also considered, and the sensitivity ${S^{\prime }}_{X_i}$ is multiplied by the mean square deviation of the random variables to obtain $S_{X_i}$. The calculation formula is as follows:

$$S_{Xi}={\displaystyle \frac{\frac{\partial F(X)}{\partial X_i}·{\sigma }_{x_i}{|}_{x_i=u{x}_i}}{{\sum }_{j=1}^n\left|\frac{\partial F(X)}{\partial X_i}·{\sigma }_{x_i}{|}_{x_i=u{x}_i}\right|}}$$

(13)

where $F(X)$ is the response surface function, and ${\sigma }_{x_i}$ is the mean square deviation of the random variable.

3. Engineering Background and Finite Element Model

The Lugou River Grand Bridge is a twin bridge, with the superstructure of the main bridge adopting a prestressed concrete continuous rigid frame, and the substructure adopting double-limb solid piers and hollow thin-walled piers with pile foundations. This bridge is representative of long-span (up to 180 m) continuous rigid-frame bridges with ultra-high piers (max. 192 m) commonly used in mountainous areas. The span arrangement of the left-line bridge is 3 × 40 + (96 + 5 × 180 + 96) + 23 × 40 m, while that of the right-line bridge is 4 × 40 + (96 + 5 × 180 + 96) + 23 × 40 m. This study uses the left-line bridge as the engineering background. The main beam is a single-box, single-chamber prestressed concrete box girder with a top slab width of 16.25 m, a bottom slab width of 8.5 m, a cantilever length of 3.875 m on each side, a beam height of 4.0 m at mid-span, and a beam height of 11.5 m at the root of the pier top. The height of the box varies according to a 1.5-degree parabolic curve. The thickness of the bottom slab of the box girder varies from 0.35 m to 1.70 m (thickness varies according to a 1.5-degree parabolic curve), and the web thickness varies between 50 cm, 70 cm, and 90 cm. The thickness of the web in the zero block of the main girder is 120 cm. The beam body is divided into the zero block, blocks 1–21, the closure segment (block 22), and the side-span cast-in-place segment (block 23). The zero block is constructed by means of a bracket, whereas blocks 1–21 are constructed through balanced cantilever construction with a formwork traveler.

The finite element analysis model of the left-line main bridge segment was established using Midas Civil software 2024, simulating the entire construction process. The main beam was simulated by adopting variable cross-section spatial beam elements, whereas the bridge piers were modeled using general spatial beam elements. The entire bridge main beam consisted of 306 nodes and 305 elements, and the entire bridge piers consisted of 202 nodes and 107 elements. The pile–soil interaction was not considered, and the bottom of the pile cap was consolidated. Finite element models that consider and do not consider structural nonlinearity were established separately. The closure segments of the bridge were constructed by means of scaffolding, with the closure sequence being side spans first, followed by middle spans, and ultimately the main spans. The bottom of the pile cap is fixed (consolidated). A total of 83 construction stages were simulated, including pier construction, form traveler installation and movement, closure of each span, and second-phase pavement. Shrinkage and creep effects were analyzed at 1, 3, 10, and 20 years after completion. The analysis used displacement control with a convergence tolerance of 1 × 10⁻⁵. In this study, closure of piers 6 and 7 was performed using a pushing force. The application of the pushing force is shown in Figure 2. To ensure force balance during application, the same pushing force was applied on both sides of the closure segment.

4. Results and Discussion

4.1. Random Response Analysis of Bridge Shrinkage and Creep

4.1.1. Response Surface Fitting

The samples of response surface design parameters are substituted into the finite element model, and 15 calculations are performed to obtain the values of mid-span deflection, horizontal longitudinal displacement at the top of the pier, and the bending moment at the base of pier #7 at each creep period. The displacement results are substituted into equation (10) to obtain the response surface calculation matrix, which is used to solve the response surface. Owing to the large number of working conditions, only the response surface coefficients for shrinkage and creep over a period of 10 years are presented in

Table 3. Response surface coefficients for shrinkage and creep over a period of 10 years.

Symbol	Bending Moment at Pier Base (103 kN)	Mid-Span Displacement (mm)	Horizontal Longitudinal Displacement at the Pier Top (mm)
			Pier #4	Pier #5	Pier #6	Pier #7	Pier #8	Pier #9
a	−667.02	−120.51	1271.02	645.20	128.87	355.69	−856.77	1535.34
b1	−12.92	−8.09	85.48	40.99	5.92	−20.07	−55.70	−101.15
b2	−15.85	−2.21	25.52	13.70	3.04	−7.53	−17.81	−30.33
b3	1054.72	201.89	2143.59	1100.39	220.43	599.39	1455.80	2580.52
b4	82.68	18.39	−210.22	−102.88	−13.98	45.18	136.52	246.24
b5	−48.00	−3.53	40.82	23.37	6.81	−14.87	−29.87	−49.29
b6	28.96	−23.79	271.54	115.73	−5.97	−17.42	−159.37	−295.23
b7	18.40	3.65	−34.30	−10.46	−2.12	8.72	16.19	43.32
c1	−1.36	0.80	−6.62	−2.93	−0.52	2.04	4.37	8.13
c2	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
c3	−486.61	−92.71	982.20	504.37	101.44	275.55	−667.32	1183.24
c4	−32.19	−5.86	67.79	33.29	4.48	−14.36	−44.00	−79.28
c5	47.23	6.41	−72.02	−38.03	−8.24	21.02	49.81	85.95
c6	−15.06	13.47	−152.96	−65.38	3.08	10.30	89.97	166.60
c7	−0.91	−0.11	2.80	1.38	0.38	−0.54	−1.56	−4.02

.

To verify the accuracy of the response prediction, the predicted values of the model are compared with the actual values. The goodness of fit is assessed using the coefficient of determination R² and the mean squared error (MSE). The coefficient of determination R² is calculated as shown in Equation (14), and the MSE is calculated as shown in Equation (15).

$$R^2=1-{\displaystyle \frac{{\sum }_i({\widehat{y}}_i-y_i{)}^2}{{\sum }_i({\bar{y}}_i-y_i{)}^2}}$$

(14)

$$MSE={\displaystyle \frac{1}{m}}{\sum }_i({\widehat{y}}_i-y_i{)}^2$$

(15)

In Equation (14), the numerator denotes the sum of the squared differences between the actual values and the predicted values, which is analogous to the MSE. Meanwhile, the denominator represents the sum of the squared differences between the actual values and the mean values. The calculation results of the coefficient of determination R² and MSE for each response surface are shown in

Table 4. Determinants and mean squared error.

Construction Stage	Parameter	Bending Moment at Pier Base	Mid-Span Displacement (mm)	Horizontal Longitudinal Displacement at the Pier Top (mm)
				Pier #4	Pier #5	Pier #6	Pier #7	Pier #8	Pier #9
Completion Stage	R2	0.99	0.96	0.97	0.97	0.93	1.00	0.99	0.95
	MSE	1.30	0.03	0.09	0.07	0.01	0.00	0.00	0.02
Shrinkage and Creep 1 Year	R2	0.98	0.94	0.98	0.98	0.97	0.93	0.96	0.97
	MSE	1.49	0.14	1.09	1.00	0.25	0.01	0.07	0.44
Shrinkage and Creep 2 Year	R2	0.99	0.93	0.98	0.98	0.97	0.95	0.96	0.97
	MSE	1.40	0.19	1.45	1.23	0.39	0.01	0.11	0.69
Shrinkage and Creep 3 Year	R2	0.99	0.92	0.98	0.98	0.97	0.96	0.97	0.97
	MSE	1.38	0.23	1.45	1.40	0.71	0.13	0.38	0.94
Shrinkage and Creep 10 Year	R2	0.99	0.89	0.98	0.98	0.98	0.97	0.97	0.98
	MSE	1.50	0.36	1.24	1.79	0.92	0.18	0.49	1.21
Shrinkage and Creep 20 Year	R2	0.96	0.86	0.98	0.98	0.98	0.97	0.98	0.98
	MSE	1.32	0.44	1.48	1.32	0.99	0.20	0.54	1.31

. The results indicate that R² is close to 1 and MSE is below 1.5, showing that the response surface equations accurately reflect the relationship between the structural response and parameters, making them suitable for structural analysis.

4.1.2. Long-Term Behavior Prediction of Bridge Responses

The probability distribution functions of each response are determined using Monte Carlo sampling based on the response surface. For a normal distribution, when the value of k is set to 2, $\mu -2\sigma$ and $\mu +2\sigma$ correspond to the 2.28% and 97.72% quantiles, respectively. Based on the probability distribution function, the 2.28% and 97.72% quantile values of the structural response are determined. The results indicate that the probability density functions of the mid-span displacement and pier-base moment are approximately normal, whereas the probability density function of the horizontal longitudinal displacement at the pier top follows an approximately log-normal distribution. Due to space constraints, the random analysis results of mid-span displacement, the bending moment at the base of pier #7, and the horizontal longitudinal displacement at the top of piers #4 and #7 are provided in Figure 3. The fitting mean and variance of each construction stage are shown in

Table 5. Determinants and mean squared error.

Construction Stage	Parameter	Bending Moment at Pier Base (103 kN)	Mid-Span Displacement (mm)	Probability Distribution of Displacement at the Top of Pier #7 (mm)
Completion Stage	Mean	14.92	−3.82	9.38
	Variance	4.42	0.71	1.38
Shrinkage and Creep 1 Year	Mean	−26.32	−8.5	−5.98
	Variance	16.41	2.67	6.88
Shrinkage and Creep 2 Year	Mean	−33.54	−10.19	−11.34
	Variance	17.13	3.35	8.84
Shrinkage and Creep 3 Year	Mean	−40.77	−11.31	−15.21
	Variance	19.3	3.86	10.31
Shrinkage and Creep 10 Year	Mean	−58.12	−14.14	−25.14
	Variance	22.86	4.7	13.83
Shrinkage and Creep 20 Year	Mean	−84.13	−15.44	−29.30
	Variance	30.61	5.28	15.45

.

4.2. Sensitivity Analysis of Creep Parameters

Based on the response surface function and using Equation (13), the sensitivity coefficients of the random factors are calculated. This paper presents the sensitivity analysis of the random factors for the displacement at the top of pier #7, the mid-span displacement, and the bending moment at the base of pier #7. The factors ${\alpha }_1$–${\alpha }_7$ represent, respectively, the creep model random factor, the shrinkage model random factor, the concrete compressive strength random factor, the concrete elastic modulus random factor, the environmental relative humidity random factor, the concrete loading age random factor, and the concrete load random factor. Figure 4 shows the sensitivity analysis results of the displacement at the top of piers #4 and #7, the mid-span displacement, and the bending moment at the base of pier #7 with respect to the random variables.

Figure 4 indicates that the creep model, concrete compressive strength, and concrete elastic modulus are the critical random variables influencing time-dependent stress. The randomness of environmental humidity should not be overlooked. In comparison, the random factor associated with the shrinkage model has a relatively minor effect on the long-term deformation of long-span continuous rigid-frame bridges with high piers.

It should be noted that the sensitivity analysis is based on a local derivative method, which may not fully capture global nonlinear effects. A global sensitivity analysis (e.g., Sobol’ indices) is recommended for future research.

4.3. Analysis of the Impact of Creep Randomness on Pushing Force

4.3.1. Determination of Pushing Stiffness and Calculation of Pier Top Displacement

In this study, the formula for calculating the pushing force is derived based on the theoretical pushing stiffness [29], and then it is computed with a finite element model. Calculations are performed using the finite element software MIDAS Civil to determine the influence of mid-span pushing force, secondary mid-span pushing force, and secondary side-span pushing force on the horizontal longitudinal displacement at the top of the piers. For this purpose, the pushing forces of spans 6 and 7 are incremented by 100 kN from 0 to 500 kN, while the pushing forces for other spans are set to 0. The horizontal longitudinal displacement at the top center of each pier under different pushing forces at the closure segments of spans 6 and 7 is shown in

Table 6. Longitudinal horizontal displacement at the pier top under various push forces.

Push Force  (103 kN)	0	1	2	3	4	5	6	7	8	9	10
Pier #4	122.99	118.3	113.61	108.92	104.23	99.54	94.85	90.16	85.47	80.78	76.09
Pier #5	71.78	67.21	62.64	58.07	53.5	48.93	44.36	39.79	35.22	30.65	26.08
Pier #6	9.38	4.8	0.22	−4.36	−8.94	−13.52	−18.1	−22.68	−27.26	−31.84	−36.42
Pier #7	−15.43	−8.99	−2.55	3.89	10.33	16.77	23.21	29.65	36.09	42.53	48.97
Pier #8	−86.91	−80.4	−73.89	−67.38	−60.87	−54.36	−47.85	−41.34	−34.83	−28.32	−21.81
Pier #9	−147.59	−140.85	−134.11	−127.37	−120.63	−113.89	−107.15	−100.41	−93.67	−86.93	−80.19

.

Figure 5 and Table 6 show that the horizontal displacement of each pier top varies linearly with the magnitude of the pushing force. The ratio of horizontal displacement to longitudinal pushing force ${\alpha }_{34}$ for pier #4 is −4.69 mm/1000 kN; ${\alpha }_{35}$ for pier #5 is −4.57 mm/1000 kN; ${\alpha }_{36}$ for pier #6 is −4.58 mm/1000 kN; ${\alpha }_{37}$ for pier #7 is 6.44 mm/1000 kN; ${\alpha }_{38}$ for pier #8 is 6.51 mm/1000 kN; and ${\alpha }_{39}$ for pier #9 is 6.74 mm/1000 kN (where − indicates displacement towards the smaller mileage direction, and + indicates displacement towards the larger mileage direction).

The horizontal displacement at the top of the pier mainly consists of the cumulative longitudinal displacement δ₁ at the pier top during the bridge formation stage, the longitudinal displacement δ₂ at the pier top due to closure temperature differences, the longitudinal displacement δ₃ at the pier top due to prestressing, and the longitudinal displacement δ₄ at the pier top due to concrete shrinkage and creep during bridge operation. In this study, the impact of shrinkage and creep randomness on the longitudinal displacement at the pier top is primarily considered [30]. Therefore, the closure construction is assumed to occur at the reference temperature, and the influence of temperature variations during closure is neglected in this study. Given that the longitudinal displacement at the pier top induced by concrete shrinkage and creep is a long-term effect, the displacement magnitude corresponding to a 10-year period under shrinkage and creep effects is used in the calculation [31,32]. If the pushing displacement is defined by the final creep value, and the pushing force fully offsets the displacement caused by shrinkage and creep, it may result in excessive reverse displacement at an early stage, leading to significant adverse bending moments at the pier base or even cracking of the pier. Therefore, 70% of the total creep displacement δ₄ is used to calculate the pushing force. This coefficient is an engineering estimate of creep development over 10 years. A lower value (e.g., 60%) would reduce the pushing force, risking under-displacement and cracking; a higher value (e.g., 80%) would increase it, risking excessive early reverse displacement. Importantly, the main conclusion—namely, that the probabilistic method outperforms the deterministic method—does not depend on the specific 70% value, because the advantage stems from the randomness of creep parameters, not the coefficient itself. Hence, the choice of 70% does not affect the comparative conclusions of this study.

Thus, the theoretical total displacement at the pier top is as follows:

$$\Delta ={\delta }_1+{\delta }_2+{\delta }_3+{\delta }_4\times 70\%$$

(16)

Using the established model, the longitudinal horizontal displacement at the bridge pier tops under the influence of various factors is simulated, and the required pushing force is calculated based on $F={\displaystyle \frac{\Delta }{K}}$.

4.3.2. Analysis of the Impact of Creep Randomness on Pushing Force

The 2.28% and 97.72% quantiles of the structural response are taken as the range of values for the horizontal longitudinal displacement of the piers. The longitudinal horizontal displacement at the top of the pier is calculated deterministically according to the CEB-FIP (1990) standard. The longitudinal horizontal displacement values at the top of the pier under the influence of various factors are shown in

Table 7. Calculated pier top longitudinal displacements for various pushing force scenarios.

Parameter	Pier #4	Pier #5	Pier #6	Pier #7	Pier #8	Pier #9
${\delta }_1+{\delta }_2+{\delta }_3$	26.35	8.10	−5.82	6.22	−12.81	−27.35
0.7 × Standard	75.00	33.55	0.86	−10.19	−46.53	−86.22
0.7 × ${\delta }_4$max	183.75	61.25	3.85	−26.25	−98	161

.

The required pushing force is calculated based on $F={\displaystyle \frac{\Delta }{K}}$, and the relationship between the horizontal displacement of piers 4–9 and the pushing force is fitted using Equation (17).

$$\begin{array}{l}{\Delta }_4={\alpha }_{14}{f}_1+{\alpha }_{24}{f}_2+{\alpha }_{34}{f}_3\\ {\Delta }_5={\alpha }_{15}{f}_1+{\alpha }_{25}{f}_2+{\alpha }_{35}{f}_3\\ {\Delta }_6={\alpha }_{26}{f}_2+{\alpha }_{36}{f}_3\\ {\Delta }_7={\alpha }_{2\prime 7}{f}_{2\prime }+{\alpha }_{37}{f}_3\\ {\Delta }_8={\alpha }_{1\prime 8}{f}_{1\prime }+{\alpha }_{2\prime 8}{f}_{2\prime }+{\alpha }_{38}{f}_3\\ {\Delta }_9={\alpha }_{19}{f}_{1\prime }+{\alpha }_{29}{f}_{2\prime }+{\alpha }_{39}{f}_3\end{array}$$

(17)

It represents the absolute value of the total displacement at the top of each pier, which can be obtained from Equation (13). It represents the pushing stiffness of pier j due to the pushing force applied at the i-th closure span. In this study, the pushing force is only applied during the closure of span 3, so all other forces are zero. The displacement at the top of each pier is substituted into Equation (17), and the average value is used as the pushing force. The deterministically calculated pushing force based on the CEB-FIP (1990) standard is 10,461 kN, rounded to 10,500 kN considering construction requirements. The pushing force obtained from the random analysis is 8814 kN, rounded to 9000 kN to take into account construction requirements. The deterministic pushing force is approximately 1.16 times the probabilistic result. This 16% difference suggests that the deterministic method may overestimate the required pushing force for high-pier bridges, leading to potentially higher strain energy and less favorable structural behavior during the early service stage.

The calculated pushing force is applied to the finite element model, and the randomness of the long-term deformation of the bridge induced by shrinkage and creep under various working conditions is recalculated [33]. This study provides the probability density function of the displacement at the top of pier #7 over a ten-year period due to shrinkage and creep, with and without pushing force, as shown in Figure 6:

It can be seen that, without pushing force during closure, significant deviations occur at piers #4 and #9 during the bridge formation stage, and large deviations occur at piers #4, #5, #8, and #9 after 10 years of shrinkage and creep [34]. This study compares the displacement at the top of pier #7 with and without the pushing force. The results shown in Figure 6 indicate that in 44.42% of the analyzed cases, the structural deformation with the pushing force applied is smaller than that without the pushing force. This means that a deterministic pushing force fails to provide consistent improvement across all random scenarios; hence, a probabilistic pushing force (e.g., based on a desired quantile) is more reliable. This finding also indicates that the randomness of concrete shrinkage and creep has a substantial influence on the pushing force of ultra-high pier continuous rigid-frame bridges. Consequently, it is essential to consider such randomness when determining a reasonable pushing force, and the use of probabilistic results is recommended for practical construction control.

4.4. Comparison Between Linear and Nonlinear Analysis Results of the Main Girder Flange

The established models take into account both structural geometric nonlinearity (P-Delta effect) and material nonlinearity (shrinkage, creep, and cracking) using Midas Civil default settings. Results from the two models indicate that for long-span continuous rigid-frame bridges with ultra-high piers, structural nonlinearity has a significant influence on the structural stress state [35]. After 10 years of shrinkage and creep, the maximum stress at the upper edge of the main beam differs by 11.4%, and the maximum stress at the lower edge differs by 15.6% between the nonlinear and linear models, as shown in Figure 7 and Figure 8. Consequently, structural nonlinearity is taken into account in all subsequent calculations. From Figure 7 and Figure 8, the stress magnitude decreases in the following order: the completed bridge stage, the completed bridge stage with nonlinear effects taken into account, the bridge after 10 years of shrinkage and creep, and the bridge after 10 years of shrinkage and creep with nonlinear effects incorporated. In accordance with the global behavior of the bridge and taking the right mid-span of the highest pier (pier #7) as a case study, the results for the top and bottom flanges of the main girder are presented in Figure 9.

5. Conclusions

Based on the CEB-FIP (1990) shrinkage and creep model and using the Monte Carlo method combined with the response surface method, this study takes the Lugou River Grand Bridge as a case study. Structural nonlinearity and the randomness of the creep model, shrinkage model, concrete compressive strength, concrete elastic modulus, environmental humidity, self-weight, and loading age are considered in order to investigate the pushing force of ultra-high pier continuous rigid-frame bridges under the influence of shrinkage and creep randomness.

(1)

For continuous rigid-frame bridges featuring ultra-high piers, structural nonlinearity exhibits a considerable influence on the mechanical performance and internal force distribution. Therefore, nonlinear effects should be rigorously considered in both structural analysis and design procedures.

(2)

The uncertainty of the concrete shrinkage and creep model leads to significant randomness in the mid-span displacement, pier base moment, and longitudinal horizontal displacement at the top of the piers during operation. Deterministic analysis based on code-specified shrinkage and creep models is insufficient to guarantee the safety and reliability of structural design. Therefore, the influence of randomness on structural response should be systematically considered in the calculation and design of pushing forces.

(3)

The creep model, concrete compressive strength, and elastic modulus of concrete are identified as key random variables that govern time-dependent stress. Additionally, the randomness of environmental humidity cannot be neglected. By contrast, the random factor associated with the shrinkage model has a relatively minor influence on the long-term deformation of long-span continuous rigid-frame bridges with high piers.

(4)

The probability density functions of mid-span section deflection and pier-base stress approximately follow a symmetric normal distribution. The horizontal longitudinal displacement at the top of the piers exhibits a significant asymmetric distribution and approximately follows a log-normal distribution. The Monte Carlo method based on the response surface can obtain the 2.28% and 97.72% percentiles of the structural response, which are used as the range of structural response values. The pushing force calculated at the value with the maximum probability distribution differs from the deterministically obtained pushing force by a factor of 1.16, and its influence cannot be ignored.

(5)

The longitudinal displacement at the top of the piers obtained after pushing with the calculated pushing force is more favorable than that without the pushing force. This indicates that the randomness of concrete shrinkage and creep has a significant influence on the pushing force of continuous rigid-frame bridges with ultra-high piers. Therefore, it is essential to account for the randomness of concrete shrinkage and creep in determining a reasonable pushing force.