The bonding interaction between steel and concrete in the civil engineering building structure is the essential mechanical mechanism for resisting external loads together. The components and the structure as a whole give full play to their strength. The research on the bonding and anchoring performance of ordinary steel bar and concrete under static load has been in-depth . After many buildings reach the aging period, the steel bars in the structure or components will rust due to specific environmental impacts. The component or structure is also subjected to dynamic loads and even impact loads, such as a car crashing into a roadside concrete highway barrier. Existing experimental studies have shown a qualitative conclusion that the bond performance of concrete and steel bars will be improved under fast loading conditions. Still, there is no quantitative analysis of the effect of loading rate on bond strength. We will first use mathematical modeling, a powerful tool, to establish a formula describing the effect of loading rate on bond bearing capacity. Combining experiments, the relevant parameters in the formula are determined, and an inference is made about the correlation between the strength of the concrete itself and the influence of the loading rate. The methods and techniques used in this article are essentially the continuous Markov process part of the mathematical random process theory. The basic idea is to treat the physical quantity studied as a random variable or a random process. Combining the physical and mechanical laws of the research object to obtain the required probability characteristic quantities . Due to the randomness of concrete materials in industrial buildings, random mathematical modeling is suitable.
Assuming that the size of a group of components, the concrete label, the type of steel bar, the degree of corrosion of the steel bar, the thickness of the protective layer, the form of stirrups, the loading rate, and other artificially controllable factors are all the same. Then the bearing of this group of components force's bonding area will certainly not be a constant . The strength of this group can be used as a representative value of the bearing capacity or strength. The rate of change of the bond stress is the rate of change of the average bond stress of the entire bond and anchor section. The viscous stress is the average bond stress of the bond anchoring section. We assume that a random variable is the average bond stress (bond load) that causes the bond area to fail under other conditions. Combining the assumptions of the Markov process to get the density distribution function of this random variable . The density distribution function obtains the expected failure stress.
1) Assume that there is an equivalent bonding area around the steel bars. There is continuous and uniform bond stress along the length of the steel bar on the inner surface of this area. The magnitude of which is
2) Under the same conditions in all aspects that human resources can control, we make the load (bonding stress) for the bond failure of the bond and anchor section to be a random variable rather than a definite value . Theoretically, its distribution interval is the open interval (0, ∞), and the bond strength should be a statistically significant expected value. From the beginning of loading to the beginning of the unloading of the components, when the bonding failure occurs in the real bonding and anchoring section, the whole process is random. Therefore, we use the failure probability
3) The space of the state probability of the bonding area at the beginning of loading is shown in equation (2). Among them, 1 is the probability of undamaged, and 0 is the probability of damage. Assume that the probability of transition from the entire state to the damaged state in a short period
Assuming that the period [0,
When the interval of each period tends to zero (i.e.
4) When formula (4) changes from a discrete Markov chain to a continuous Markov process, it cannot be directly used to perform effective calculations. Now write the probability of the transition from the undamaged state to the damaged state in a small period
5) According to the brittle damage model, the stress expression of the damage rate is as follows
The above formula is obtained by taking the damage threshold
6) The law of the influence of steel bar corrosion on the bond performance and the law of the influence of the loading rate on the bond performance is independent of each other. The effect of loading rate on bonding performance under different corrosion rates of steel bars obeys the same form of law. We can express the effect of strain rate or loading rate on bond strength in the same form. However, the parameters may vary with the corrosion rate of steel bars.
Integrate equation (7). There is an initial condition that the probability of no bond failure in the bonding area when the loading time is zero is 1, and there is
Change the form of expression
Equation (11) shows that the failure probability can be expressed as a function of the bond stress. Therefore, the density function of the failure probability expressed by the cohesive force is
The fracture bond stress distribution interval is (0 + ∞), so the expected bond failure stress is
The detailed calculation process of the points is as follows
In equation (15), suppose
Substituting equation (15) can completely separate the stress rate influence from the integral number
Assuming that the loading rate under static load is
(18) Compared with (19), the bond bearing capacity is affected by the stress rate, and the expression is (20)
The author of this paper conducted a rapid loading test with multiple rust rates and multiple loading rates in the disaster prevention and load reduction laboratory . The concrete strength of the components is 25Mpa, and the rebar is threaded steel with a diameter of 16mm. The basic form of the component is shown in Figure 1. Assume that the average bond stress model in 1 is shown in Figure 2.
In the test, the ratio of the measured dynamic and static load peak values of the components with rust rates of 2% and 4%, respectively, and the test fitting curves obtained according to equation (19) are shown in Figs. 3 and 4. It can be seen that the influence of the loading rate on the bond strength can be approximated by the power function of the loading rate. The test data is in good agreement with the fitted curve.
Comparing the above two figures, it can be seen that the influence index b value of the ratio of dynamic and static strength under different rust rates is not much different. The b value with a rust rate of 2% is 0.0136. The b value with a rust rate of 4% is 0.0143. The difference between the two is about 5%. Therefore, it can be concluded that the influence index b of the rust rate on the loading rate is very small in the smaller rust rate range . Then the value of b can be written as a constant. For the member in this test, b is 0.014. Since the b value in formula (20) has nothing to do with the degree of corrosion, the following method can be used to introduce the influence of concrete material strength on the dynamic and static strength ratio with the loading rate.
Since the damage modulus
The concrete itself is random. The failure of the bonded area is a complex physical process. The effect of loading rate on the bearing capacity of the bonded area is a more complex issue. This paper uses the method of random mathematics to establish the overall failure model based on the bearing strength. After mathematical analysis and calculation, it is expressed as a power function of the loading rate ratio. It represents the effect of loading rate on bearing capacity. The curve fitted according to formula (20) better reflects the changing trend of the bearing capacity of the bonding area as the loading rate changes in the test. And the relative influence of concrete strength and loading rate on the bond bearing capacity obtained by this method is completely consistent with the current test results and qualitative conclusions.
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