Fatigue Strain and Damage Analysis of Concrete in Reinforced Concrete Beams under Constant Amplitude Fatigue Loading

Concrete fatigue strain evolution plays a very important role in the evaluation of the material properties of concrete. To study fatigue strain and fatigue damage of concrete in reinforced concrete beams under constant amplitude bending fatigue loading, constant amplitude bending fatigue experiments with reinforced concrete beams with rectangular sections were first carried out in the laboratory. Then, by analyzing the shortcomings and limitations of existing fatigue strain evolution equations, the level-S nonlinear evolution model of fatigue strain was constructed, and the physical meaning of the parameters was discussed. Finally, the evolution of fatigue strain and fatigue damage of concrete in the compression zone of the experimental beam was analyzed based on the level-S nonlinear evolution model. The results show that, initially, fatigue strain grows rapidly. In the middle stages, fatigue strain is nearly a linear change. Because the experimental data for the third stage are relatively scarce, the evolution of the strain therefore degenerated into two phases. The model has strong adaptability and high accuracy and can reflect the evolution of fatigue strain. The fatigue damage evolution expression based on fatigue strain shows that fatigue strain and fatigue damage have similar variations, and, with the same load cycles, the greater the load level, the larger the damage, in line with the general rules of damage.


Introduction
Concrete fatigue strain can be a true reflection of the variation of material deformation under fatigue loading.If the relationship between the curve and the cycles is known, we can give a qualitative and quantitative description of the material fatigue strain evolution, thus providing the basis for the evaluation of material behavior.Much of the amplitude fatigue test showed that, whether ordinary concrete, lightweight aggregate concrete, high strength concrete, or fiber reinforced concrete, but also whether compression fatigue, tension fatigue, bending fatigue, uniaxial fatigue, or multiaxial fatigue, concrete longitudinal total deformation and residual deformation exhibit a very stable three-stage development law [1][2][3][4][5][6], for the rapid growth stage, the stable growth stage, and again the rapid growth stage, and this law is universal.Chen et al. [7] used a cubic polynomial fitting curve to attain correlation coefficients above 0.937, but different stress levels have different coefficients, a difference of nearly an order of magnitude.Cachim et al. [8] determined that when the concrete was under compression fatigue loading, the second phase of the concrete maximum strain rate and the logarithm of the fatigue load cycles form a linear relationship.Xie et al. [9] also fitted the second phase of fatigue strain and gained an experienced index formula.Wang et al. [10] fitted concrete experimental strain data for amplitude compressive fatigue to a two-stage nonlinear formula.Through the evolution of fatigue strain and current analytical methods of analysis, we found that the analysis methods used for fatigue strain now have low accuracy and poor adaptability, and the meaning of the parameters in the model is not clear.
Based on the analysis above, the level-S nonlinear evolution model of fatigue strain was constructed and the physical meaning of the parameters was discussed.A constant amplitude bending fatigue experiment with reinforced  concrete beams with a rectangular section was carried out in the laboratory.Then, the evolution of fatigue strain and fatigue damage of concrete in the compression zone of the experimental beam was analyzed based on the level-S nonlinear evolution model (Figure 2).and JL-2 to JL-4 beams were used for fatigue testing under different load levels to determine fatigue strain information throughout the life cycle.Specific test conditions are shown in Table 3.

Experimental Device and Loading
Method.The static load test and fatigue test were both carried out using the online mode of a computer and electrohydraulic servo loading equipment.The test machine control system is shown in Figure 1.The static load test has two purposes, determining the cracking load  cr and the ultimate load   and judging whether the reinforcement of the beam is suitable.Before the formal static load test, the beams should first be preloaded three times.The formal static load test begins after completion of the preload test.The static load test is conducted by grading the load, with a load increment of 20 kN.At each loading, concrete strain values were observed by the controlling software of YE2539 and the high-speed static strain meter.To check that the data acquisition system is working properly and the testing load control system is safe and reliable, the beams should first also be preloaded three times before the formal fatigue test.Because the test was controlled by load amplitude, first, load to load by ( max +  min )/2 in the form of a graded load was tested and then the fatigue tests began after setting the amplitude to ( max −  min )/2 and entering the load frequency and the number of the next cycle.The concrete strain was collected when the fatigue cycle ranged to 1 time, 1000 times, 10000 times, 20000 times, 50000 times, 100000 times, 200000 times, and every 200000 times thereafter.Loading frequency is 5 Hz.The cracking load  cr is 40 kN, and the ultimate load   is 220 kN.Through static load test, the reinforcement of the beam is suitable.
In formula (1),  0 and   are initial strain and the fatigue strain when fatigue load cycle has moved  cycles.The parameters ,   are the fatigue load cycles and fatigue life, respectively., , and  are equation parameters.
If you take   as the maximum fatigue strain   max or fatigue residual strain   res after  cycles and take  0 as the initial maximum strain  0 max or initial residual strain  0 res , then formula (1) becomes ( 2) and (3) as follows: Formula ( 2) is the maximum fatigue strain evolution model, and formula (3) is the fatigue residual strain evolution model.

Physical Meaning and the Range of Model Parameters.
The initial maximum strain  0 max and initial residual strain  0 res are caused mainly by initial defects in the material and the preloading of the member (preloading a general load to the fatigue upper limit).Generally, distinguishing between the two is difficult, so, in this paper, the two are, respectively, represented by fatigue maximum strain  1 max and residual strain  1  res when stress reaches the upper limit stress the first time, and the values are generally obtained by experiment.According to the actual experimental data, Wang et al. [10] determined that the relationship between the two is  1 res = 0.25( 1 max / unstab ) 2 , where  unstab is the total strain of the concrete unstable state.
To study the range of parameters , , and  and their impact on the evolution of fatigue strain curves, formulas ( 2) and ( 3) can be divided by ultimate fatigue strain on both sides at the same time, so the two formulas become as follows: In formula (4),   max is the extreme of the maximum fatigue strain.In formula (5),   res is the extreme of fatigue residual strain.Formulas (4) and ( 5) are the normalized evolution model of maximum fatigue strain and fatigue residual strain, respectively.
Parameter , called the destabilizing factor, is not an independent parameter and is relevant to the parameters  and .When the circulation ratio /  is 1, the curve of formulas (4) and ( 5) passes the point (1.1).If the coordinate is put into formulas (4) and ( 5), we will obtain parameter  calculation formulas for the fatigue strain evolutionary models (4) and (5) as follows: The paper takes the maximum fatigue strain evolution model (4) as the example to study the influence of parameters on the fatigue strain evolution curve.First, we take  0 max /  max and /  max as fixed values to study the impact of different  on the curve and then to study the different effects of  on the curve by taking the set  0 max /  max and .The result shows that parameter  affects the curve convergence speed of the level-S nonlinear strain model.Specifically, the greater the  value is, the faster the convergence of the third phase of fatigue strain curve is.Therefore,  is called the instability speed factor.From the paper,  ranges in the recommendations for [2,8].The result shows that  affects the proportion of the third stage (convergence stage) of fatigue strain in the total  fatigue life.Specifically, the larger the  value, the smaller the proportion of the acceleration phase.

Evolutionary Analysis of Concrete Fatigue Strain.
In this paper, concrete fatigue strain obtained in the experiment was analyzed by a level-S nonlinear strain evolution model.The fatigue maximum strain and fatigue residual strain of different load levels were fitted according to formulas (2) and (3).Fitting results and testing results are shown in Figures 3-5.Table 4 shows the coefficient of fatigue strain evolution equation at each stress level.
From Figures 3-5 and Table 4, fatigue strain evolution formulas (2) and ( 3) can be good fit to the experimental data.Correlation coefficients are above 0.98.Because experimental strain data of the third stage (near destruction) are relatively small, evolution of maximum fatigue strain and fatigue residual strain both occurs by the threestage reduced to a two-stage variation.The first phase of both is rapid growth; the second phase is nearly linear change.The model can be adapted to specific forms of change.

Evolutionary Analysis of Fatigue Damage Based on Fatigue
Strain.Generally, the preloading test should be done before performing the experiment, and the preload is often the upper limit of the fatigue load.This preload will cause damage to the beam.The initial damage contemplated herein consists of two parts, the initial damage caused by material defects and the damage caused by the process of the load first loaded to the upper fatigue limit.After understanding this point, the damage based on maximum fatigue strain is defined as follows: In ( 8),  0 is the initial damage.If we take the maximum fatigue strain evolution formula (2) into formula (8), the damage evolution equation based on the maximum fatigue strain is obtained.The damage evolution formula is as follows: To facilitate data fitting, if   /  max = , then the above equation becomes Similarly, the definition of damage based on fatigue residual strain can be expressed as follows: If we take the fatigue residual strain evolution formula (3) into formula (11), the damage evolution equation based on the fatigue residual strain is obtained.The expression is as follows: To facilitate data fitting, if   /  res = , then the above equation becomes In this paper, concrete damage evolution in three beams was analyzed using fatigue damage evolution formulas (10) 5.
Seen from Figures 6-11, from the whole process of damage development, the concrete damage increases as the circulation ratio increases; however, initially, the concrete damage grows rapidly, and when in the middle stage, it is a nearly linear change.Similar to strain evolution, it has a twostage variation.From Table 5, we know that the model has high accuracy.With the same load cycles, the greater the load level, the larger the damage, in line with the general rules of damage.

Conclusions
The conclusions are as follows: (1) By analyzing the shortcomings and limitations of the existing fatigue strain evolution equations, the level-S nonlinear evolution model of fatigue strain was constructed, and the physical meaning of the parameters was discussed.
(2) Test data analysis by the level-S nonlinear strain evolution model shows that, initially, fatigue strain grows rapidly and that, when in middle stage, it is a nearly linear change.The model has strong adaptability and high accuracy and can reflect the evolution of fatigue strain.
(3) The fatigue damage evolution expression based on fatigue strain shows that fatigue strain and fatigue damage have similar variations and that, with the same load cycles, the greater the load level is, the

Figure 3 :
Figure 3: Testing and fitting results of fatigue strain of JL-2.

Figure 4 :Figure 5 :
Figure 4: Testing and fitting results of fatigue strain of JL-3.

Figure 6 :
Figure 6: Damage result of JL-2 based on maximum fatigue strain.

Table 4 :
Coefficients of evolution equation.

Table 5 :
Fatigue damage evolution equation coefficients.