With the help of fractional differential equations, this article studies the failure morphology, fatigue strength, and fatigue life of corroded reinforced concrete beams under fatigue loading. Studies have shown that the mid-span deflection of low-corrosion reinforced concrete beams is smaller than that of uncorroded reinforced concrete beams. The corrosion-fatigue coupling effect accelerates the fatigue crack growth rate of steel bars. This reduces the fatigue modulus of concrete and causes the stiffness of the beam to degrade. The research results provide a theoretical basis for the fatigue performance evaluation of corroded reinforced concrete beams.

#### Keywords

- Reinforced concrete
- Fractional differential equation
- Corrosion
- Restoring force model
- Fatigue load

#### MSC 2010

- 74A10

A certain degree of damage will inevitably occur to the structure due to the environment and load within the design service life. The most common ones are steel corrosion, concrete carbonation, freeze-thaw cycles, etc. These factors lead to a reduction in the seismic performance of the structure. The dynamic response of reinforced concrete structures under the action of corrosion and earthquake affects the comfort of occupants and directly affects the safety and durability of the structure. The structure is subjected to reciprocating loads in an earthquake, and the restoring force model is the basic element to describe the ability of reinforced concrete structures to resist deformation under this action. China is a country prone to earthquakes, and many structures are in a severely corrosive environment at the same time. Many tests and studies have shown that corroded reinforced concrete structures’ bearing capacity, stiffness, flexibility, and energy consumption have been greatly reduced. The shape of the skeleton curve of corroded reinforced concrete members is the same as that of intact members, but the various parameters of the skeleton curve are reduced [1]. The current research results are based on the experimental research of the specific force state or characteristic geometric specimens. The results are different. Therefore, to some extent, the existing research has certain limitations.

This experiment aims to obtain the restoring force model of corroded reinforced concrete members through experimental research [2]. The size, reinforcement, and loading form of the 6 specimens are the same, but the degree of steel corrosion is different. The size and reinforcement of the test piece are shown in Figure 1.

The corrosion of steel bars adopts the electrochemical corrosion method. After pouring and curing, the test piece is soaked in 5% sodium chloride solution for 20 days, and then the DC power supply is applied to accelerate the corrosion electrochemically. The measured corrosion rates of specimens L1-L6 were: 0, 2.76%, 5.47%, 8.63%, 9.81%, 11.59%, respectively. Both ends of the test piece are hinged, and vertical repeated loads are applied to the end of the middle column.

When loading, the load control cycle is first used to load until the specimen enters the yielding state. After the specimen yields, the displacement control cycle loading is adopted according to the yield displacement. The load control cycle level is 0.5kN, the displacement control cycle level is the yield displacement Δy, and each level is cycled three times until the specimen fails. We divide the loading state of the specimen into cracking state, yield state, limit state, and failure state according to the pre-loading system of the test. The failure load should be less than 80% of the peak load [3]. After the specimen reaches the failure state, it can be determined that the loaded specimen has been destroyed. At this time, we will stop the experiment.

The internal force, deformation, concrete cracks, and slip deformation between reinforced concrete structures of corroded reinforced concrete structures undergo reciprocal changes under the action of an earthquake. It is necessary to have the constitutive relationship of the material or section performance under repeated loads. This is also called the resilience model [4]. The restoring force model is the basis of the nonlinear analysis of structures under earthquake action. It can be divided into two types: curve type and broken line type. The stiffness given by the curvilinear restoring force model is continuously changing, which is closer to the actual engineering. Still, it is insufficient in the determination of stiffness and the calculation method. At present, the steel structure mostly adopts the double-line type. The degenerate three-line restoring force model is often used in reinforced concrete structures [5].

When the steel bar begins to yield, the concrete has already passed the linear elastic stage (0 − 0.4 _{c}_{c}

According to the assumption that the concrete strain of the normal section of the member conforms to the assumption of the plane section, we take the neutral axis height conversion factor _{1} = 0.8. The article can obtain _{0} = 0.001456.

According to the test results of corroded steel bars, it can be seen that the mechanical properties of the steel bars have changed after corrosion [6].

In formulas (2)–(4), _{y}_{u}_{s}

We need to determine the critical point of steel corrosion rate when the yield platform of the corroded steel stress-strain curve is degraded [7]. After statistical analysis of the tensile load-displacement curve of corroded steel bars and the corresponding corrosion rate, the article concludes that the critical point of the cross-sectional loss rate of different types of steel bars is 20%. According to the test results, the change law of strengthening strain _{shc}_{suc}

In the formula, _{sy}_{0}, _{sh}_{0} is the yield strain and strengthening strain of the uncorroded steel bars.

The mathematical model of the stress-strain relationship of the corroded steel can be obtained by using the calculation formulas of the characteristic parameter change law.

It can be seen from Fig. 3 that as the corrosion rate increases, the stiffness of the test piece gradually decreases as a whole [8]. Corroded specimens have relatively low initial stiffness due to rust damage and cracks, and stiffness degradation is relatively slow. With the continuous increase of displacement, the decay rate of the stiffness of the rusted specimen and the uncorroded specimen decreases and finally stabilizes.

We perform a fitting analysis on the average value of the relative stiffness attenuation rate of the rusted specimen (see Figure 4) to obtain the relative decay rate of the loading and unloading stiffness of the rusted specimen. See equations (8) and (9) for details:

It can be seen from Figure 4 that the stiffness of the specimen continues to decrease with the increase of displacement. The stiffness degrades more rapidly after the specimen is cracked, especially after reaching the yield load [9]. When the load reaches the peak load, the attenuation of stiffness tends to be gentle.

First, assume that the cross-section of the structure remains flat after deformation. The concrete strain on the section is distributed in a straight line. The tensile strength of concrete is not considered, but the slippage of corroded steel bars and concrete should be considered.

The relationship between the yield load of the test member and the yield moment of the section is:

_{y}_{y}

1) Brittle failure and corroded reinforced concrete members have Σ

From Figure 5, we can assume that the ultimate compressive strain of concrete is taken as _{cc}

2) The damage component of ductile failure has Σ

_{yc}

The non-slip strain
_{sc}

Where _{yc}_{sc}ε_{scy}_{c}_{c}ξ_{scy}

_{sc}ε_{scy}_{E}_{sc}_{c}_{sc}_{sc}_{0}),

The test specimen can be simplified as a cantilever beam for calculation. The relationship between bending moment and curvature adopts an ideal two-fold line model. The yield curvature of the specimen is expressed as _{yc}_{scy}_{0}]. According to its bending deformation distribution, it can be obtained:

According to the test results of corroded reinforced concrete members, it is found that when the concrete is crushed, it is taken as the limit state of the structure. From Figure 8, the ultimate bending moment is:

Considering the influence of rust on the bond-slip performance, the rotation angle in the limit state can be expressed as:

According to the balanced equation, the peak displacement can be solved:

The experiment found that the corrosion degree of the stirrup in the reinforced concrete specimen is more serious than that of the longitudinal reinforcement. The stirrups are prone to severe corrosion or rust breakage, especially at the intersection of the stirrup and the longitudinal reinforcement. This is also very similar to the test results of many actual engineering structures. The severe corrosion of the stirrups leads to a decrease in the shear bearing capacity of the structure. We assume that the ductility reduction factor of the uncorroded specimen is 1. We perform regression analysis on the relationship between the rusted specimen's ductility reduction coefficient and the rust rate change to obtain equation (23).

The failure load of the reinforced concrete structure is taken as the state. Where the peak load of the component is reduced by 15%,

The failure displacement can be determined according to the peak displacement method, and the rotation angle in the failure state can be expressed as

Section curvature in failure state:

We consider the influence of stirrup corrosion and introduce the ductility reduction factor to obtain:

We compare and analyze the experimental value and the calculated value. The purpose is to verify the accuracy of the earthquake damage-based restoring force model of corroded reinforced concrete compression-bending members established in this study. The failure form of the specimen or structure with a corrosion rate of less than 10% shall be considered a ductile failure. The failure mode of the specimen or structure with a corrosion rate greater than 10% shall be considered as brittle failure. Figure 6 shows the comparison between the calculated value and the experimental value of the characteristic points of the skeleton curve of each specimen using the model in this article.

It can be seen from Figure 6 that the calculated skeleton curve described by the research model in this paper is generally in good agreement with the experimental results of the references. Some calculated values are different from experimental values. This is due to the existence of test errors. As the number of repeated load cycles increases for each specimen, the hysteresis loop of the component gradually decreases, and the energy consumption capacity decreases. As the corrosion rate increases, the components’ bearing capacity, stiffness, ductility, and energy consumption gradually decrease. The above phenomenon is consistent with the test. This also shows that it is feasible to determine the restoration force model of corroded reinforced concrete compression-bending members based on earthquake damage according to the method in this paper.

(1) Rebar corrosion greatly influences the hysteretic performance of concrete members under repeated loads. As the corrosion degree of each specimen increases, the hysteresis performance of the specimen decreases. Especially the severely corroded specimens are more likely to be brittle failures in earthquakes. (2) Corrosion of steel bars changes the damage pattern of the structure to a certain extent. The failure mode of structures or components with a high corrosion rate under repeated loads should consider the influence of corroded steel bars and analyze without the assumption of a flat section.

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