Experimental study on shrinkage and creep of prestressed simply supported box girder for high-speed railway

Abstract

To investigate the long-term performance of a 32 m prestressed simply supported box girder, a 1:4 scale prestressed concrete simple supported box girder was cast. The casting procedure adheres to the principle of stress equivalence within the concrete in the middle span after tensioning of prestressed tendons. Utilizing the CEB-FIP 90 model as a foundation, we established a long-term deformation calculation model for the box girder. Subsequently, the reliability of the long-term deformation model was confirmed by employing data from a 96 d long-term deformation test conducted on the box girder. Meanwhile, a new database was created by integrating shrinkage and creep experiment data with the shrinkage and creep database developed by Bazant. The shrinkage and creep uncertainty coefficients were introduced to complete the modeling of concrete shrinkage creep uncertainty calculations. The results demonstrate that the long-term deformation prediction model can effectively characterize the tendency of the mid-span upward deflection in the box girder. At 988 d, the upward deflection at the mid-span of the 1:4 scale model was expected to reach approximately 3.67 mm. It is worth noting that the CEB-FIP 90 model tends to slightly overestimate long-term deformation compared with experimental results. Additionally, it significantly underestimates the shrinkage strain observed in the test results. The uncertainty associated with the long-term deformation prediction of the structural system increased as the prediction time extended.

Graphical abstract

Keywords

shrinkage / creep / box girder / quantization of uncertainty / long-term deformation / mid-span upper arch

1 Introduction

China has achieved historic breakthroughs in railway science and technology, advancing to the forefront of global innovation. In particular, its independent advancements in high-speed rail technology lead the world, establishing a uniquely Chinese path of railway innovation. Until 2024, China’s high-speed railway network had a total operational mileage exceeding 45000 km. This vast network accounted for more than two-thirds of the global high-speed rail mileage and represented the largest high-speed railway system worldwide. Meanwhile, the ballasted track system is gradually being replaced by the ballastless track system, addressing the issues of high noise levels and increased maintenance costs. This transition has also significantly enhanced the durability of the railway system. To meet the high safety and stability requirements of high-speed railways for both domestic and abroad most high-speed railways employ ballastless tracks. In China, more than 90% of the constructed high-speed railways utilize ballastless track structures [1]. Ensuring the long-term safety and stability of the extensive high-speed railway infrastructure has become a pressing issue in the current stage of high-speed railway development in China. To achieve the high flatness, stability, and safety required for high-speed railways, a large number of elevated girders with high stiffness are widely employed in the substructure support system of China’s high-speed rail network. Compared to embankment settlements, elevated girders offer greater stability and control, making them better suited to ensure the safety and flatness of train operations. In newly opened high-speed rail systems such as Japan’s Shinkansen, has a girder component proportion of approximately 45% [2]. In the case of Taiwanc, China 345 km high-speed rail network, girders constitute 73% of the total infrastructure. Moreover, for the Beijing-Tianjin intercity railway, Beijing−Shanghai high-speed railway, and Guangzhou−Zhuhai high-speed railway, the respective percentages of girder components are notably high, standing at 87.7%, 80.5%, and 94.2% [3,4]. As stated above, elevated girders have become the primary supporting structure for high-speed rail systems, and 32 m precast prestressed concrete (PC) simply supported girders are extensively used in the lower supporting structures of high-speed railways. Consequently, scholars from various countries have conducted extensive research on PC simply supported girders. Due to the influence of time and external factors, PC girders undergo long-term deformations. These deformations can negatively impact the normal functionality and safety of the girders, as well as affect the ride comfort and operational safety of high-speed rail systems. Therefore, in-depth research into the long-term deformations of PC girders caused by concrete creep becomes particularly necessary.

Concrete creep is induced by constant loads applied to concrete structures, exhibiting a degree of uncertainty and unpredictability. In recent years, scholars across the globe have proposed various models to predict the shrinkage and creep behavior of concrete structures. The Comité Européen du Béton (CEB) has successively introduced several models, i.e., the CEB-FIP series [5–8]. These models are extensively incorporated into numerous national standards. The ACI 209R-92 model is commonly applied in North America [9,10]. With more detailed and microscopic studies on the shrinkage and creep behavior of concrete, computational models have become increasingly effective in capturing the physical mechanisms of creep behavior [11–14]. The models mentioned above consider various parameters. Among them, the models that involve parameters that are more readily obtainable are the CEB-FIP 78, 90, and 2010 models. Due to the easy accessibility of these parameters, such models have been widely employed. In terms of prediction accuracy, the CEB-FIP 90 outperforms the CEB-FIP 78, compared to CEB-FIP 2010, the creep coefficient calculated using CEB-FIP 1990 aligns more closely with the experimental values in an outdoor environment [15–17]. At the same time, the existing literature shows that the prediction accuracy of CEB-FIP models is better than Gardner and Lockman (GL) 2000 [13] for precast box girder C50 concrete [16]. The above-mentioned prediction models are developed through extensive fitting of experimental data. Concrete shrinkage and creep exhibit significant randomness and are influenced by numerous factors [18], which results in notable discrepancies between the predicted outcomes of these models and actual measurements. Researchers have assessed and adjusted these models. For example, Pan [19] found that the CEB-FIP 90 was relatively conservative in comparison to actual shrinkage and, as a result, made coefficient corrections to the model. Lam [20], Al-Manaseer and Lam [21] conducted assessments of the ACI 209R, CEB-FIP 90, B3 and GL2000 models. Their research revealed that the B3 model underestimated shrinkage strain, while the GL2000 model overestimated it. The GL 2000, B3, and CEB-FIP 90 models exhibited relatively small deviations from actual creep deformation.

Due to the numerous factors influencing concrete creep and the inherent randomness associated with these factors [20], predicting concrete creep in structural components is challenging and often imprecise [21]. Therefore, it is more appropriate to consider a range of values rather than specific values for long-term deformation calculations in concrete structures. Researchers [22–29] have conducted similar experimental studies on the long-term deformation of concrete structures. The aforementioned studies have predominantly employed deterministic methods to analyze the long-term deformation of concrete structures. However, these deterministic predictions often exhibit significant disparities when compared to actual measurements. Therefore, it can be concluded that deterministic methods are inherently limited in accurately predicting structures’ long-term deformation. A probabilistic approach is required to analyze the long-term deformation of structures [15,16]. Bai [30] introduced the Monte Carlo method with a segmented response surface to perform stochastic analyses of the long-term deformations of reinforced girders. By comparing the results obtained from Monte Carlo methods, response surface Monte Carlo methods, and segmented response surface Monte Carlo methods, it was found that the stochastic nature of creep calculations should not be overlooked. Sousa et al. [31] provided a mathematical probabilistic approach based on short-term bridge deformation measured data to predict its long-term deformation. Yang [32–33] and Han et al. [34] established an uncertainty model for structural shrinkage creep based on Bayesian inference. The computational cost of the large number of finite element analyses required by this method was addressed by proposing different proxy models. The findings indicate that stochasticity is significant in concrete girder deformation calculations, and deterministic analysis methods are insufficient for accurately predicting the creep effects in concrete structures [35]. Currently, the prediction of long-term deformation in high-speed railway box girders still predominantly relies on deterministic methods. However, these approaches overlook the impact of uncertain factors, such as environmental conditions. Consequently, the influence of material property variability cannot be accurately accounted for in the prediction of concrete long-term deformation.

This research developed a probabilistic computation model for concrete shrinkage and creep, integrating the Bazant shrinkage and creep database [36,37] with empirical results from concrete specimens used in prefabricated high-speed railway box girders. By introducing uncertainty coefficients, this model was incorporated into the ABAQUS software with User-Material (UMAT) computational model. Subsequently, the model’s reliability was validated using experimental data obtained from a 1:4 scale model of the box girder. Furthermore, field measurements were based on, providing additional validation for the reliability of the proposed calculation method. These research findings address the uncertainty associated with the long-term deformation of box girders commonly used in high-speed railway systems due to material properties, and they provide a foundation for stochastic overarching monitoring of high-speed railways. The research framework is listed as Fig.1.

2 Long-term deformation experiment of 1:4 scale box girder

2.1 Design parameters for the 1:4 scale box girder

The 32 m precast box girder utilized in high-speed railways is a post-tensioned concrete simply supported girder designed for double-track lines. The span, overall length, height and top width of girder are 31.5, 32.6, 3.052, and 12.6 m, respectively. Material parameters are the same as those of the 1:4 scale box girder. The prestressing tendons in the girder consist of 22-hole 9-1 × 7Φ15.2 and 5-hole 8-1 × 7Φ15.2 steel strands. The reinforcement within the girder consists of Hot-rolled Ribbed Bars 400 grade deformed steel bars. The longitudinal reinforcement for the top slab, bottom slab, and web comprises 12 mm diameter bars arranged in two layers. While the transverse reinforcement includes bars with diameters of 18 and 16 mm. Shear reinforcement and stirrups are of 16 and 12 mm diameter, respectively [38].

A span of PC simply supported box girder with 1:4 scale geometry was cast. The casting procedure adheres to the principle of stress equivalence within the concrete in the middle span after tensioning of prestressed tendons. Due to limitations in the construction, vibration, and grouting processes for prestressing ducts, the dimensions of the model did not strictly adhere to the requirements of theoretical geometric similarity. To maintain consistency in material parameters between the box girder in high-speed railways and the prototype structure, the dimensional analysis method [39] is applied, resulting in the similarity constants shown in Tab.1. Considering the arrangement and anchorage of prestressed steel strands, the bottom slab thickness of the 1:4 scale girder was slightly larger than one-fourth of the prototype girder. The span, overall length, height and top width of 1:4 scale girder were 7.87, 8.15, 0.76, and 3.16 m, respectively. The mid-span web thickness measured 160 mm, while the top and bottom slab thicknesses were 110 and 150 mm, respectively. The reinforcement ratio in 1:4 scale girder was roughly the same as the prototype girder. According to the principle of similarity and the equivalent stress in mid-span concrete, the 1:4 scale girder had 7 bundles of 3-1 × 7Φ15.2 prestressing steel strands, and the final tensile strength was 920 MPa. The ordinary steel bars reinforcement used was HRB400 grade, with longitudinal and transverse bar diameters of 8 and 12 mm, respectively. Stirrups and ties were configured with 12 and 8 mm diameter, respectively, as required by the design. Fig.2 illustrated the relevant structural details of the 1:4 scale girder. Tab.2 provided the pertinent data and mix ratio of the concrete. Nine standard concrete specimens were cast simultaneously in accordance with relevant specifications [40]. These specimens were then placed alongside the girder in the hall for natural curing over a period of 28 d. The average humidity and temperature during curing were 57.2% and 18.4 °C, respectively. The nine concrete specimens were divided into three groups, i.e., S1, S2, and S3. The 28 d compressive strength of the concrete was measured, and the results were given in Tab.3 [40].

2.2 1:4 scale box girder tensioning and long-term deformation test

Prestressing of the 1:4 scale box girder was carried out when the concrete reached an age of 28 d. Before applying tension, Linear Variable Differential Transformer (LVDT) displacement transducers were mounted on the girder’s top slab, and data collection commenced immediately. Subsequent to tensioning, grouting and anchoring were carried out without delay, and the elastic uplift of the girder was measured. After completing these tests, a 95 d long-term deformation test was conducted. The displacement measurement points on the top slab of the girder were illustrated in Fig.3.

2.3 Long-term deformation test results and analysis of 1:4 scale box girder

The temperature and humidity of the box girder hall during the test are shown in Fig.4(a). To clearly observe the variations at each measurement point of the box girder, the time-dependent displacement curves were plotted, as illustrated in Fig.4(b).

As illustrated in Fig.4(b), the displacement of measuring points S1, S2, S5, and S6 grew slowly, with a maximum average vertical displacement of 0.178 mm, which could be considered negligible. Based on the slope of the evolution curves at measuring points S3 and S4, the long-term deformation observed in this test was divided into three distinct stages. First, from 0 to 17 d after the girder prestressing (corresponding to a concrete age of 28 to 45 d). During this initial phase, the average vertical displacement at measuring points S3 and S4 increased rapidly, reaching 1.592 mm, this accounted for 56.23% of the total measured vertical displacement. Secondly, from 17 to 43 d after girder prestressing (corresponding to a concrete age of 45 to 71 d). In this stage, the growth of vertical displacement at measuring points S3 and S4 gradually decelerated. After 43 d, the average vertical displacement at measuring points S3 and S4 increased to 2.377 mm, representing 83.96% of the total measured vertical displacement. Thirdly, from 43 to 95 d after girder prestressing (corresponding to a concrete age of 71 to 123 d), the rate of increased in vertical displacement at measuring points S3 and S4 further diminishes. Ultimately, after 95 d of prestressing, the average vertical displacement at measuring points S3 and S4 reached 2.831 mm.

3 Finite element analysis of 1:4 scale box girder

3.1 Concrete shrinkage and creep model

To predict concrete creep, various concrete creep prediction models have been developed. Concrete creep is influenced by several factors, with concrete strength and environmental relative humidity being the most significant contributors to concrete creep [15–17]. In a previous study [15–17], we delved into the behavior of C50 concrete utilized in box girders designed for high-speed railway applications. Our findings unequivocally demonstrate that the predictive accuracy of the uncertainty model established by CEB-FIP 90 surpasses that of the Model Code (MC) 2010 model when analyzing C50 concrete specimens. Meanwhile, the CEB-FIP 90 model was used by China’s standard until now [41]. The B3/B4 and MC 2010 models have demonstrated high accuracy, but they were primarily developed based on data sets from European and North American concrete mixes, which may differ in composition, curing conditions, and humidity exposure compared to China’s high-speed railway structures. The Chinese standard model incorporates empirical parameters derived from local experimental data, making it more suitable for direct application. Therefore, the CEB-FIP 90 was chosen to predict the long-term deformation of the concrete girder. The model formula was shown below

(1)\phi \left(t,t_0\right)={\varphi }_0{\beta }_{\mathrm{c}}\left(t-t_0\right),

(2){\varphi }_b={\varphi }_{\mathrm{R}\mathrm{H}}\beta (f_{\mathrm{c}\mathrm{m}})\beta (t_0),

(3){\varphi }_{\mathrm{R}\mathrm{H}}= 1+\frac{1-RH/R{H}_0}{0.46{\left(h/h_0\right)}^{1/3}},

(4)\beta \left(f_{\mathrm{c}\mathrm{m}}\right)=\frac{5.3}{{\left(f_{\mathrm{c}\mathrm{m}}/f_{\mathrm{c}\mathrm{m}0}\right)}^{0.5}} ,

(5)\beta \left(t_0\right)=\frac{1}{0.1+{\left(t_0/t_1\right)}^{0.2}},

(6){\beta }_{\mathrm{c}}(t-t_0)={\left[\frac{(t-t_0)/t_1}{{\beta }_{\mathrm{H}}+(t-t_0)/t_1}\right]}^{0.3},

(7){\beta }_{\mathrm{H}}=150\left[1+{\left(1.2\frac{RH}{R{H}_0}\right)}^{18}\right] \left(\frac{h}{h_0}\right)+ 250\le 1500,

where β(f_cm) represents strength modification factor of concrete; β(t₀) denotes loading age modification factor; β_c denotes creep development function; β_H represents relative humidity correction factor for creep; ϕ₀ denotes initial value of the creep coefficient; ϕ_RH denotes environmental relative humidity correction factor; t₀ represents the loading age (d); t is the calculation age (d); \phi \left(t,t_0\right) denotes the creep coefficient; f_{\mathrm{c}\mathrm{m}} denotes the cylinder compressive strength (MPa) at 28 d, f_cm = 0.8 f_cu.k + 8 MPa; f_cu.k denotes the compressive strength of concrete cubes (MPa), f_cu.k for this study can be found in Tab.3; RH indicates the relative humidity (%); RH₀ = 100%; h represents the theoretical thickness of the component (mm); h₀ = 100 mm; t₁ = 1 d; f_cm0 = 10 MPa.

The expression used to compute concrete shrinkage is given as follows

(8){\epsilon }_{\mathrm{c}\mathrm{s}}\left(t,t_{\mathrm{s}}\right)={\epsilon }_{\mathrm{c}\mathrm{s}0}{\beta }_{\mathrm{s}} \left(t-t_0\right),

(9){\epsilon }_{\mathrm{c}\mathrm{s}0} ={\epsilon }_{\mathrm{s}} \left(f_{\mathrm{c}\mathrm{m}}\right){\beta }_{\mathrm{R}\mathrm{H}},

(10){\epsilon }_{\mathrm{s}} \left(f_{\mathrm{c}\mathrm{m}}\right)=\left[160+10{\beta }_{\mathrm{s}\mathrm{c}}\left(9-f_{\mathrm{c}\mathrm{m}}/f_{\mathrm{c}\mathrm{m}0}\right)\right]\times {10}^{-6},

(11){\beta }_{\mathrm{R}\mathrm{H}}=1.55 \left[1-{\left(\frac{RH}{R{H}_0}\right)}^3\right],

(12){\beta }_{\mathrm{s}}(t-t_{\mathrm{s}})= {\left[\frac{(t-t_{\mathrm{s}})/t_1}{350{\left(h/h_0\right)}^2+(t-t_{\mathrm{s}})/t_1}\right]}^{0.5},

where t is the age of concrete shrinkage that needs to be calculated (d); t_s is the age of concrete shrinkage under actual conditions (d); {\epsilon }_{\mathrm{c}\mathrm{s}} \left(t,t_{\mathrm{s}}\right) denotes the shrinkage strain; {\beta }_{\mathrm{s}} denotes the time-dependent development coefficient of concrete shrinkage; {\beta }_{\mathrm{s}\mathrm{c}} denotes the coefficient related to the type of cement used; β_RH denotes relative humidity correction factor for shrinkage. The remaining parameters are identical with those used in the creep model.

3.2 Iterative calculation of creep coefficient

The nonlinear optimization function LSQCURVEFIT was employed to calibrate the creep coefficient and shrinkage strain, from which the corresponding fitting parameters were obtained [42]. To accurately simulate the shrinkage and creep deformation of concrete structures, the initial step involved importing the pertinent model parameters into CEB-FIP 90 to obtain calculation outcomes. Subsequently, the calculation of CEB-FIP 90 was transformed into a recursive format using the Dirichlet series representation [43]. This facilitated the recursive computation of shrinkage and creep phenomena within the ABAQUS environment. The relevant calculation parameters were listed in Tab.4. The creep coefficient was provided in Eq. (13).

(13)\phi (t,\tau )= \sum _{j=1}^m{\varphi }_j(\tau )\left[1-{\mathrm{e}}^{-r_j(t-\tau )}\right],

where {\varphi }_j(\tau ) is the fitting parameter associated with the loading age; r_j is the empirically determined constant; m = 5.

Based on the aforementioned parameters and computational methods, the comparison results of the fitted creep coefficient and shrinkage strain were obtained, as illustrated in Fig.5.

3.3 The calculation process of shrinkage creep in finite element method

Since the shrinkage strain of concrete is independent of the iterative calculation of the concrete structure system’s strain, it can be directly added to the total strain of the structure, thereby reducing the computational cost. The creep effect of concrete is dependent on the stress state of the concrete structure after each iteration. Therefore, the creep behavior of concrete structures could be computed using the stepwise integration method. The uniaxial strain increment of concrete under uniaxial stress was expressed by Eqs. (14)–(16).

(14)\epsilon (t)={\epsilon }^{\mathrm{m}}(t )+{\epsilon }^{\mathrm{c}}(t )+{\epsilon }^{\mathrm{s}\mathrm{h}}(t) +{\epsilon }^{\mathrm{t}}(t) ,

(15){\epsilon }^{\mathrm{m}}(t) =\frac{\mathrm{\Delta }{\sigma }_0}{E\left(t_0\right)} +{\int }_{t_0}^t\frac{1}{E(\tau )}\frac{\mathrm{d}\sigma }{\mathrm{d}\tau }\mathrm{d}\tau ,

(16){\epsilon }^{\mathrm{c}}(t) =\mathrm{\Delta }{\sigma }_{0}C\left(t,t_0\right)+ {\int }_{t_0}^{t}C\left(t,t_0\right)\frac{\mathrm{d}\sigma }{\mathrm{d}\tau }\mathrm{d}\tau ,

where {\epsilon }^{\mathrm{m}}(t) is the mechanical strain; {\epsilon }^{\mathrm{c}}(t) denotes the creep strain of concrete; {\epsilon }^{\mathrm{s}\mathrm{h}}(t) represents the shrinkage strain; {\epsilon }^{\mathrm{t}}(t) denotes the thermal strain.

Thus, \mathrm{\Delta }{\epsilon }^{\mathrm{m}}(t ) and \mathrm{\Delta } {\epsilon }^{\mathrm{c}}(t) can be represented by the following formula

(17)\mathrm{\Delta }{\epsilon }^{\mathrm{m}}(t) ={\epsilon }^{\mathrm{m}}(t_n)-{\epsilon }^{\mathrm{m}}(t_{n-1})=\frac{\mathrm{\Delta }\sigma }{E(t_{n-0.5})},

(18)\mathrm{\Delta }{\epsilon }^{\mathrm{c}}(t )={\epsilon }^{\mathrm{c}}\left(t_n\right)-{\epsilon }^{\mathrm{c}}\left(t_{n-1}\right)=C\left(t,t_{n-0.5}\right)\mathrm{\Delta }{\sigma }_n,

(19)\mathrm{\Delta }{\epsilon }^{\mathrm{c}}= {\epsilon }^{\mathrm{c}}\left(t_n\right)-{\epsilon }^{\mathrm{c}}\left(t_{n-1}\right)={\eta }_n+ C\left(t_n,t_{n-0.5}\right)\mathrm{\Delta }{\sigma }_n,

(20){\eta }_n=\sum _{j=1}^m\left[1-{\mathrm{e}}^{-r_j\Delta t_n}\right]{\omega }_{jn} ,

(21){\omega }_{jn}={\omega }_{j,n-1}{\mathrm{e}}^{-r_j\mathrm{\Delta }{t}_{n-1}}+ \mathrm{\Delta }{\sigma }_{n-1}{\varphi }_j\left(t_{n-1-0.5}\right){\mathrm{e}}^{-0.5r_j\mathrm{\Delta }{t}_{n-1}},

(22){\omega }_{j1}=\mathrm{\Delta }{\sigma }_0{\varphi }_j\left(t_0\right).

Expanding the aforementioned uniaxial strain calculation method to spatial strain calculations results in the following

(23)\left\{\mathrm{\Delta }{\epsilon }_n^{\mathrm{e}}\right\}=\frac{1}{E\left(t_{n-0.5}\right)}[A]\left\{\mathrm{\Delta }{\sigma }_n\right\},

(24)\left\{\mathrm{\Delta }{\epsilon }_n^{\mathrm{c}}\right\}=\left\{{\eta }_n\right\}+C\left(t_n,t_{n-0.5}\right)[A]\left\{\mathrm{\Delta }{\sigma }_n\right\},

(25)\left\{{\eta }_n\right\}=\sum _{j=1}^m\left[1-{\mathrm{e}}^{-r_j\Delta t_n}\right]\left\{{\omega }_{jn}\right\},

(26)\left\{{\omega }_{jn}\right\}=\left\{{\omega }_{j,n-1}\right\}{\mathrm{e}}^{-r_j\mathrm{\Delta }{t}_{n-1}} +[\mathit{A}]\left\{\mathrm{\Delta }{\sigma }_{n-1}\right\}{\varphi }_j\left(t_{n-1-0.5}\right){\mathrm{e}}^{-0.5r_j\mathrm{\Delta }{t}_{n-1}},

(27)\left\{{\omega }_{j1}\right\}=[\mathit{A}]\left\{\mathrm{\Delta }{\sigma }_0\right\}{\varphi }_j\left(t_0\right),

(28)[\mathit{A}]=\left[\begin{array}{cccccc}1& -v& -v& 0& 0& 0\\ -v& 1& -v& 0& 0& 0\\ -v& -v& 1& 0& 0& 0\\ 0& 0& 0& {\displaystyle \frac{1+v}{2}}& 0& 0\\ 0& 0& 0& 0& {\displaystyle \frac{1+v}{2}}& 0\\ 0& 0& 0& 0& 0& {\displaystyle \frac{1+v}{2}}\end{array}\right] ,

[ \mathit{A}] is the matrix associated with the Poisson’s ratio. ν is the Poisson’s ratio of concrete. By integrating the above calculation methods, the shrinkage and creep of the concrete structure can be computed by utilizing the UMAT during the Abaqus analysis.

3.4 Model establishment

The prestressed tendons of the box girder were embedded within the concrete. Both ends of the box girder bottom were constrained by simply supported boundary conditions, consistent with those of the test model. The load on the box girder consisted of the self-weight of the concrete and steel reinforcement, as well as the prestress. The self-weight was simulated by applying a gravity field, while the prestress was introduced by assigning an initial stress to the prestressed tendons. The initial stress of each prestressed bar was 920 MPa. Since the 1:4 scale box girder was not in service during the long-term deformation, it could be considered to be in the elastic stage and plasticity was not considered, the relevant parameters were presented in Tab.5 and Tab.6. Fig.6 showed the finite element model of the box girder.

3.5 Calculated results

To validate the reliability of the aforementioned calculation model, displacement–time curves for vertical displacements at the midspan were drawn to compare experimental data with model calculations, as shown in Fig.7(a). The vertical displacements of the girder midspan obtained in Fig.7(a) were calculated by deducting the elastic strains caused by prestressing from the total deflection. Fig.7(b) was the overall deflection cloud diagram of the box girder under shrinkage and creep at 2068 d, which included the elastic upper arch after prestressed tension.

Fig.7(a) presented the finite element analysis results, which could be categorized into two distinct phases. In the first phase, characterized by rapid growth, the midspan vertical displacement of the girder experienced a substantial increase. This phase spanned from the time immediately following prestressing until 140 d later (corresponding to a concrete age of 281 to 168 d). During this period, the midspan vertical displacement escalated to 2.83 mm, representing approximately 76.85% of the vertical displacement at concrete age of 2068 d. The subsequent phase, referred to as the slow growth phase, covered the interval from 140 to 988 d after prestressing (equivalent to a concrete age of 168 to 1016 d). Within this stage, the midspan vertical displacement of the girder exhibited gradual growth, reaching 3.56 mm, accounting for around 96.85% of the vertical displacement at concrete age of 2068 d. It reached 3.67 mm at the age of concrete 2068 d. Fig.7(b) showed that the overall upper arch deflection of box girder with elastic upper arch value was 6.99 mm at 2068 d.

As the long-term deformation test on the 1:4 scale girder lasted for 95 d, this study only compared the data within the first 100 d after prestressing. Fig.7(a) revealed that the trend of midspan vertical displacement in the experimental and computational results closely aligns. Due to the inherent stochastic nature of concrete creep, the discrepancies observed in the computational model’s results fall within reasonable limits. Consequently, the long-term deformation calculation model for the girder was considered reliable. The late incremental deformation of the box girder decreased during the long-term deformation of the last 3 months. At 95 d after the box girder was tensioned, the deformation reached 2.80 mm, which was about 76% of the total predicted deformation.

4 Shrinkage and creep testing of C50 concrete

To investigate the stochastic nature of long-term deformation in 32 m high-speed railway girders and to ensure the predictive accuracy of the calculation model mentioned earlier, shrinkage and creep tests were carried out on C50 concrete. The same batch of concrete used for the 1:4 scale girder long-term deformation test was employed for these experiments. A total of 72 test specimens were casted on the same day as the concrete placement of the girders, with the relevant parameters presented in Tab.7. After casting, the specimens were covered with a layer of plastic film to maintain adequate humidity. The test blocks were transferred to a standard curing room for 28 d the next day.

A total of eight creep testing machines were used in this experiment, with each machine simultaneously conducting creep tests on two test blocks. Specifically, two creep testing machines were employed for testing in an outdoor hall environment, and these tests were performed on concrete specimens at 28 d of age (S1–S4). The remaining 6 machines were used for indoor testing at different concrete ages: 38 d (S5–S8), 48 d (S9–S12), and 58 d (S13–S16).

4.1 Test method

4.1.1 Test method for compressive strength of concrete prisms

To determine the creep test loads, three sets of concrete prism compressive strength tests were designed, each consisting of three specimens. According to existing standards [45], the creep test loads were set at 40% of the minimum strength values obtained from the three sets of specimens to ensure that the concrete remains within the linear creep range.

The loading procedure for the concrete strength tests was illustrated in Fig.8.

4.1.2 Test method for creep of concrete prisms

The creep tests were conducted using a spring-loaded compression creep meter, with displacement measurements made using micrometer gauge accurate to 0.0001 mm. The creep tests were performed using 40% of the minimum strength values obtained from the concrete prism tests as the test loads. During test block loading, the loading system was carefully adjusted using a high-precision alignment mechanism to minimize eccentricity. A spirit level and laser alignment tools were used to ensure uniform force distribution across the test specimens.

Fig.9(a) represented the indoor environmental condition creep test, while Fig.9(b) represented the creep test conducted in an outdoor hall environment.

4.1.3 Test method for shrinkage of concrete prisms

Three groups of concrete shrinkage tests were designed for two different environmental conditions, with each group consisting of two specimens. These tests were designed to account for the concrete shrinkage strain in the creep tests. Two groups were placed in an indoor environment, while the remaining group was placed in an outdoor hall environment. The size of the concrete shrinkage specimen was 100 mm × 100 mm × 515 mm, and the shrinkage displacement on the concrete surface with a length of 400 mm was measured using micrometer gauge. The concrete shrinkage tests were illustrated in Fig.10.

4.1.4 Relative humidity of environment and temperature

The variation curves of RH and T indoor and outdoor with respect to the loading age of the concrete were shown in Fig.11.

As shown in Fig.11(a), the RH in the outdoor environment exhibited noticeable differences compared to the indoor environment. The outdoor RH fluctuated significantly, whereas the indoor RH remained relatively stable. The average RH in the outdoor and indoor environments were 60.71% and 62.08%, respectively. Fig.11(b) indicated that the T variation trends in both the outdoor and indoor environments were generally consistent, with similar mean values.

4.2 Test results

4.2.1 Analysis of concrete strength test results

The results of the prism compressive strength test were shown in Tab.8. There were nine specimens in this test. The specimens were divided into three groups, named d, e, and f, with three specimens in each group. Tab.8 showed the average compressive strength of each group from the tests.

4.2.2 Results of the creep test

To assess the predictive accuracy of the adopted creep model, a comparison curve was plotted between the model’s predicted results and the experimental creep data. The concrete strength in CEB-FIP 90 was the measured value in Subsection 3.2, i.e., f_cm = 53.2 MPa, as shown in Fig.12. To remove the effect of temperature effect on thermal expansion and contraction. The measurement results of the creep test block include creep deformation, shrinkage deformation, and temperature-induced deformation due to the thermal expansion and contraction of the concrete test block. Similarly, the measurement results of the shrinkage test block account for shrinkage deformation and temperature-induced deformation. Therefore, the creep deformation of the test block can be determined by subtracting the measured results of the corresponding shrinkage test block. The results are presented in Fig.12. The shrinkage and creep test procedures refer to standard [45].

As shown in Fig.12, the creep coefficients under the mentioned conditions grew relatively quickly during the initial loading stage, then gradually decreased as time progressed. Due to larger humidity fluctuations in the outdoor environment compared to the indoor one, the data in Fig.12(a) exhibited significant fluctuations. However, the data from the other three groups showed a relatively stable development of creep coefficients. The creep coefficient development patterns for the specimens in each group were essentially consistent with the creep model. There was a clear trend of decreasing creep coefficients as the concrete loading age increased in all data sets. The experimental results from each group exhibited a certain level of fluctuation and unpredictability. However, the general pattern of creep coefficient evolution remained largely in agreement with the predictions of the creep model. Consistent with the findings reported in Ref. [15], the creep coefficient estimated using the CEB-FIP 90 creep model remained in good agreement with the experimental results, despite notable fluctuations in outdoor ambient humidity. However, the CEB-FIP 2010 model exhibits a certain deviation from the experimental results in outdoor conditions [15]. Therefore, the CEB-FIP 90 model provides more accurate predictions for the long-term deformation behavior of C50 concrete structures.

5 Uncertainty quantification of shrinkage and creep

5.1 Shrinkage and creep database

The shrinkage and creep test database utilized in this study was derived from the latest database developed by Bazant et al. [36,37]. Considering the research focus on concrete creep in girder and building structures, experiments conducted under extreme environmental conditions were excluded. Specifically, data from experiments with RH levels exceeding 95% or falling below 20% were removed. Tests carried out under environmental temperatures above 80 °C or below 0 °C were likewise omitted. The optimized shrinkage and creep data set was combined with the previously mentioned experimental results to establish a revised shrinkage and creep test database. The new creep database consists of 143 test sets, accumulating a total of 5315 data points. The new shrinkage database incorporates 305 test sets, accumulating a total of 39616 data points.

To further explore the representativeness of the creep database, the distribution of key parameters for the concrete specimens contained within the database is illustrated in Fig.13.

In Fig.13, the creep database consisted of a total of 143 sets of experimental data. Among them, 81 sets of samples had 28 d compressive strength f_cm values ranging from 30 to 70 MPa, which accounted for 56.64% of the total. Additionally, the database included 112 sets of samples with 28 d elastic modulus E₂₈ falling within the range of 24000 to 42000 MPa, representing 78.32% of the total. Creep tests were typically conducted on specimens in standard indoor environmental conditions, resulting in relatively smaller dimensions and theoretical thickness. Furthermore, 136 sets of samples in the database had RH ranging from 48% to 72%, making up 95.1% of the total. In summary, this creep database covered fundamental concrete properties and working conditions commonly encountered in daily life, allowing for more accurate predictions of concrete creep effects.

To better assess the applicability of the shrinkage database, the main parameter distribution of concrete in the shrinkage database was presented in Fig.14.

As depicted in Fig.14, the shrinkage database encompassed a total of 305 sets of experimental data. Among them, 262 sets of samples had 28 d compressive strength f_cm values ranging from 20 to 80, accounting for 85.9% of the total. Additionally, the database included 292 sets of samples with 28 d elastic modulus E₂₈ within the range of 20000 to 50000, representing 95.7% of the total. Similar to the creep tests, the shrinkage tests were typically conducted on specimens in standard indoor environmental conditions, resulting in relatively smaller dimensions and theoretical thickness. Furthermore, 300 sets of samples in the database had a RH ranging from 40% to 70%, making up 98.3% of the total. The parameter ranges mentioned above were in good alignment with the performance and working conditions of concrete in actual structures. Therefore, this shrinkage database effectively characterized the shrinkage performance of commonly used concrete.

5.2 Creep function and shrinkage strain uncertainty coefficient

Drawing on the CEB-FIP 90 model predictions and the established shrinkage and creep database, the uncertainty coefficients of the prediction model were introduced as shown in Eqs. (29) and (30).

(29)u_{\mathrm{c}}=\frac{J_{\mathrm{c},\mathrm{e}}}{J_{\mathrm{c},\mathrm{m}}},

(30)u_{\mathrm{s}\mathrm{h}}=\frac{{\epsilon }_{\mathrm{s}\mathrm{h},\mathrm{e}}}{{\epsilon }_{\mathrm{s}\mathrm{h},\mathrm{m}}},

where u_c and u_sh denote the creep and shrinkage uncertainty coefficients, respectively; J_c,e and J_c,m represent the creep functions obtained from the experimental database and the predictive creep model, respectively; ε_sh,e and ε_sh,m represent the experimental shrinkage strain from the shrinkage database and the predicted shrinkage strain from the shrinkage model, respectively. Each uncertainty coefficient was defined as the ratio of the experimental creep coefficient or shrinkage strain to the corresponding value predicted by the CEB-FIP 90 model under varying parameter conditions.

Uncertainty coefficients were determined using Eqs. (29) and (30). The distribution of u_c and u_sh was obtained through statistics, as illustrated in Fig.15. By applying Gaussian fitting to the probability distribution, the uncertainty coefficient distributions were u_c~N (0.85, 0.1225) and u_sh~N (1.54, 0.5184), respectively. These findings suggest that the creep model overpredicted the deformation, whereas the shrinkage model considerably underpredicted the strain.

As depicted in Fig.15, the uncertainty coefficient u_c for creep exhibited a relatively small level of dispersion. Whereas the uncertainty coefficient u_sh for shrinkage displayed a comparatively larger degree of dispersion. The 2.5%–97.5% range of u_c and u_sh was chosen to establish their 95% confidence intervals. These upper and lower bounds of the confidence intervals for u_c and u_sh were subsequently introduced into the calculation model in Subsection 3.3, as shown in Eqs. (31) and (32).

(31)\epsilon (t)={\epsilon }^{\mathrm{m}}(t) +u_c{\epsilon }^{\mathrm{c}}(t)+u_{\mathrm{s}\mathrm{h}} {\epsilon }^{\mathrm{s}\mathrm{h}}( t).

In space, the creep strain increment was

(32)u_{\mathrm{c}} \left\{\mathrm{\Delta }{\epsilon }_n^{\mathrm{c}}\right\}=u_{\mathrm{c}} \left\{{\eta }_n\right\}+u_{\mathrm{c}}C\left(t_n,t_{n-0.5}\right)[\mathit{A}]\left\{\mathrm{\Delta }{\sigma }_n\right\}.

Shrinkage strain did not require iterative calculations. Therefore, it could be directly multiplied by u_sh.

5.3 Long-term deformation uncertainty analysis for 1:4 scale box girder modeling

To evaluate the reliability of the aforementioned uncertainty calculation model, displacement–time curves of mid-span vertical displacements with 95% confidence intervals for model-calculated results were generated, as shown in Fig.16.

As evident from Fig.16, the mid-span long-term deformation test data for the 1:4 scale model fall largely within the 95% confidence interval of the aforementioned established uncertainty prediction model. The 95% confidence interval for the long-term upper arch deflection of the 1:4 scale girder model ranges from 2.60 to 5.17 mm.

6 Long-term deformation uncertainty analysis of 32 m box girders

6.1 Model establishment

Based on the above research results, the uncertainty analysis is introduced into 32 m box girder. The span, overall length, height and top width of girder are 31.5, 32.6, 3.052, and 12.6 m, respectively. The prestressing tendons in the girder consist of 22-hole 9-1 × 7Φ15.2 and 5-hole 8-1 × 7Φ15.2 steel strands. The reinforcement within the girder consists of HRB400 grade deformed steel bars. The longitudinal reinforcement for the top slab, bottom slab, and web comprises 12 mm diameter bars arranged in two layers. While the transverse reinforcement includes bars with diameters of 18 and 16 mm. Shear reinforcement and stirrups are of 16 and 12 mm diameter, respectively [38]. The corresponding parameters were provided in Tab.4 and Tab.5, the relevant calculation parameters of CEB-FIP 90 are consistent with the 1:4 scale model, except for the theoretical thickness of the component h, which for the full-scale model is 400 mm.

Load selection was shown in the standard [41]. The prestress age of the box girder, dead load age of the second phase, and rail laying age were 28, 70, and 70 d, respectively. The ballastless track on the upper part of the box girder of the high-speed railroad was converted into a homogeneous load applied.

6.2 Finite element results

The whole-time diagram of mid-span displacement with time was indicated in Fig.17.

According to Fig.17, the 95% confidence interval for the long-term upward deflection of the 32 m full-scale box girder varied between 1.27 and 2.30 mm. This interval was determined by the uncertainty calculation model mentioned earlier. The age of the girder concrete was 1480 d, and this calculation involved subtracting the box girder mid-span displacement before track laying.

7 Conclusions

Drawing upon the findings of this study, the following key conclusions were reached.

1) The 1:4 scale model box girder, prepared based on the principles of similarity and the equivalent stress in mid-span concrete, exhibited a mid-span vertical displacement of 2.83 mm 95 d after post-tensioning. UMAT computational model for long-term deformation based on the CEB-FIP 90 model effectively represents the trend of mid-span upper arch deflection in the experimental box girder. The trends of vertical displacement of the box girder in the experiments and calculations roughly align. It is projected that the mid-span long-term upward deflection of the 1:4 scale model will reach approximately 3.67 mm 988 d after post-tensioning.

2) Utilizing the simplified and consolidated data from the shrinkage and creep database, and applying the CEB-FIP 90 formula for calculations, we obtained uncertainty coefficients for creep, which exhibit a distribution of u_c~N (0.85, 0.1225). Likewise, the uncertainty coefficients for shrinkage were established with a distribution of u_sh~N (1.54, 0.5184). These findings suggest that the creep model overpredicted the deformation, whereas the shrinkage model considerably underpredicted the strain. However, the shrinkage model considerably underestimates the observed concrete shrinkage strain.

3) The 95% confidence interval for the long-term upper arch deflection of 32 m full-scale box girder ranged from 1.27 to 2.30 mm at the age of concrete 1480 d (subtracting the box girder mid-span displacement before track laying). The uncertainty associated with the prediction model for long-term mid-span deformation of the girder increased progressively with the extension of the prediction period.

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