Fatigue Damage Study of Reinforced Concrete T-Beam Bridge Considering Bearing Defect

Abstract

Vehicle load is one of the primary factors influencing fatigue damage in reinforced concrete beam bridges, and diseases of bearings will accelerate the fatigue damage of beam bridges, but most of the current studies only consider the influence of vehicle load on the fatigue damage of beam bridges. In order to evaluate the fatigue damage of T-beam bridges more comprehensively, a numerical analysis method for the entire fatigue damage process of T-beam bridges is proposed, which considers the impact of bearing diseases and vehicle loads. The research results indicate the following: (1) During the fatigue calculation process of the T-beam, the maximum tensile fatigue damage value grows rapidly in the early stages. The effects of different bearing diseases and severities on the T-beam are generally similar. In the mid-stage, the growth of the maximum tensile fatigue damage value slows down, and the deeper the bearing disease, the greater the impact on the T-beam. In the later stage, the damage value grows rapidly until failure. (2) The displacement of bearings leads to stress concentration in the beam and the steel plates of bearings. The larger the slippage amplitude, the greater the stress value, which leads to a shortened lifespan of the concrete beam. (3) Bearing maintenance should focus on the damage of the beam. If displacement of bearings occurs, measures should be taken as early as possible (preferably within the mid-stage, 10–65 years); otherwise, it will result in a significant decrease in the beam’s life.

1. Introduction

In recent years, with the frequent accidents of failure collapse caused by fatigue accumulation damage under the action of frequent overload, environmental erosion, and other factors, the problem of highway bridge fatigue failure has attracted more and more attention from scholars from all over the world [1]. In the past, people mainly studied the fatigue damage failure problem of reinforced concrete structures through fatigue tests. However, because the fatigue test is time-consuming and laborious and subject to the limitation of experimental conditions, in the beginning, some scholars tried to use numerical methods to simulate this kind of complex structural fatigue damage and even failure process. In 2017, Izabela [2] used the plastic damage of concrete (Concrete Damaged Plasticity, CDP) model to study the seismic performance of cable footbridges under earthquakes, and the study shows that the implementation of the concrete damage plastic model allows the detailed analysis of the seismic performance of the reinforced concrete bridge deck. In 2021, Liu [3], from the perspective of concrete damage, reasonable modeling, and analysis, proposed methods of unloading damage behavior and the use performance of shield tunnels. He introduced a new strategy and established a double scalar damage constitutive model of concrete in order to consider the degradation behavior of the asymmetric tensile pressing material of concrete. In 2023, Mona [4], using the proposed CDP model, studied the influence of shear span ratio, opening size, opening position, concrete compressive strength, main ratio, and web reinforcement ratio on the plasticity of concrete damage. The results show that the proposed simulation model can be used to simulate the behavior of reinforced RC deep beams effectively. The above studies show that the CDP model has good authenticity and reliability.

On the other hand, bridge inspection statistics show that rubber bearings are widely used in highway-reinforced concrete beam bridge systems. Under environmental conditions and the coupling of operating loads, a large number of rubber bearings experience typical diseases, such as sliding, emptiness, deformation, aging, mat stone crushing, and steel corrosion [5]. The aging of bearing material performance, performance degradation, and changes in the contact interface will affect bearing normal deformation, shear performance, and friction performance, which will cause local and overall structure stiffness distribution, uneven force distribution, and local bearing damage. Existing studies often focus solely on the impact of fatigue on the concrete bridge itself. These changes can also change the stress deformation characteristics of adjacent structures and increase the complexity and uncertainty of structural damage [6,7]. Therefore, in order to accurately evaluate the safety of the bridge structure, clarifying the influence of the rubber bearing defect on the T-beam is necessary [8].

In highway bridges, issues such as bearing sliding and other forms of damage frequently occur. To assess the performance changes of the bridge under the influence of fatigue while considering the impact of bearing damage, it is necessary to quantify the bearing damage by building upon traditional damage models. In this work, based on the concrete CDP model, fatigue strength degradation, and rubber aging evolution, through the secondary development of the ABAQUS (v6.10.1) subsystem, the vehicle load and bearing defect realize the life prediction of the prestressed reinforced concrete bridge structure under high weekly fatigue load and bearing defect. The results showed that the severity of the bearing distress had a large effect on the rate of performance degradation of the concrete girder bridge. The disease of bearings can significantly affect the lives of concrete bridges, which provides new ideas and references for the maintenance and fatigue performance analysis of bridges.

2. Vehicle Load Model

This section introduces a random vehicle load model that considers parameters such as vehicle type, vehicle weight, vehicle distance, axle spacing, and axle weight. The simulation of the random traffic flow is achieved by random sampling of the vehicle parameter probability model. The results lay the foundation for analyzing the fatigue damage of the bridge under the influence of vehicle load. The specific simulation procedure is shown in Figure 1.

First, the relevant parameters of each variable are obtained from the data collected by the dynamic weighing system (Weight in Motion, WIM) so that the simulated random traffic flow is very similar to the actual traffic flow; second, the random traffic flow is generated in MATLAB; finally, the generated information is transmitted into the ABAQUS subroutine DLOAD using the Fortran language. The DLOAD subroutine is mainly used to define complex load conditions and is used to define changes in non-uniform loading with respect to coordinates, time, number of cells, number of integration points, and other related factors.

2.1. Vehicle Load Parameters

Based on the probabilistic statistical analysis of vehicle parameters (such as model, body quality, workshop distance, etc.) provided by WIM of a bridge in Beiyang Line, Tianjin, the time period of thebselected data is 2022, and the total number of vehicles is 6,178,720. Limited to the acquisition capacity of WIM, the loading speed is 70 km/h of the mean speed in literature [7].

2.2. Model Ratio

According to the number of axles, the models are divided into two-axle, three-axle, four-axle, five-axle, and six-axle vehicles. Among them, two-axle vehicles accounted for 58.30%; three-axle for 8.76%; four-axle for 7.41%; five-axle for 3.46%; and six-axle for 22.07%. Since the proportion of seven-axle cars and above is small, it is not considered.

2.3. Vehicle Weight, Workshop Spacing, and Axle Spacing

The statistical analysis of the probability characteristic parameters of vehicle weight and workshop distance is shown in Figure 2.

According to the operating characteristics of the automobile fleet, it can be divided into general operation state and intensive operation state [9]. General running state refers to the usual running state of the road, which can directly reflect the overall level of actual vehicle load. The intensive running state mainly considers the special state, such as large traffic flow, road congestion, or parallel driving, and the actual probability of occurrence in the installation position of the WIM system is almost zero. Since the research object of this paper is the off-city highway, the general operation state is mainly considered here. The minimum value of the workshop distance sample counted in this paper is 3.12 m, and the maximum value is 12,430.17 m. Considering that when the distance between vehicles before and behind in the same lane exceeds 1000 m, the superposition effect is very small, and there is a small amount of sample data of more than 1000 m in the original data, only the workshop distance data within 1000 m are fitted.

The sampling of axle spacing and axle weight of different models is too complicated and insignificant. In order to facilitate the subsequent simulation, the axle spacing and axle weight of different models are simplified by using the vehicle load data regression analysis method, as shown in

Table 1. Axle weight and axle spacing.

Vehicle Model	Number of Axles	Axle Load Ratio (%)	Distance Between Shafts (m)
Two-axle vehicle	1	52.17%	3
	2	47.83%
Three-axle vehicle	1	25.33%	2.2
	2	25.43%	4.5
	3	49.24%
Four-axle vehicle	1	22.79%	2.1
	2	21.38%	5
	3	25.31%	1.4
	4	30.52%
Five-axle vehicle	1	18.74%	3.5
	2	25.27%	6.5
	3	19.27%	1.8
	4	18.36%	1.5
	5	18.36%
Six-axle vehicle	1	13.85%	2.5
	2	14.19%	2
	3	19.25%	7
	4	18.75%	1.3
	5	18.10%	1.3
	6	15.86%

.

2.4. Lane Distribution

The vehicles passing on the road are generally uneven in the lanes, which may cause fatigue loading concentration. Therefore, it is necessary to study the distribution characteristics of vehicles of different models along different lanes [10].

The position of vehicles on the bridge is shown in Figure 3. As we can see from the figure, two-axle small vehicles generally drive in the first lane, and the vast majority of trucks with more than four axles drive in Lanes three and four, among which Lane three has more dense traffic.

3. Finite Element Model

For subsequent fatigue analysis, a prestressed reinforced concrete T-beam bridge of the Beiyang Line in Tianjin was selected as the model object in this section for model building.

3.1. Concrete Constitutive Model

3.1.1. Fatigue Stiffness Is Degraded

The degradation of concrete elastic modulus is an important index for measuring the degradation of concrete stiffness, which is related to the characteristics of the constituent concrete materials, the mix ratio of concrete, the fatigue load amplitude, and the loading times. Wang Shiyue [11] performed the fatigue test of concrete static load and load in three stages along with an increase in fatigue loading times: in the first stage, the elastic modulus of concrete is fast and unstable, accounting for about 10% of the total fatigue life; in the second stage, about 80% of the total fatigue life; in the third stage, the elastic modulus is rapidly reduced until the fatigue failure, about 10% of the total fatigue life. When the concrete is close to fatigue failure, the ratio of the elastic modulus and the initial elastic modulus is between 0.474 and 0.757. Holmen [12] also proposed the linear degradation of the concrete fatigue elastic modulus through the following test:

$$E_N=(1-0.33N/N_f)E_0$$

(1)

In the formula, $E_N$ is the residual elastic modulus after N fatigue loading, and E₀ is the initial elastic modulus of the concrete.

3.1.2. Concrete Plastic Damage Model

For reinforced concrete components or structures, a correct and reasonable constitutive model is the key to the nonlinear analysis of components or structures [13]. ABAQUS provides three kinds of concrete constitutive relationship models: the brittle cracking model, the dispersion cracking model, and the CDP model. The CDP model combines isotropic elastic damage and tensile and compression plasticity to describe the inelastic behavior of concrete. It is suitable for the standard and explicit solving modules, it can be used to simulate concrete under arbitrary load force, and it considers the elastic stiffness caused by pull, plastic strain degradation, and cyclic load stiffness recovery with good convergence [14].

The CDP model in ABAQUS is a continuous, plasticity-based damage model for concrete. The two main failure mechanisms are tensile cracking and compression crushing of concrete material. The model assumes that the uniaxial stretching and compression responses of concrete are characterized by damage plasticity, and in the case of uniaxial stretching, the stress–strain response follows a linear elastic relationship until failure is reached 4. The failure stress corresponds to the occurrence of microscopic cracks in the concrete material. After exceeding the failure stress, the formula of microscopic cracking adopts the softening stress–strain response to make the macroscopic representation so that strain localization is produced in the concrete structure. Although this representation is simplified, it captures the main characteristics of the concrete response. The results of the C50 concrete CDP model parameters measured by Xue [15] in 2023 are shown in Figure 4.

3.2. Rubber Constitutive Model

Rubber adopts the first-order strain energy function Mooney-Rvilin model, which is expressed as follows:

$$U=C_{10}(\overline{I_1}-3)+C_{01}(\overline{I_2}-3)$$

(2)

In Equation (2), U is the strain energy per unit volume; C₁₀ and C₀₁ are ultraelastic coefficients; and $\overline{{\mathrm{I}}_i}$ are the first- and second-order strain invariants. For most of the rubber materials, when C₀₁/C₁₀ ≈ 0.05~0.2, a reasonable simulation can be obtained within 200% strain. Specifically, for C10 and C01, this article uses data obtained through experimental fitting by Dong Jianhua [16].

3.3. Bearing Disease Simulation

According to the special maintenance project of ordinary national and provincial highway bridges in Tianjin in 2022, the general national and provincial highway bridges in Tianjin were investigated, and 75 bridges had sliding, which is a large number, so this paper mainly simulates sliding and aging diseases.

(1)

Sliding disease simulation

The technical standard of Tianjin: 25% of the corresponding side length and the corresponding side length, the bearing position is serious; when the bearing length is >25% of the corresponding side length, the bearing position is particularly serious; considering the influence of extreme disease on the T-beam, the bearing increases by 50%. The specific sliding simulation method is shown in Figure 5.

(2)

Aging disease simulation

Ma Yuhong [17] fits the change rule of the material constant within 90 d of aging (80 °C aging box), as shown in Equations (3) and (4).

$$C_{10}=0.0015n+0.4054$$

(3)

$$C_{01}=0.0004n+0.2165$$

(4)

In the formula, n is the actual environmental aging time (years). According to Equations (3) and (4), the variation curves of C₁₀ and C₀₁ are obtained, as shown in Figure 6. As shown in the figure, C₁₀ and C₀₁ increase linearly with an increase in aging time.

The operating conditions set in this article are shown in

Table 2. Working condition settings.

Number	Disease Type
Working condition 1	Only affected by aging diseases
Working condition 2	Aging, sliding 10%
Working condition 3	Aging, sliding 25%
Working condition 4	Aging, sliding 50%

. The model built in this article consists of two T-beams and four bearings. All four bearings in the working condition simulate aging diseases, and only one bearing simulates serial motion diseases.

3.4. Geometric Model

The bridge is a 25 m simply supported beam bridge with a width of 18 m. The T-beam is 1.4 m high, the flange plate is 0.2 m high and 1.8 m wide, and the web plate is 1.2 m high and 0.2 m wide. To improve computational efficiency, the model includes a lane and two beams.

The bearing adopts ordinary plate rubber bearing, with a bearing model of GBZJ200 × 300 × 41 (CR), a bearing height of 41 mm, a length of 300 mm, a width of 200 mm, and four embedded layers of steel plates with a length of 290 mm, a width of 190 mm, and a thickness of 3 mm. There is one bearing on each side of the T-beam. The specific parameters of the bridge are shown in Figure 7.

We established a finite element model of rubber bearing T-beam coupling using ABAQUS software. In the model, the expansion angle is taken as 30°, the eccentricity is taken as 0.1, the biaxial strength/uniaxial strength is taken as 1.16, the invariant stress ratio is taken as 0.66667, and the viscosity coefficient is taken as 0.0005. Concrete adopts the eight-node linear hexahedral element C3D8R, and rubber adopts the C3D8RH element [18]. To improve the quality of the grid, the thickness of the flange plate of the T-beam is cut, and the element shape is hexahedral. After verifying the convergence of the grid, we took the T-beam grid size as 50 mm and the rubber bearing grid size as 50 mm.

4. Fatigue Performance Analysis

4.1. The Calculation Process

The fatigue loading times are usually in the millions, and if calculated gradually, the workload is enormous. On the other hand, some scholars [19] are more concerned about the key points of structural fatigue failure, such as the initiation of fatigue damage, the stable development of fatigue damage, and the number of critical loading cycles for fatigue failure. Therefore, in numerical simulations, different cyclic jumping steps can be used according to the different stages of the linear development of fatigue damage, simplifying the continuous fatigue damage process into a discrete damage process in order to significantly reduce the computational workload without affecting the accuracy of the calculation.

In the first stage of fatigue deformation, according to the characteristics of fatigue deformation, the step size should be selected as small as possible, and the jump step size of adjacent simulation calculations can be different; the second stage of fatigue deformation exhibits a linear growth characteristic, and when selecting the step size, a relatively large step size is used for cyclic jumping. According to the relevant interpolation functions in mathematics, the fatigue performance of concrete materials under any number of fatigue loading cycles between adjacent numerical simulations is linked so that the entire fatigue damage accumulation process of concrete materials is continuous, and the fatigue performance of concrete materials under the entire fatigue load is obtained.

The main calculation process of this article is as follows (Figure 8): according to the simplified principle of fatigue loading times, the fatigue loading times are skipped. When simulating the i-th fatigue loading time (a random traffic flow passing time of 5 min for one fatigue loading), the changes in material parameters of materials such as concrete and rubber in the first N fatigue loading must be considered, and the structural response under the i-th fatigue loading action must be simulated and calculated; after simulating and analyzing the fatigue loading of any cyclic jump, according to the failure criteria of the concrete CDP model, determine whether the structure has fatigue failure. If the structure has a fatigue failure, the cyclic jump and numerical simulation calculation are completed. If the structure is not damaged, according to the principle of simplifying the fatigue loading times, calculate the fatigue constitutive models of concrete and rubber under the fatigue loading times after the next cycle jump step, update the parameters related to concrete fatigue damage and rubber, and return to continue the simulation calculation until the structure is fatigue damaged.

4.2. T-Beam Damage Situation

The calculation results are shown in Figure 9, where DAMAGEC represents compressive damage, and DAMAGET represents tensile damage in the legend; 0~1 represents the damage rate of concrete. In the figure, 0 represents no damage, and 1 represents complete damage and failure. For quasi-brittle materials such as concrete, it is generally believed that the standard for concrete failure is a damage coefficient of 0.9 [20]. Fatigue failure occurs after 80 years in working condition four, 90 years in working condition three, 95 years in working condition two, and 98 years in working condition one.

From Figure 9a–d, it can be seen that in the early stage of fatigue, the compression damage area is mainly concentrated near the contact area between the beam bottom and the bearing, and the area is small with small values, causing less damage to the beam body. The tension damage area is mainly concentrated in the contact area between the beam bottom and the bearing, as well as the middle part of the beam top, and the area is large with large values, causing greater damage to the beam body. This is because concrete materials have strong compressive performance and weak tensile performance. At this stage, there is not much difference in the damage values of the beam caused by the displacement of the bearing and the non-displacement of the bearing. However, the larger the displacement amplitude, the larger the high damage value area (red area) in the contact area between the bearing and the beam bottom. This indicates that the displacement of the bearing will cause stress concentration in this area. From Figure 9e–h, it can be seen that in the mid-fatigue stage, the compressive damage area develops upwards from the contact area between the beam bottom and the bearing and begins to appear in the middle of the beam top, with the damage value gradually increasing. The tensile damage area develops upwards and towards the mid-span from the contact area between the beam bottom and the bearing and rapidly develops from the middle of the beam top to the mid-span. The damage area increases significantly compared to the early fatigue stage. This is because after a certain number of fatigue loads, as the concrete material degrades, the damage area begins to increase rapidly. The difference in beam damage caused by serially moving bearings and non-serially moving bearings begins to emerge, and the greater the amplitude of serially moving bearings, the greater the damage value caused. From Figure 9i–l, it can be seen that in the late stage of fatigue, the compression damage area of the T-beam without shear damage continues to develop upwards from the contact area between the beam bottom and the bearing to the beam top, and the compression damage value increases significantly. The compression damage area of the T-beam with shear damage decreases with the increase in shear amplitude. The compression damage area of the T-beam with shear damage of 50% is limited to the lower half of the T-beam, which is due to stress concentration in the contact area between the T-beam and the bearing caused by shear damage. The tensile damage area rapidly develops from the middle of the beam top to the middle of the span and to both sides of the bridge deck.

When fatigue failure occurs in working condition one, the areas where the tensile damage reaches 0.9 are region A and region C (see Figure 9 for specific area division); when working condition four fatigue failure occurs, the areas where the tensile damage reaches 0.9 are A, B, and C. When the beam is damaged, with the increase in the range of motion, the failure area of A and C in the area where the tensile damage reaches 0.9 gradually decreases, while the failure area of B gradually increases. This is due to stress concentration in area B caused by the movement of the bearings, resulting in a larger area of tensile failure in that region.

From Figure 10, it can be seen that in the early stage of fatigue calculation (0–10 years), the maximum tensile fatigue damage value of all working conditions increases rapidly, and the difference between different working conditions is not significant; in the mid-term of fatigue calculation (10–65 years), the maximum tensile fatigue damage value of all working conditions increases slowly, and differences begin to appear between different working conditions. Among them, the maximum tensile damage value increases with the increase in the range of motion, and in the later stage of fatigue calculation (65–100 years), the maximum tensile damage value increases rapidly, and the larger the range of deformation, the faster the fatigue failure. This indicates that the damage caused by sliding movement has a significant impact on the service life of T-beams, and measures should be taken in a timely manner.

4.3. Bearing Damage Situation

This article summarizes the stress cloud maps of the steel plate inside the bearing under different working conditions, as shown in Figure 11. From Figure 11a,e, it can be seen that when the bearing does not undergo any displacement, the force is mainly borne by the edge area, and the central area is the low-stress area (blue area in the stress diagram). Although stress concentration occurs in the corner of the bearing along the bridge, the stress value is relatively low. From Figure 11b,f, it can be seen that when the bearing undergoes 10% displacement, the low-stress area decreases, and the decrease in the low-stress area increases with the increase in fatigue life. From Figure 11c,g, it can be observed that when the bearing undergoes 25% displacement, the stress concentration area experiences a rapid increase in stress values, while the low-stress area shifts from the center to the outer edge. This is because as the displacement amplitude increases, the outer side of the bearing is no longer under stress. From Figure 11d,h, it can be seen that the stress values in the stress concentration area continue to increase rapidly, while the low-stress area develops from the outer edge to the center, accounting for nearly 50% of the total. This indicates that after the bearing is moved in series, the outer 50% of the area is subjected to less stress.

Based on the comprehensive analysis of Figure 11, it can be concluded that the steel plate of the bearing that has suffered from displacement damage experiences stress concentration at the corner along the bridge direction. The greater the displacement amplitude, the greater the stress concentration value. As the fatigue time increases, the low-stress area gradually decreases due to the decrease in concrete fatigue strength and changes in the rubber material of the bearing for the unstressed bearing.

4.4. Bearing Maintenance Suggestions

From the perspective of beam damage, in the early stage of bearing displacement (0–10 years), there is not much difference in the damage value of the beam caused by the displacement of bearings compared to the non-displacement of bearings. Therefore, no measures can be taken at this stage. During the mid-stage of bearing displacement (10–65 years), the damage to the beam caused by the displacement of bearings begins to be greater than that caused by non-displacement bearings, but the difference is small, and the growth rate is slower. In the later stage of bearing displacement (65–100 years), the damage value of the beam increases rapidly, and the growth rate of beam damage caused by displacement of bearings is much faster than that of non-displacement bearings. Therefore, measures should be taken as early as possible in the middle term (10–65 years) or before a bearing has moved in series. Otherwise, it will lead to a significant decrease in the service life of the beam.

From the point of view of bearing damage, sliding will lead to the stress concentration of the internal steel plate of the bearing. According to the plate rubber bearing of the Highway Bridge (JT/T 4-2019), the internal stiffening steel plate of the plate rubber bearing shall be of Q235C or above, the yield strength is 235 MPa, and the stress concentration value caused by the sliding of the bearing is less than 235 MPa.

In summary, bearing maintenance should be started from the perspective of beam damage. If bearing sliding disease occurs, measures should be taken in the middle period (10–65 years) or before; for example, the bearing sliding will cause stress concentration in the contact area of the bearing and the beam base, and reinforcement measures can be taken in this area. Otherwise, the life of the beam body will decrease significantly.

5. Conclusions

Based on the CDP model of concrete, the evolution law of fatigue strength degradation and rubber aging, and using the ABAQUS subroutine, a numerical analysis method for the whole process of fatigue damage of the T-beam affected by vehicle load and bearing defect is proposed in this paper. The research results are as follows:

(1) In the early stage of T-beam fatigue calculation, the maximum tensile fatigue damage value of all working conditions increases rapidly, and the gap between different working conditions is not large; in the middle stage of fatigue calculation, the maximum tensile fatigue damage value of all working conditions increases slowly, and the greater the motion range, the faster the damage growth; in the later stage of fatigue calculation, the damage value increases rapidly until destroyed.

(2) Sliding disease will lead to stress concentration in the contact area between the bearing and the T-beam and the bearing steel plate along the corner of the bridge. The greater the sliding amplitude, the greater the stress value, and this reduces the life of the T-beam. Sliding disease can cause low stress in the sliding part of the bearing steel plate.

(3) Bearing maintenance should be from the perspective of beam body damage. If bearing sliding disease occurs, measures should be taken in the middle period (10–65 years) or before. Otherwise, the life of the beam body will decline significantly.

6. Limitations of the Study and Future Work

Although this study considers the impact of bearing deterioration on bridge performance and demonstrates significant potential in predicting the deterioration damage of bridges with bearing defects, the model does not account for environmental factors. In future research, environmental factors will be incorporated to refine the damage model, and its applicability will be further validated with results from real-world engineering projects.