Harsh environments such as alternating wet and dry conditions and cyclic loading cause erosion in steel bars, leading to severe damage to structures. Steel bars are also prone to bending under long-term fatigue loads. This paper establishes the bending performance test of corroded reinforced concrete (RC) beam under fatigue load based on the mathematical–statistical algorithm. It investigates the influence of heavy load on the fatigue performance of damaged RC beams, the influence of corrosive environment on the fatigue performance of RC beams, the degree of corrosion of steel bars in concrete beams under fatigue loading, and the distribution of chloride ions. The study results found that the stress ratio has a significant effect on the maximum crack width of the specimen beam. The greater is the stress ratio, the longer the fatigue life. This directly affects the performance of the specimen beam under fatigue loading. For this reason, we should pay special attention to the impact of corrosion on the fatigue performance of a structure during the design of steel bars.
Keywords
- reinforced concrete beam
- mathematical
- statistical algorithm
- fatigue load
- bending performance
MSC 2010
- 03B48
Reinforced concrete (RC) beams are essential components of civil engineering structures such as buildings and bridges. Affected by corrosive environmental effects such as concrete carbonisation and chloride ion erosion, RC beams often suffer from steel corrosion and concrete expansion and cracking, which reduce their shear capacity. This affects the safety of the structure [1]. Therefore, the study and establishment of a calculation model for the shear capacity of corroded RC beams are of great significance for the safety assessment and bearing capacity redesign of in-service RC structures.
The shear mechanism of corroded RC beams is complex and has many influencing factors. At present, research done on it has a few shortcomings: first, the introduced steel corrosion correction coefficient is usually determined according to engineering experience or test data fitting analysis. Second, the research model usually takes the critical oblique crack inclination angle to be approximately 45°, ignoring the impacts of factors such as shear span ratio, augmentation ratio, counterbearing ratio, and steel corrosion on the critical oblique crack inclination angle. This leads to limited calculation accuracy [2]. Therefore, it is necessary to comprehensively consider the influence of steel corrosion on the yield augmentation of counterbearings, counterbearing ratio, augmentation ratio, effective shear cross-sectional area of RC beams, and other essential factors. At the same time, we also need to consider the impact of objective uncertainty and subjective uncertainty to establish a probability model for calculating the shear capacity of corroded RC beams.
This paper first comprehensively considers the influence of steel corrosion on the yield augmentation of counterbearings, augmentation ratio, counterbearing ratio, critical oblique crack inclination, beam effective shear cross-sectional area, and other essential factors. At the same time, we established a deterministic model for calculating the shear capacity of corroded RC beams by combining the modified pressure field theory (MCFT) and the critical oblique crack inclination model, considering the influence of the shear span ratio [3].
Corrosion of steel bars reduces the yield augmentation and effective cross-sectional area of counterbearings, leading to a decrease in the contribution of the counterbearing shear capacity and often causes concrete corrosion, expansion, cracking or cracking spalling [4]. The following comprehensive consideration of the influence of steel corrosion on the yield augmentation of counterbearings, augmentation ratio, counterbearing ratio, effective shear cross-sectional area of beams and other important factors will help to establish a deterministic model. Based on the MCFT, the shear capacity
Combining formulas (4) and (5) we obtain as follows:
The values of the parameter
According to the engineering experience or existing test data, the prior distribution of the probability model parameter
Basic parameters of corroded RC beams and test values of shear capacity test
100 | 175 | 1.5 | 1.94 | 0.44 | 49.71 | 52.87 | 23.68 | 27.07 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 50.71 | 48.93 | 21.47 | 38 | |
100 | 175 | 1.5 | 1.94 | 0.44 | 59.92 | 50.16 | 22.78 | 26.69 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 52.44 | 48.96 | 20.21 | 37.92 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 54.05 | 51.79 | 20.75 | 38.84 | |
100 | 175 | 1.5 | 1.94 | 0.44 | 61.91 | 58.13 | 22.99 | 28.22 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 53.28 | 54.07 | 21.05 | 39.91 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 53.25 | 49.48 | 20.22 | 38.7 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 52.79 | 48.96 | 20.21 | 38.21 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 53.65 | 50.5 | 20.5 | 38.7 | |
100 | 175 | 1.5 | 1.94 | 0.44 | 62.65 | 65.86 | 22.74 | 29.84 | |
100 | 175 | 1.5 | 1.94 | 0.44 | 63.53 | 67.28 | 22.99 | 30.02 | |
100 | 175 | 1.5 | 1.94 | 0.44 | 65.29 | 74.67 | 22.99 | 31.97 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 53.73 | 48.96 | 20.21 | 38.98 | |
100 | 175 | 1.5 | 1.94 | 0.44 | 66.06 | 76.02 | 23.19 | 32.12 | |
100 | 175 | 1.5 | 1.94 | 0.44 | 66.76 | 77.86 | 23.36 | 33.26 | |
120 | 200 | 2 | 1.92 | 0.32 | 73.84 | 52.86 | 30.01 | 39.34 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 58.13 | 56.18 | 21.66 | 41.39 | |
100 | 175 | 1.5 | 1.94 | 0.44 | 68.06 | 77.52 | 23.36 | 33.53 | |
120 | 200 | 2 | 1.92 | 0.32 | 77.74 | 73.12 | 30.32 | 42.34 |
Based on the experimental data in Table 1 combined with formula (13) and the MCMC method, we can determine the posterior distribution information of each probability model parameter. The paper first uses
Probability model parameter
0.032 | 0.047 | 0.049 | 0.05 | |
0.067 | 0.053 | 0.058 | 0.059 | |
0.055 | 0.506 | 0.541 | 0.56 | |
0.031 | 0.039 | 0.045 | 0.048 | |
0.03 | 0.037 | 0.036 | 0.042 | |
× | 0.038 | 0.064 | × | |
× | 0.068 | 0.048 | × | |
× | 0.586 | 0.473 | × | |
× | 0.037 | 0.053 | × | |
0.031 | 0.035 | 0.034 | 0.025 |
The distribution type with the smallest
Based on determining the posterior distribution information of the probability model parameter
It can be seen from Figure 4 that nearly 1/2 of the measured values are within the 50% confidence interval, and almost all of the measured values are within the 95% confidence interval. This shows that the established probability model can better describe the probability distribution characteristics of the beam's shear capacity.
Taking the deterministic shear capacity calculation model as an example, the distribution of the calculated values of each model and the experimental test values within the confidence interval of the probability model is as shown in Figure 5.
Although nearly half of the calculated value of the model
In addition, we can use the probability model of this paper to calibrate the calculation accuracy of the traditional deterministic model. Choose a corroded beam from Table 1 as an example. The mean value and standard deviation of the shear capacity determined by the probability model are 52.64 kN and 8.77 kN, respectively. After the K–S test, the shear bearing capacity does not refuse to obey the normal distribution, and its probability density distribution is as shown in Figure 6. The calculated values of models
This paper combined the modified pressure field theory, Bayesian theory and MCMC method to establish a probability model for calculating the shear capacity of corroded beams. The established probability model for calculating the shear capacity of corroded beams not only has a rigorous theoretical basis but also comprehensively considers the effects of subjective and objective uncertainties. It has good applicability and calculation accuracy, which can reasonably describe the probability distribution characteristics of the shear capacity of corroded beams.
Basic parameters of corroded RC beams and test values of shear capacity test
100 | 175 | 1.5 | 1.94 | 0.44 | 49.71 | 52.87 | 23.68 | 27.07 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 50.71 | 48.93 | 21.47 | 38 | |
100 | 175 | 1.5 | 1.94 | 0.44 | 59.92 | 50.16 | 22.78 | 26.69 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 52.44 | 48.96 | 20.21 | 37.92 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 54.05 | 51.79 | 20.75 | 38.84 | |
100 | 175 | 1.5 | 1.94 | 0.44 | 61.91 | 58.13 | 22.99 | 28.22 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 53.28 | 54.07 | 21.05 | 39.91 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 53.25 | 49.48 | 20.22 | 38.7 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 52.79 | 48.96 | 20.21 | 38.21 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 53.65 | 50.5 | 20.5 | 38.7 | |
100 | 175 | 1.5 | 1.94 | 0.44 | 62.65 | 65.86 | 22.74 | 29.84 | |
100 | 175 | 1.5 | 1.94 | 0.44 | 63.53 | 67.28 | 22.99 | 30.02 | |
100 | 175 | 1.5 | 1.94 | 0.44 | 65.29 | 74.67 | 22.99 | 31.97 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 53.73 | 48.96 | 20.21 | 38.98 | |
100 | 175 | 1.5 | 1.94 | 0.44 | 66.06 | 76.02 | 23.19 | 32.12 | |
100 | 175 | 1.5 | 1.94 | 0.44 | 66.76 | 77.86 | 23.36 | 33.26 | |
120 | 200 | 2 | 1.92 | 0.32 | 73.84 | 52.86 | 30.01 | 39.34 | |
100 | 175 | 2.5 | 1.94 | 0.44 | 58.13 | 56.18 | 21.66 | 41.39 | |
100 | 175 | 1.5 | 1.94 | 0.44 | 68.06 | 77.52 | 23.36 | 33.53 | |
120 | 200 | 2 | 1.92 | 0.32 | 77.74 | 73.12 | 30.32 | 42.34 |
Probability model parameter K − S test D value
0.032 | 0.047 | 0.049 | 0.05 | |
0.067 | 0.053 | 0.058 | 0.059 | |
0.055 | 0.506 | 0.541 | 0.56 | |
0.031 | 0.039 | 0.045 | 0.048 | |
0.03 | 0.037 | 0.036 | 0.042 | |
× | 0.038 | 0.064 | × | |
× | 0.068 | 0.048 | × | |
× | 0.586 | 0.473 | × | |
× | 0.037 | 0.053 | × | |
0.031 | 0.035 | 0.034 | 0.025 |
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