Fatigue performance analysis of precast segmental assembled concrete beams

Abstract

In recent years, assembled bridges have become widely utilized in bridge construction, raising concerns about durability-related bridge diseases over time. These issues significantly impact the fatigue life of assembled bridges, necessitating an in-depth exploration of their fatigue performance. While existing research primarily concentrates on the transverse connection of multiple longitudinal beams, there is a notable dearth of studies on longitudinal precast segmental assembled bridges. This paper addresses this gap by establishing a fatigue benchmark finite element model for segmental assembled concrete beams, building upon existing experiments. The study employs numerical simulation to analyze the entire fatigue process, examining stress distribution, damage development, and considering the influence of reinforcement corrosion. Furthermore, a fatigue life prediction method, based on fatigue residual strength (R), is proposed for predicting the fatigue life (N) of concrete in precast segmental assembled beams. Results reveal that prestressed and ordinary reinforcements experience increasing stress with loading cycles, peaking around 100,000 cycles. Throughout fatigue loading, compressive stress in concrete remains low, preventing fatigue compression failure. However, tensile stress near joints gradually rises, initiating cracks at the mid-span beam’s bottom. With continued cyclic loading, these cracks propagate towards the loading point. The upper and lower limits of fatigue life predicted by the fatigue life prediction method closely align with the compressive fatigue test values of concrete, proposed fatigue life prediction method is efficient and accurate.

Keywords

precast segmental assembly fatigue life numerical simulation P-R-N curve durability damage

Introduction

In recent years, precast segmental assembly technology has gained widespread adoption in bridge construction. In comparison to traditional on-site pouring construction methods, precast segmental assembly technology allows synchronous construction independent of pier construction progress. It boasts advantages such as standardized production, rapid construction speed, and energy efficiency, making it an environmentally friendly option (Liang et al., 2022). However, the collapse of the 2022 Castro prefabricated segmental assembled bridge in Spain due to fatigue issues has underscored the insufficient research on bridge fatigue performance. Assembled concrete bridges can be broadly categorized into transverse assembly and longitudinal segment assembly. While some studies have investigated the fatigue performance of transverse assembled bridges, such as the work by Lantsoght et al. (2019), who conducted fatigue loading tests on a semi-scale model of a prestressed concrete T-beam, indicating improved punching capacity under cyclic loading due to the pressure film effect of the bridge deck slab between T beams, research on the fatigue performance of longitudinal precast segmental assembled bridges is still limited. Zheng (2019) explored the fatigue performance of such bridges, highlighting that fatigue failure under cyclic loading is attributed to brittle failure caused by fatigue fracture of prestressed reinforcements. To mitigate fatigue crack development and extension, increasing the prestressing level is recommended. While these studies contribute valuable insights into the fatigue properties of assembled concrete bridges, there remains a need for a more comprehensive analysis of the fatigue performance of precast segmental concrete beams throughout their entire lifespan. This paper addresses this gap by conducting a thorough analysis of the entire fatigue process of precast segmental concrete beams.

To comprehensively investigate the fatigue performance of PC beams, it is crucial to examine the fatigue behaviors of both reinforcements and concrete. The fatigue performances of reinforcements and concrete are primarily influenced by factors such as fatigue strength degradation, fatigue stiffness degradation, residual strain accumulation, and the fatigue S-N curve. The initiation of failure in reinforced concrete (RC) bridges commonly stems from the failure of reinforcements due to factors like corrosion and fatigue (Su et al., 2022). The fatigue life of a well-designed RC beam is contingent upon the dimensions and quantity of reinforcements (Prashanth et al., 2019). Regarding the fatigue performance of reinforcements, Han et al. (2015) identified three stages of residual strain in ordinary reinforcements in partially precast (PC) beams under fatigue loading, proposing a model for predicting fatigue residual strain. Precast segmental assembled beams share fundamental characteristics with PC beams. Consequently, numerous scholars have delved into research on the fatigue performance and fatigue life of PC beams. Du et al. (2020) identified fatigue fracture of tensile reinforcement at the bottom of the pure bending section as the fatigue mode of PC beams through fatigue testing. Additionally, the synergistic impact of corrosion and fatigue accelerates fatigue damage accumulation, diminishing the fatigue life of concrete beams. Zhang et al. (2012) and Ma et al. (2018) considered the effects of corrosion-induced cracking and the reduction in fatigue stiffness, noting more complex stress redistribution and shorter residual fatigue life in corroded concrete beams. Given that certain assembled concrete beams are exposed to chloride environments for extended periods, it becomes imperative to consider the fatigue performance of corroded precast segmental assembled concrete beams. Furthermore, concerning the fatigue performance of concrete, its stiffness degradation exhibits a three-stage developmental pattern (Du et al., 2020; Wang and Li, 2022; Yadav and Thapa, 2020). To enhance the fatigue life of concrete beams, various researchers (Hosseini et al., 2017; Rojob and El-Hacha, 2018; Wu et al., 2022) have proposed methods employing materials such as carbon fiber-reinforced polymer (CFRP) and iron-based shape memory alloy (Fe-SMA). As both reinforcements and concrete play pivotal roles in precast segmental concrete beams, the studies on PC beams lay a crucial foundation for exploring the fatigue performance of precast segmental concrete beams.

The Stress-Fatigue Life Curve (S-N) method is commonly employed for evaluating the fatigue life of PC beams. Researchers such as Du et al. (2020), Wang and Li (2022), and Deng et al. (2022) have proposed expressions for the S-N curve of concrete and reinforcements under diverse conditions. However, owing to the discrete nature of the mechanical properties of reinforcements and concrete, some scholars have utilized probability statistics and reliability theory to analyze fatigue strength and life, predicting fatigue life based on fatigue strength. Gao et al. (2019) established the Generalized Density Evolution Equation (GDEE) for residual fatigue life by integrating the Probability Density Evolution Method (PDEM). Su et al. (2022) proposed a fatigue life prediction method for corroded PC beams, incorporating the coupling effect of corrosion and fatigue. The accurate prediction of fatigue life is essential to ensure the safety and durability of bridges. Previous studies have primarily focused on fatigue life prediction models for bridges. However, these models often neglect the influence of different failure probabilities. Additionally, a fatigue life prediction model specific to precast segmental concrete beams has yet to be proposed.

The objective of this paper is to propose a fatigue life prediction model considering failure probability for precast segmental concrete beams. The subsequent sections are organized as follows: In the section “Theoretical basis”, an analysis of the fatigue parameters required for building the finite element model (FE model) is presented. The paper proposes the fatigue strength degradation equation for steel and introduces the compressive fatigue constitutive model for concrete. In the section “Numerical simulation” outlines the establishment of a fatigue benchmark finite element model for precast segmental concrete beams. This model is based on the combination of 16 groups of fatigue parameter degradation models. The proposed model undergoes a comprehensive examination to analyze stress, strain, and fatigue damage. Therefore, in the section “Fatigue life prediction probability model” of this paper, considering the discrete distribution of fatigue life in the joint concrete of prefabricated prestressed beams, a method is proposed based on the probability distribution of fatigue residual strength to estimate the fatigue life and its probability distribution of failure in the joints of prefabricated prestressed beams. Utilizing probability statistical methods, the residual strength distribution at low cyclic loading cycles is employed to estimate fatigue life and obtain the distribution of fatigue life failure probability. The effectiveness of this method is verified through existing experiments, confirming its validity.

Theoretical basis

Fatigue degradation model

Fatigue stiffness degradation

The fatigue stiffness degradation of concrete primarily manifests as a reduction in elastic modulus, gradually diminishing with an increase in cyclic loading times. This attenuation of the elastic modulus adheres to a three-stage law (Lei and Song, 2013). This law is contingent upon the ratio of cyclic loading times to fatigue life (N/N_f) and remains independent of load stress amplitude and fatigue life. The life ratios (N/N_f) for the three degradation stages are approximately 10%, 80%, and 10%, respectively. Holmen (1982) derived the concrete degradation equation for elastic modulus through fitting concrete fatigue test data, as represented in equation (1).

E_{c} (N) = (1 - 0.33 \frac{N}{N_{f}}) E_{0}

(1)

where E_c (N) is the unloading elastic modulus of concrete after N cyclic loading, and E₀ is the initial elastic modulus of concrete.

Building on this foundation, Lei and Song (2013) derived a new stiffness degradation model by applying logarithms to fatigue loading times and fatigue life in equation (1). This transformation aligns with the S-N curve of concrete, which adheres to the double logarithm law. To identify an appropriate fatigue stiffness degradation model, a comparison of fatigue test data for concrete proposed by various scholars (Li et al., 2014; Ou and Lin, 1999; Tang et al., 2007; Wang et al., 2003) was conducted, and the outcomes are consolidated in Figure 1.

Figure 1.

Attenuation law of fatigue elastic modulus of concrete.

As depicted in Figure 1, Wang’s fatigue test results (Wang et al., 2003) indicate that the ratio of the elastic modulus to the initial elastic modulus of concrete specimens near fatigue failure ranges between 0.474 and 0.757, with an average value of 0.63. For Li’s fatigue test results (Li et al., 2014), the elastic modulus ratio falls between 0.678 and 0.725, averaging at 0.69. In Tang’s fatigue test results (Tang et al., 2007), the elastic modulus ratio spans from 0.677 to 0.725, with an average value of 0.7. While Li’s model (Li et al., 2014) captures the three-stage law of stiffness degradation, there is a considerable deviation from the analytical prediction and test results. On the other hand, Holmen’s model (Holmen, 1982), characterized by linear decay, yields an elastic modulus ratio of 0.67 near fatigue failure. It is evident that the Holmen model aligns more closely with the test values compared to the Lei model and effectively predicts the attenuation pattern of the elastic modulus during the concrete fatigue process. Therefore, the Holmen model is selected and applied in this context.

Scholars have reached a consensus on the law of fatigue stiffness degradation of reinforcements (Feng et al., 2006; Li et al., 2006; Wang et al., 2016). According to their collective understanding, the fatigue stress experienced by reinforcements during fatigue tests is typically significantly lower than their yield strength and predominantly falls within the elastic phase. Consequently, the degradation of stiffness is generally not considered during the cyclic loading process.

Fatigue residual strength model of concrete

Similar principles governing the degradation of strength can be discerned through a comparative analysis of different fatigue residual strength models for concrete (Meng, 2006; Zhu, 2010). The initial and fatigue failure boundary conditions of the residual strength model are characterized by the stresses corresponding to the material’s static load and the fatigue upper limit load, respectively. Additionally, the residual fatigue strength experiences a gradual decline with an increase in cyclic loading times, degrading slowly in the initial stages and declining rapidly as the material approaches failure. Balaguru and Shah (1982) proposed a straightforward equation for the fatigue residual tensile strength of concrete, with the initial tensile strength (f_t) as the main parameter, as depicted in equation (2). Subsequently, Zhu (2010) introduced equations for calculating the fatigue residual tensile and compressive strengths of concrete, as shown in equations (3) and (4), respectively. In equations (3) and (4), x (N) is a function related to the fatigue loading times, considering both the initial tensile strength (f_t) and the initial compressive strength (fc) of concrete.

σ_{t, N} = f_{t} (1 - \frac{1}{10.953} \log_{10} N)

(2)

σ_{r, c} (N) = f_{r, c} \frac{x (N)}{α_{d} {[x (N) - x (1)]}^{2} + x (N)}, x (N) \geq 1

(3)

σ_{r, t} (N) = f_{t} \frac{x (N)}{α_{t} {[x (N) - x (1)]}^{1.7} + x (N)}, x (N) \geq 1

(4)

where σ_r,t (N) and σ_r,c (N) means the fatigue residual tensile and compressive strength of concrete under cyclic loading N times, respectively; α_d and α_t are the parameter values of the concrete uniaxial compression and tension

σ - ε

curves, respectively; x (N) is an equation related to the fatigue loading times, and the value is x (1) = 1.

Fatigue residual strength model of reinforcements

As the fatigue failure of reinforcements typically manifests through a reduction in the effective cross-sectional area (Balaguru and Shah, 1982; Wang et al., 2016; Zhu, 2010), this reduction is commonly employed as an indicator for investigating the fatigue residual strength of reinforcements. Zhu (2010) derived the equation for the fatigue residual strength of reinforcements under arbitrary fatigue loading, utilizing the envelope curve of fatigue residual strength and the S-N double logarithmic curve of reinforcements, as presented in equation (5). Notably, equation (5) reveals that the fatigue residual strength is not equal to the maximum stress value (σ_s,max) corresponding to the fatigue upper limit load when the cycle loading times reach the fatigue life. Wang et al. (2016) also proposed a fatigue strength degradation equation for reinforcements based on the reduction of the effective cross-sectional area, as illustrated in equation (6). However, it is apparent from equation (6) that Wang overlooks the fact that the reinforcement S-N curve follows a double logarithmic form. In this paper, an improved fatigue residual strength model for reinforcements is introduced, combining the Wang and Zhu models into a double logarithmic curve, as outlined in equation (7). The accuracy and applicability of equation (7) are demonstrated by establishing the fatigue benchmark finite element model in the section “Fatigue benchmark finite element model”.

f_{y} (N) = (σ_{\min} + Δ σ) [1 - \frac{\log_{10} N}{\log_{10} N_{f}} (1 - \frac{Δ σ + σ_{\min}}{f_{y}})]

(5)

f_{y} (N) = f_{y} [1 - \frac{N}{N_{f}} (1 - \frac{σ_{s, \max}}{f_{y}})]

(6)

f_{y} (N) = f_{y} [1 - \frac{\log_{10} N}{\log_{10} N_{f}} (1 - \frac{σ_{s, \max}}{f_{y}})]

(7)

where f _y (N) means the fatigue residual strength of reinforcement under cyclic loading N times; σ_min is the stress of reinforcement corresponding to the fatigue lower limit load; Δσ is the allowable stress amplitude for fatigue action of reinforcement; f _y is the initial yield strength of reinforcement.

Fatigue residual strain of concrete

The attenuation pattern of concrete strain mirrors that of concrete stiffness degradation, following the three-stage law (Lei and Song, 2013). Matsushita and Tokumitsu (1979) proposed a fatigue residual strain equation considering the stress ratio, presented in equation (8), with constants applicable for concrete grades ranging from C20 to C60. Wang et al. (1991) conducted a fitting of experimental data based on Holmen’s (1982) residual strain equation and introduced the fatigue residual strain equation for concrete, as outlined in equation (9).

Δ ε_{r} (N) = \frac{f_{c u}}{E_{0}} l o {g_{10}}^{- 1} (3.92 \frac{σ_{\max} - σ_{\min}}{f_{c u} - σ_{\min}} - 4.66) N^{0.29}

(8)

Δ ε_{r} (N) = Δ ε_{r} (1) + \frac{0.00105 ε_{\max}^{1.98} {(1 - ε_{\min} / ε_{\max})}^{5.27}}{ε_{u n s t a b}^{1.41}} N^{0.395}

(9)

where Δε_r (1) and Δε_r (N) are the residual strain after the first loading and the residual strain after cyclic loading N times; ε_max and ε_min are the concrete strain corresponding to the upper limit stress and the lower limit stress, respectively; ε_unstab is the concrete strain corresponding to the peak stress at one loading failure.

Fatigue S-N curves

In consideration of the fatigue parameters degradation equation for steel and concrete (i.e., equations (1) and (5)–(7) in the section “Fatigue degradation model”), it is imperative to calculate the fatigue life (N_f). In scenarios where a fatigue test is unavailable, the fatigue life is typically derived from the fatigue S-N curve.

Fatigue S-N curve of reinforcements

In this section, we present a compilation and analysis of fatigue S-N curves for ordinary reinforcements as per specifications and those proposed by researchers (Raithby, 1983; Rojob and El-Hacha, 2018; Shi and Zheng, 1989; Wang et al., 2003; Xu, 2000; Zeng and Li, 1999). It is observed that these S-N curves are derived through linear regression of test data under specific conditions, introducing certain limitations. The fatigue S-N curves, along with relevant test data for fatigue life (Jiang et al., 2010; Lu et al., 2017; Zhong et al., 2005; Zhu, 2011), are summarized in Figure 2.

Figure 2.

Fatigue S-N curves of ordinary reinforcements.

The S-N curves of steel reinforcement estimated under ground stress amplitudes by Moss, Tilly, JREA, and Zhong Ming exhibit excessively long fatigue life, while the estimation of steel reinforcement fatigue life under high-stress amplitudes is overly conservative. Therefore, existing models fail to accurately predict the fatigue life of retrofitting materials. This paper addresses this issue by proposing fatigue S-N curve equations for common retrofitting materials through fitting test data, as shown in equation (10), to solve this problem. The suggested fatigue life estimation of deformed reinforcement S-N curves in this paper avoids the issue of overly conservative predictions under both low and high-stress amplitudes.

\log_{10} N = 11.97 - 2.68 \log_{10} Δ σ

(10)

To demonstrate the precision of the proposed S-N curve equation, the curve is also presented in Figure 2. As depicted in the figure, the proposed S-N curve exhibits the highest prediction accuracy, with the estimated fatigue life of reinforcements closely aligning with the test data under both high-stress and low-stress amplitudes.

Fatigue S-N curve of concrete

The equations of concrete fatigue S-N curves are divided into the following two common forms according to whether the stress ratio K is considered, which can be expressed as equations (11) and (12).

S_{\max} = A - B \log_{10} N_{f}

(11)

S_{\max} = 1 - β (1 - K) \log_{10} N_{f}

(12)

where A, B and β are constants, which can be obtained by fitting the test data.

Similar to ordinary reinforcements, the aforementioned S-N curves of concrete are also derived through linear regression of the test data under specific conditions, and they possess certain limitations. The concrete fatigue S-N curves along with the uniaxial fatigue test data (Cao, 2004; Jin-Keun and Yun-Yong, 1996; Meng, 2006; Song et al., 2008; Wu et al., 1994; Yang, 2015; Zhao et al., 1993) are illustrated in Figure 3 (with the stress ratio K and the minimum stress level Smin set at 0.1).

Figure 3.

Some common fatigue S-N curves of concrete.

As can be seen from Figure 3, the fatigue S-N curves and the test data have a clear boundary at S_max = 0.75. Therefore, in this paper, S_max = 0.75 is selected to take as the dividing point to describe the S-N curve, and the recommended concrete fatigue S-N curve is proposed and given by equation (13).

\begin{array}{l} S_{\max} = 1 - 0.0565 l o g_{10} N_{f}, S_{\max} > 0.75 \\ S_{\max} = 1.038 - 0.064 l o g_{10} N_{f}, S_{\max} \leq 0.75 \end{array}

(13)

Fatigue constitutive model

Fatigue constitutive model of concrete

Zhu and Zhu (2012) and Wang et al. (2016) successively established the fatigue constitutive model of concrete. However, the version of the specification used by Zhu and Zhu (2012) is outdated, and some parameters in Wang’s model (Wang et al., 2016) are inconsistent with the new specification, as expressed in equation (14). Therefore, based on the concrete uniaxial static load constitutive relationship in the Design Code for Concrete Structure (GB 50010-2010, 2015) and Zhu’s model (Zhu and Zhu, 2012), a new concrete fatigue constitutive model is proposed in this paper, presented in equation (15). The applicability and correctness of the new formula in equation (15) are demonstrated by establishing the fatigue benchmark finite element model in the section “Fatigue benchmark finite element model”.

σ = {\begin{cases} \frac{ρ_{c}^{n}}{n - 1 + x^{n}} E_{N} [ε - ε_{f, r} (N)], x_{c} \leq 1 \\ \frac{ρ_{c}}{α_{c} {(x - 1)}^{2} + x} E_{N} [ε - ε_{f, r} (N)], x_{c} > 1 \end{cases}

(14)

in which

ρ_{c} = \frac{f_{c N}}{E_{N} ε_{c r}}, x_{c} = \frac{ε - ε_{f, r} (N)}{ε_{c r}}, n = \frac{E_{N} ε_{c r}}{f_{c N}}

σ (N) = {\begin{cases} E_{c} (N) ε & Δ ε_{r} (N) \leq ε \leq ε_{e} (N) \\ \frac{ρ_{c} n E_{c} (N) ε}{n - 1 + {(ε / ε_{c, r} (N))}^{n}} & ε_{e} (N) < ε \leq ε_{c, r} (N) \\ \frac{f_{c, r} (N) (ε / ε_{c, r} (N))}{α_{c} {[(ε / ε_{c, r} (N)) - 1]}^{2} + ε / ε_{c, r} (N)} & ε > ε_{c, r} (N) \end{cases}

(15)

in which

ρ_{c} = \frac{f_{c, r} (N)}{E_{c} (N) ε_{c, r} (N)}, n = \frac{E_{c} (N) ε_{c, r} (N)}{E_{c} (N) ε_{c, r} (N) - f_{c, r} (N)}

Constitutive relationship of reinforcement, considering the residual strain of reinforcement. Therefore, based on the double linear model and the double diagonal model, fatigue constitutive models for ordinary reinforcements and prestressed reinforcements can be established, as expressed in equations (16) and (17), respectively.

σ (N) = {\begin{cases} E_{s} ε (N) & Δ ε_{r} (N) < ε (N) \leq ε_{y} (N) \\ f_{y} (N) & ε (N) > ε_{y} (N) \end{cases}

(16)

σ (N) = {\begin{cases} E_{s} ε (N) & Δ ε_{r} (N) < ε (N) \leq ε_{y} (N) \\ f_{y} (N) + \frac{f_{u} (N) - f_{y} (N)}{ε_{u} (N) - ε_{y} (N)} (ε (N) - ε_{y} (N)) & ε_{y} (N) < ε (N) \leq ε_{u} (N) \end{cases}

(17)

where σ (N) and ε (N) are fatigue stress and fatigue strain of reinforcement, respectively; E_s is the elastic modulus of reinforcement; f_y (N) and f_u (N) are the fatigue residual yield strength and ultimate strength of reinforcement under cyclic loading N times, respectively; ε_y (N), ε_u (N) and Δε_r (N) are the yield strain value, ultimate strain value and residual strain value of the reinforcement under cyclic loading N times, respectively.

Numerical simulation

To investigate the fatigue performance of precast segmental assembled concrete beams, a fatigue benchmark finite element model is constructed to simulate the entire fatigue process. The finite element model under static loading is created using ABAQUS software, and its validity is confirmed by comparing the simulated results with experimental data from tests (Zheng, 2019). Subsequently, based on the fatigue parameters degradation model, the S-N curves, and the fatigue constitutive models presented in the section “Theoretical basis”, the fatigue benchmark finite element model is established, and the complete fatigue process is numerically simulated.

Finite element model

The experimental beam employed in the study (Zheng, 2019) comprises three prefabricated segment joints bonded with adhesive. For clarity, these segments are designated as Segment L, Segment M, and Segment R. Each side of the joint features a single-key shear key, with key tooth dimensions referenced from completed practical engineering projects, designed in a trapezoidal structure. The specimen and shear key dimensions are depicted in Figure 4. The experiment follows a multistage loading procedure, with an initial load of 20 kN per stage until reaching 60 kN. Subsequent loading increments by 10 kN per stage until specimen failure, with the failure load recorded. A 1-min holding period is maintained at the end of each loading stage, ensuring sufficient stress on the beam before advancing to the next stage. Cracks in the components are observed during loading, with timely records made. The progression of cracks and the mode of failure in the components are meticulously observed.

Figure 4.

Dimensional diagram of specimen and shear key. (a) Elevation. (b) I-I. (c) Shear key.

In this study, a 3-segment precast segmental assembled concrete beam is chosen for finite element modeling. The dimensions of the 3-segment precast segmental assembled concrete beam are 0.8 m × 0.12 m × 0.22 m, as illustrated in Figure 5. The concrete grade is C55, and the protective layer thickness is 15 mm. The constitutive relationship of concrete is represented by the plastic damage model. The ordinary reinforcements are HRB335 with a diameter of 8 mm, and the prestressed reinforcements are PSB785 with a diameter of 25 mm. The constitutive relationship of ordinary and prestressed reinforcements is modeled by the double linear model and the double diagonal model, respectively. The unit type for concrete and reinforcements is brick units C3D8R and truss units T3D2, respectively.

Figure 5.

Configuration of segmental assembled concrete beam. (a) Elevation. (b) Side view. (c) Plan.

From the observations of static and fatigue crack initiation and failure modes in the segment precast glued beams, it is evident that there is no significant slip on both sides of the glued joint, and the joint itself exhibits good overall integrity. It can be assumed that the structural adhesive remains in an elastic state throughout the entire process of static and cyclic loading. Additionally, Zheng (2019) indicates that the average thickness of the adhesive joint is 0.48 mm with a standard deviation of 0.23 mm. Relative to the dimensions of the test beam itself, the thickness of the adhesive joint can be neglected. Considering these factors, the structural adhesive elements are not considered in the numerical simulation, and the tie constraint functionality in ABAQUS is employed to simulate the interaction between segments.

The interaction between the reinforcement and the concrete is simulated using the “embedded” command, with neglect of bond slip. To prevent local damage near the load application point, support positions, and the two ends of the prestressed reinforcement, steel pads are placed for elastic transition. The boundary conditions incorporate simply supported constraints, where the displacements in three directions at one end are individually constrained, along with the longitudinal and transverse angular displacements. At the other end, the linear displacements in the transverse and vertical directions, as well as the angular displacements in the longitudinal and transverse directions, are constrained. The mesh size of the finite element (FE) model is set at 10 mm to ensure sufficient accuracy, as depicted in Figure 6. It is crucial to note that the elastic modulus of the plain concrete near the joint decreases within 20 mm on both sides of the joint in the finite element model. The elastic modulus reduction coefficient at the joint is determined as 0.72 based on the concrete strain values at the midspan and the joint in the conducted test (Zheng, 2019). In the context of concrete under compression, the maximum plastic damage factor, as determined through the energy method, is 0.907, while the minimum is 0.092. For concrete under tension, the maximum plastic damage factor obtained is 0.939, and the minimum is 0.286.

Figure 6.

Schematic diagram of finite element model.

For the plastic damage model, there are five key input parameters: the expansion angle is 30°, the eccentricity is 0.1, f_b0/f_c0 is 1.16, K is 0.6667, and the viscous parameter is 0.0005.

The validation of finite element model

The static load analysis involves applying an external load to the finite element model, and the load-deflection simulation output results are compared with the test results, as illustrated in Figure 7(a) and (b). To ensure model accuracy, two simulation curves are plotted and compared with the test curve. Simulation curve I represents the load-deflection simulation curve without considering the decrease in elastic modulus of concrete near the joint, while simulation curve II represents the load-deflection simulation curve considering this decrease (refer to Figure 7(a)). The accuracy of numerical simulation results is directly influenced by the plastic damage factor of concrete. To determine the appropriate calculation method for the damage factor, three simulation curves are plotted and compared with the test curve, as presented in Figure 7(b). The damage factors for these simulation curves are calculated using the graphic method (Abaqus user conference), the Zhang Formula (Zhang et al., 2008), and the energy method (Yang et al., 2017), respectively.

Figure 7.

Load-deflection curves. (a) Load-deflection curves of static load simulation results. (b) Load-deflection curves considering different plastic damage factors.

As depicted in Figure 7(a), the three load-deflection curves exhibit the same trend and can be delineated into two stages. In the initial stage, the load increases linearly from 0 to the point of cracking, showing a rapid ascent in the load-deflection curves. Upon reaching the cracking load and transitioning to the second stage, the curves exhibit a gradual slowing down, accompanied by the generation of concrete cracks. During this stage, the tensile stress is gradually shouldered by the reinforcements. Furthermore, it is evident that simulation curve II aligns closely with the test curve, with simulation curve II outperforming simulation curve I.

The maximum discrepancy between the values of simulation results in curve I and the experimental results is 17.7%. On the other hand, the maximum error between the values of simulation results in curve II and the experimental results is 7.6%, falling within the acceptable threshold of 10%. This observation highlights that finite element models incorporating the reduction factor of elastic modulus are more precise than those neglecting this factor, rendering them suitable for numerical simulation analyses across the entire fatigue process. Given the diverse elastic moduli of different materials, an inappropriate choice may introduce errors in predicting stress concentration regions. Accounting for the reduction in elastic modulus enhances the accuracy of predicting stress concentration areas. Furthermore, the elastic modulus directly influences the material’s deformation response, emphasizing the significance of a correct modulus selection for accurate predictions of deformation distribution.

As depicted in Figure 7(b), among the three curves, the load-deflection curve derived from the energy method exhibits the highest degree of agreement with the test curve. Consequently, the energy method is chosen for calculating the damage factor in the numerical simulation analysis across the entire fatigue process.

Fatigue benchmark finite element model

In Chapter 2, leveraging the fatigue constitutive models for concrete and steel proposed by scholars, along with various fatigue parameter degradation models, we systematically arranged combinations of fatigue parameter degradation models necessary for the fatigue constitutive model. This process led to the generation of multiple fatigue constitutive models. Utilizing the simplified fatigue loading cycles, each model was input into the Abaqus finite element software, and computations were executed to obtain concrete and steel stress results under varying loading cycles. Subsequently, these results were meticulously compared with experimental data to discern the optimal combination, thereby establishing the benchmark finite element analysis model for fatigue as shown in Figure 8.

Figure 8.

The building process of fatigue benchmark finite element model.

Due to the inherent characteristic of steel, where its elastic modulus remains essentially unchanged during cyclic loading, and considering the significantly higher elastic modulus of steel compared to concrete, the variations in strain are more pronounced in concrete than in steel. Consequently, when evaluating the accuracy of numerical simulations, it is more appropriate to assess the degree of accuracy based on errors in concrete strain. The fatigue finite element models derived from the 16 combinations underwent fatigue loading analysis. Figure 9 presents the comparison errors between the simulated concrete strain values under different fatigue loading cycles and the experimental results.

Figure 9.

Errors of compressive strain of concrete under different cyclic loading times.

As depicted in Figure 9, the 10th group exhibits the smallest comprehensive error value in the calculated compressive strain of concrete among the 16 groups. Consequently, the fatigue benchmark finite element model was constructed using the fatigue parameters and constitutive models of the 10th group. The load-deflection curves of the fatigue benchmark finite element model at the midspan under various cyclic loading times (50,000, 100,000, 300,000, 500,000, 800,000) are presented in Figure 10.

Figure 10.

Load-deflection curves in the midspan under different cycle loading times.

As illustrated in Figure 10, the load-deflection curves at the midspan, obtained through finite element simulation, closely align with the test curves under varying cyclic loading times. The slope of the curves and its changing trend mirror the test results, indicating the accuracy of the fatigue benchmark finite element model. In the static load test, the initial load-deflection curve exhibits nearly linear growth. When the load reaches approximately 80 kN, the growth rate of the curve slows down, signifying a reduction in the beam’s stiffness. This behavior results from stress redistribution after the model reaches the cracking load. Concurrently, concrete cracks undergo cyclic states of opening and shrinking under cyclic loading. When the cyclic loading times reach 500,000 and 800,000, the slope of the load-deflection curves shows no significant change at around 80 kN, and the entire curves are approximately linear. This observation suggests that, after reaching certain cyclic loading times, the internal cracks in concrete gradually stabilize, and the phenomenon of stress redistribution weakens.

Numerical simulation analysis of whole fatigue process

Stress analysis of reinforcements

Under the influence of fatigue loading, the tensile stress in prestressed reinforcements, as well as the tensile and compressive stresses in ordinary reinforcements, at different cyclic loading times (n = 0, 50,000, 100,000, 300,000, 500,000) are depicted in Figure 11. Observing Figure 11, the three stress curves for reinforcements exhibit similar trends. Initially, the stresses in both ordinary and prestressed reinforcements experience a rapid increase with fatigue cycles, followed by a slower development when the cyclic loading times reach 100,000, aligning with the three-stage law of fatigue damage in reinforcements. The tensile and compressive stresses in ordinary reinforcements remain relatively small, indicating the elastic stage.

Figure 11.

The simulation values of reinforcement stress under different cyclic loading times.

Stress analysis of concrete

The principal tensile and compressive stresses in concrete under cyclic loading for the fatigue benchmark finite element model are illustrated in Figure 12(a) and (b). Examining Figure 12(a), during the initial stage of fatigue loading (1 to 100,000 cycles), the concrete tensile stress gradually increases from the bottom of the midspan towards the middle and upper zones of the beam ends, accentuated by the tensile stress near the joint. In the middle stage of fatigue loading (100,000 to 500,000 cycles), the tensile stress in the midspan concrete slightly increases in the middle and lower zones. The upward propagation of concrete cracks is impeded by prestressed reinforcements, manifesting the phenomenon of stress redistribution. In the final stage of fatigue loading (500,000 to 800,000 cycles), the concrete tensile stress in the middle and lower zones of the midspan continues to rise, and the zone of tensile stress in the beam gradually disperses.

Figure 12.

Principal stress of concrete under different fatigue cycles (unit: MPa). (a) Tensile stress. (b) Compressive stress.

As depicted in Figure 12(b), the compressive stress in the concrete’s compressed zone gradually decreases with the increasing cyclic loading times. This phenomenon arises from the progressive degradation of concrete strength and stiffness under fatigue loads. Notably, the principal compressive stress in the concrete’s compressed zone, under various cyclic loading times, has not reached the standard compressive strength value, remaining essentially in the elastic stage. Additionally, the transfer of pressure follows a pattern from the loading zone at the top of the midspan to the bottom support, aligning with real-world scenarios. As cyclic loading times increase, the tensile zone expands from the bottom of the midspan to the upper ends of the beam.

Subsequently, the maximum tensile stress in prestressed reinforcements and the maximum compressive stress in concrete are transformed into simulated values for the maximum tensile strain in prestressed tendons and the maximum compressive strain in concrete using the fatigue constitutive model under varying cyclic loading times. The comparison between simulated and experimental values for the maximum strain of the materials is illustrated in Figure 13, where it can be observed that the concrete strain diminishes before reaching a cyclic loading times count of 800,000.

Figure 13.

The strain values of concrete and prestressed reinforcement under different cyclic loading times.

As depicted in Figure 13, the strain simulation curve for the material aligns with the strain test curve, exhibiting an increase with cyclic loading times. Initially, the strains of prestressed reinforcement and concrete experience rapid growth with fatigue cycles, followed by a slower development when the cyclic loading times reach 100,000, consistent with the three-stage law of fatigue damage for reinforcements and concrete. Moreover, the maximum error in tensile strain for prestressed reinforcement between simulated and test values is 13%, and the maximum error in compressive strain for concrete between the two values is 12%.

Fatigue damage analysis

Under different cyclic loading times, the compressive damage of concrete is slight, and no fatigue damage occurs. The tensile damage of concrete under different cyclic loading times is shown in Figure 14.

Figure 14.

Fatigue damage of concrete under different cyclic loading times.

As depicted in Figure 14, during the initial stage of cyclic loading (n = 50,000 cycles), cracks initially appear in the tensile zone of the midspan concrete. With the progression of cyclic loading times, the concrete cracks in the midspan tension zone gradually increase and develop upward. At n = 300,000 cycles, cracks manifest at the joint. As the loading cycles continue to increase until 800,000 cycles, midspan cracks persist in vertical development, and the cracks at the joints extend towards the loading points on the top of the beam. A comparison of the crack distribution in the simulated beam with the experimental beam (Zheng, 2019) reveals identical crack locations and lengths, affirming the correctness and validity of the finite element model.

The schematic diagram of crack distribution in the experimental beam is depicted in Figure 15 (Zheng, 2019). Observations reveal that at 0 thousand cycles of static loading, under a 60 kN load, vertical bending cracks emerged near the right side of the mid-span on the front face of the test beam. Upon increasing the load to 80 kN, the cracks extended upward, reaching a length of approximately 6 cm. Almost complete closure of the cracks occurred after unloading. At 50 thousand cycles of static loading, during the loading process, the vertical cracks slightly extended, accompanied by an increase in crack width. Post-unloading, the cracks almost completely closed. Over 300 thousand cycles of cyclic loading, two short vertical cracks appeared at the midpoint of the front face of the beam, each with a length of about 3 cm. By 500 thousand cycles, under sinusoidal cyclic loading, existing cracks opened and closed, and vertical cracks on the back face extended further without the formation of new cracks. At 690 thousand cycles, new cracks emerged within a 2 cm range on the left side of the front face of the beam near the joint. These cracks extended upward to 5 cm on the front face of the test beam, traversing the width of the beam. Cracks on the back face, approximately 1 cm away from the joint, extended upward to 6 cm from the bottom of the beam. The original mid-span vertical cracks continued their upward extension. At 910 thousand cycles, the beam experienced sudden failure, and the test force was automatically unloaded.

Figure 15.

Schematic diagram of crack distribution in the experimental beam. (a) Diagram of front cracks development. (b) Frontal failure pattern. (c) Diagram of back cracks development. (d) Backside failure pattern. (e) General distribution of cracks of test beam.

Upon comparing the finite element simulation of crack locations (Figure 15), it is evident that the finite element results align well with the experimental results. The occurrence of cracks near the joint during the finite element simulation may be attributed to the lower tensile strength of the bond layer between the structural adhesive and the surface of the segment beam concrete, resulting in cracks when the bond layer fails. Additionally, the concrete around the joint, being plain concrete with lower strength, is susceptible to damage and cracking. Furthermore, in the finite element simulation, the cracking time of the concrete near the joint occurred earlier than in the experimental process, attributed to the simulation not accounting for the actual structural adhesive strength being higher than the concrete strength.

Fatigue life prediction probability model

Fatigue life prediction method based on fatigue residual strength

According to Section 3.4.3, the fatigue damage analysis of concrete reveals that the plain concrete near the joint is the fatigue-vulnerable region, and its fatigue life may impact that of the precast segmental assembly beam. To precisely predict the fatigue life of precast segmental assembly beams, this section presents a method for predicting the fatigue life and probability distribution of concrete at joints based on fatigue residual strength.

The fatigue S-N curve of concrete exhibits significant discreteness. This is attributed to the fact that the fatigue life determined by the S-N curve represents the mean value of fatigue life under various stress levels or the fatigue life with a fatigue failure probability of 50%. Consequently, the cyclic loading number N is selected as the abscissa of the failure probability-residual strength-fatigue life (P-R-N) curve rather than the stress amplitude S.

Application of fatigue life prediction probability model

This approach can be employed to predict both the compressive and tensile fatigue lives of concrete. The study utilizes the compressive fatigue life as an illustration, demonstrating more pronounced numerical variations. In the compressive fatigue test (Meng, 2006), stress level S₁ = 0.750 σ0 was cyclically loaded 10,000 times and 20,000 times, while stress level S₂ = 0.850 σ0 was cyclically loaded 500 times and 1000 times, respectively, where σ0 represents the mean of the initial compressive strength.

P-R-N curve

The strength of concrete is typically assumed to follow a normal distribution (Meng, 2006). This implies that the mean and standard deviation of the sample can be utilized to compute the P-R expressions. Consequently, five groups of specimens were tested for initial compressive strength and fatigue residual compressive strength under various stress conditions and fatigue loading times, and the results were employed to calculate the sample means and standard deviations. The formulas of the P-R relationship for different cycle loading times at maximum stress levels S₁ and S₂ can be used to determine the fatigue residual strength of concrete for various failure probabilities, as shown in Table 1, where S₀ represents the initial strength.

Table 1.

Experimental date and P-R relationship expressions.

Group	Stress level	Cycle loading times	Mean of sample	Variance of sample	P-R relationship
1	S ₀	0	24.47	1.74	$P (σ_{r} \leq r) = Φ (\frac{r - 24.47}{1.319})$
2	S ₁	1 × 10⁴	23.61	2.15	$P (σ_{r} \leq r) = Φ (\frac{r - 23.61}{1.466})$
3	S ₁	2 × 10⁴	22.77	2.39	$P (σ_{r} \leq r) = Φ (\frac{r - 22.77}{1.546})$
4	S ₂	5 × 10²	23.57	2.07	$P (σ_{r} \leq r) = Φ (\frac{r - 23.57}{1.439})$
5	S ₂	1 × 10³	22.89	2.19	$P (σ_{r} \leq r) = Φ (\frac{r - 22.89}{1.480})$

The fatigue residual strength (R) of concrete can be calculated by setting p = .1, .2, …, .8, .9 in the P-R relationship expressions in Table 1. Subsequently, using the R-N fitting curves (refer to Section 2.1.2), the fatigue life of concrete at different failure probabilities for the stress levels S₁ and S₂ can be obtained. The results are presented in Table 2.

Table 2.

Fatigue life for different failure probabilities under stress levels S₁ and S₂.

Failure probability	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
Fatigue life under stress level S₁	15,401	18,214	20,562	22,806	25,632	28,181	31,275	35,101	41,004
Fatigue life under stress level S₂	998	1245	1356	1611	1747	1877	2046	2303	2724

Taking the stress level S₂ as an example, the P-R-N fitting curves with the values of probabilities taking as p = .1, .2, .7 and .8 are depicted in Figure 16. In Figure 16, when σ_r = 20.80 MPa (S₂ = 0.850 σ₀ = 20.80 MPa), N denotes the fatigue life under different failure probabilities in the P-R-N curves.

Figure 16.

P-R-N curves under stress level S₂.

Distribution curve of fatigue life

Lognormal is the most common distribution for the compressive stress of concrete (Meng, 2006; Wang et al., 1991). Therefore, the probability distribution of fatigue life can be obtained by fitting the P-N relationship expressions. Let

Y = \log_{10} N

represent the logarithm value of the fatigue life, and

X = Φ^{- 1} (p)

represents the inverse standard normal distribution of different failure probabilities. The conversion values under the stress levels S₁ and S₂ are shown in Table 3.

Table 3.

X, Y conversions values under stress levels S₁ and S₂.

X	−1.282	−0.842	−0.524	−0.253	0	0.253	0.524	0.842	1.282
Y₁ (S₁)	4.188	4.260	4.313	4.358	4.409	4.450	4.495	4.545	4.613
Y₂ (S₂)	2.999	3.095	3.132	3.207	3.242	3.273	3.311	3.362	3.435

The fitting method of least squares linear regression is used to derive the X-Y relationship, and the expressions are depicted in equations (17) and (18), where R² is the fitting error.

\begin{array}{l} Y_{1} = 0.166 X + 3.228 & R^{2} = 0.992 \end{array}

(17)

\begin{array}{l} Y_{2} = 0.168 X + 4.403 & R^{2} = 0.997 \end{array}

(18)

Then, the probability distribution function of compressive fatigue life of concrete can be derived using equations (17) and (18) and the value of $μ = \hat{a}, σ = \hat{b}$ (Y = aX + b). The probability distribution function of the compressive fatigue life of concrete under the stress levels S₁ and S₂ are established as shown in equations (19) and (20). The failure probability distribution curves of compressive fatigue life from equations (19) and (20) are displayed in Figure 17.

P (N_{f} \leq x) = Φ (\frac{\log_{10} x - 4.403}{0.168})

(19)

P (N_{f} \leq x) = Φ (\frac{\log_{10} x - 3.228}{0.166})

(20)

Figure 17.

Failure probability distribution curves of compressive fatigue life of concrete under different stress levels. (a) Stress level S₁. (b) Stress level S₂.

As shown in Figure 17, the failure probability of compressive fatigue life of concrete increases with an escalation in cycle loading times at a given stress level, and the likelihood of failure rises rapidly from 5% to 95%. Taking a 50% failure probability as an example, the fatigue lives of stress levels S₁ and S₂ are 25,293 and 1,690, respectively, with the fatigue life of stress level S₁ being approximately 15 times that of stress level S₂. Moreover, for varying stress levels, the cycle loading times required to achieve the same failure probability under high-stress levels are significantly shorter than those under low-stress levels. This observation highlights that high stress can more readily induce fatigue failure in structures.

The fatigue lives corresponding to failure probabilities of 0.05 and 0.95 are considered as the upper and lower limits of fatigue life, respectively. For stress level S₁, the maximum and minimum values of fatigue life are 47,790 and 13,386, respectively. For stress level S₂, the maximum and minimum values of fatigue life are 3170 and 901, respectively. Subsequently, the upper and lower limits of fatigue life predicted by the fatigue life prediction model closely align with the fatigue life test values of concrete, with the maximum error being less than 4.6%. This indicates that this method can be utilized to forecast the compressive fatigue life of concrete.

Conclusion

In this study, a comprehensive numerical simulation of the complete fatigue process utilizing experimental data was conducted to investigate the fatigue performance of precast segmental concrete bridges. The variations in stress and strain, as well as the fatigue damage development laws of both reinforcement and concrete, were experimentally examined. Building upon the fatigue residual strength of concrete, a fatigue life prediction model was established to predict the fatigue life of the concrete. The following conclusions can be drawn:

(1) A new fatigue residual strength model of reinforcement and a fatigue constitutive model for concrete are proposed by taking into account previously developed fatigue parameter degradation models. The fatigue S-N curves for concrete and reinforcement are determined using the aforementioned curves. Based on the numerical analysis results, optimizing the design of prefabricated segments to enhance fatigue performance may necessitate adjustments in beam geometry, material selection, prestress design, and other aspects.

(2) From the load-deflection curves, the model that considers the decrease in the elastic modulus of concrete at the joint exhibits better agreement with the test results compared to the model without considering the decrease. Consider the impact of construction processes on fatigue performance to mitigate the potential generation of defects and cracks.

(3) The stress in both ordinary reinforcement and prestressed reinforcement increased gradually with cyclic load times, and no fatigue failure occurred. Concrete in the compressive area shows slight damage, and the vulnerable parts of concrete include the midspan and the joints. Concrete in the joints is prone to cracking and develops toward the loading point.

(4) Both the stress in ordinary reinforcement and prestressed reinforcement exhibits an increasing trend with the number of cyclic loading, characterized by a rapid initial growth phase followed by a more gradual stabilization. This observed pattern aligns with the three-stage law of fatigue damage in steel reinforcement. Notably, the simulated values of tensile strain in prestressed bars demonstrate a commendable agreement with the experimental values.

(5) Considering the impact of durability damage, such as corrosion in prestressed bars, an analysis of the influence of different degrees of corrosion on the fatigue performance of prefabricated prestressed concrete beams was conducted. The results indicate that, when the corrosion rate is relatively low, the change in concrete tensile stress is minimal, resulting in a negligible impact. However, when the corrosion rate equals or exceeds 9%, a significant increase in tensile stress is observed at the mid-span and joint locations.

(6) The proposed method for predicting the fatigue life of precast segmental concrete beams is based on the fatigue residual strength of concrete. Using this prediction method, a probability distribution curve of the failure life of concrete was presented to predict fatigue failure at different failure probabilities. The analytical results demonstrated good agreement with the experimental results. Based on the numerical analysis results, develop a proactive maintenance strategy encompassing routine inspections, continuous monitoring of crack propagation at critical locations, and implementation of corrective measures as needed.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the Annual Basic Research Foundation for Young Teachers, China (Grant No. JC22547025) and the China Postdoctoral Science Foundation (Grant No. 2023M743217).

ORCID iD

Shun-En Ren

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