Simulation method of pre-peak shear curve of concrete interface based on Haldane distribution

Abstract

The present invention discloses a method for simulating a pre-peak shear curve of a concrete interface based on a Haldane distribution, comprising obtaining a distribution function of the Haldane distribution; applying the distribution function to the pre-peak segment of the concrete interface shear curve to obtain an initial simulation function of the pre-peak shear curve; substituting the peak point coordinates of the pre-peak shear curve into the parameter values ​​of the initial simulation function of the pre-peak shear curve; correcting the pre-peak shear curve using an exponential function to obtain a final simulation function of the pre-peak shear curve; and performing actual simulation of the pre-peak shear curve using the pre-peak shear curve simulation function. The method for simulating a pre-peak shear curve of a concrete interface based on a Haldane distribution provided by the present invention establishes a simulation model based on the Haldane distribution and an exponential correction coefficient, and the established model can characterize the shear deformation of three pre-peak curves, and has good versatility, high precision, and is scientific and reasonable.

Description

Technical Field

[0001] The present invention belongs to the field of concrete engineering, and in particular relates to a method for simulating a pre-peak shear curve of a concrete interface based on Haldane distribution. Background Art

[0002] Concrete, owing to its advantages such as plasticity, integrity, and durability, has been widely used in civil engineering. Pouring new concrete over old concrete structures is a common method for reinforcing concrete structures, effectively mitigating the aging and damage of concrete in engineering structures such as dams, chambers, and bridges. Under these conditions, where both new and old concrete bear a load, the shear mechanical properties of the interface are crucial for structural reinforcement. Furthermore, the shear mechanical properties between concrete and the surrounding medium are also crucial factors influencing the mechanical stability of concrete structures.

[0003] On the indoor test scale, extensive theoretical, experimental and numerical studies have been conducted on the shear deformation behavior of concrete interfaces, and fruitful results have been achieved. The shear stress-displacement curve is the most important consideration for evaluating its shear mechanical properties, and provides a basis for the numerical calculation of similar shear failure problems. Based on existing research, researchers have fully realized that the shear stress-displacement curve can be divided into two key stages: pre-peak and post-peak by the peak point. Since the shear test requires specific test conditions, the test results themselves have a certain degree of discreteness. In practical applications, it is difficult to select the most appropriate test results to evaluate the shear constitutive relationship. Theoretical modeling is the basis for describing the shear characteristics of the interface. According to the typical failure curve of interface shear, in addition to the peak stress, the stress thresholds for crack compaction, cracking and damage are all located in the pre-peak stage, which is closely related to the development of cracks. Therefore, the study of the pre-peak shear curve is particularly important.

[0004] There are three types of pre-peak shear stress-displacement curves of interfaces, namely concave curves, quasi-linear curves and convex curves, such as Figure 1 Currently, numerous researchers have studied the modeling of these three types of pre-peak shear stress-displacement curves. However, current research results are unable to accurately simulate all three types of pre-peak shear curves simultaneously. This results in limited applicability, low simulation accuracy, and limited practicality. Summary of the Invention

[0005] One of the purposes of the present invention is to provide a method for simulating the pre-peak shear curve of a concrete interface based on Haldane distribution with good versatility, high precision and scientific rationality.

[0006] The method for simulating the pre-peak shear curve of a concrete interface based on the Haldane distribution provided by the present invention comprises the following steps:

[0007] S1. Obtain the distribution function of Haldane distribution;

[0008] S2. The distribution function obtained in step S1 is applied to the pre-peak segment of the concrete interface shear curve to obtain the initial simulation function of the pre-peak shear curve;

[0009] S3. Substitute the coordinates of the peak point of the pre-peak shear curve into the initial simulation function of the pre-peak shear curve obtained in step S2 to obtain the parameter values ​​of the initial simulation function of the pre-peak shear curve;

[0010] S4. Substitute the parameter values ​​obtained in step S3 into the initial simulation function of the pre-peak shear curve and correct it using an exponential function to obtain the final pre-peak shear curve simulation function;

[0011] S5. Using the pre-peak shear curve simulation function obtained in step S4, perform actual pre-peak shear curve simulation.

[0012] The step S1 of obtaining the distribution function of the Haldane distribution specifically includes the following steps:

[0013] The distribution function F(x) of the Haldane distribution is obtained as Where x is the independent variable of the distribution function.

[0014] Step S2, in which the distribution function obtained in step S1 is applied to the pre-peak segment of the concrete interface shear curve to obtain an initial simulation function of the pre-peak shear curve, specifically includes the following steps:

[0015] The obtained distribution function F(x) is applied to the pre-peak section of the shear curve to obtain the initial simulation function τ(u) of the pre-peak shear curve: Where α is the first parameter; u is the shear displacement.

[0016] Substituting the peak point coordinates of the pre-peak shear curve into the initial simulation function of the pre-peak shear curve obtained in step S2 to obtain parameter values ​​of the initial simulation function of the pre-peak shear curve in step S3 specifically includes the following steps:

[0017] The peak point coordinates of the pre-peak shear curve (u p ,τ p ) is substituted into the initial simulation function τ(u) of the pre-peak shear curve, and the first parameter α is calculated as

[0018] Step S4, in which the parameter values ​​obtained in step S3 are substituted into the initial simulation function of the pre-peak shear curve and corrected using an exponential function to obtain the final simulation function of the pre-peak shear curve, specifically comprises the following steps:

[0019] A. Substitute the parameter values ​​obtained in step S3 into the initial simulation function of the pre-peak shear curve to obtain the function τ(u):

[0020] B. Correction coefficient using exponential function The function τ(u) obtained in step A is corrected to obtain the final pre-peak shear curve simulation function τ: Where γ is the second parameter, which is determined by the statistical method of successive approximation in practical applications.

[0021] The method for simulating the pre-peak shear curve of the concrete interface based on the Haldane distribution provided by the present invention establishes a simulation model based on the Haldane distribution and the exponential correction coefficient. The established model can characterize the shear deformation of the three pre-peak curves, and has good versatility, high accuracy and scientific rationality. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] Figure 1 Schematic diagram of the existing pre-peak shear stress-displacement curve.

[0023] Figure 2 Schematic diagram of the process flow of the present invention.

[0024] Figure 3 Schematic diagram of shear stress-displacement curves corresponding to anisotropic shear test results of various samples under different normal stresses according to the embodiment of the method of the present invention.

[0025] Figure 4 Schematic diagram of comparative test results of an embodiment of the method of the present invention.

[0026] Figure 5 It is an enlarged schematic diagram of comparative test results of an embodiment of the method of the present invention.

[0027] Figure 6 Schematic diagram of the determination coefficient of the comparative test results of the embodiments of the method of the present invention. DETAILED DESCRIPTION

[0028] like Figure 2 The method flow diagram of the present invention is shown as follows: The method for simulating the pre-peak shear curve of a concrete interface based on Haldane distribution provided by the present invention comprises the following steps:

[0029] S1. Obtaining the distribution function of the Haldane distribution; specifically comprising the following steps:

[0030] The distribution function F(x) of the Haldane distribution is obtained as Where x is the independent variable of the distribution function;

[0031] S2. Applying the distribution function obtained in step S1 to the pre-peak segment of the concrete interface shear curve to obtain an initial simulation function of the pre-peak shear curve; specifically comprising the following steps:

[0032] The obtained distribution function F(x) is applied to the pre-peak section of the shear curve to obtain the initial simulation function τ(u) of the pre-peak shear curve: Where α is the first parameter; u is the shear displacement;

[0033] S3. Substitute the coordinates of the peak point of the pre-peak shear curve into the initial simulation function of the pre-peak shear curve obtained in step S2 to obtain the parameter values ​​of the initial simulation function of the pre-peak shear curve; specifically comprising the following steps:

[0034] The peak point coordinates of the pre-peak shear curve (u p ,τ p ) is substituted into the initial simulation function τ(u) of the pre-peak shear curve, and the first parameter α is calculated as

[0035] S4. Substituting the parameter values ​​obtained in step S3 into the initial simulation function of the pre-peak shear curve, and correcting it using an exponential function to obtain the final pre-peak shear curve simulation function; specifically comprising the following steps:

[0036] A. Substitute the parameter values ​​obtained in step S3 into the initial simulation function of the pre-peak shear curve to obtain the function τ(u):

[0037] B. Correction coefficient using exponential function The function τ(u) obtained in step A is corrected to obtain the final pre-peak shear curve simulation function τ: Where γ is the second parameter, which is determined by the statistical method of successive approximation in practical applications;

[0038] S5. Using the pre-peak shear curve simulation function obtained in step S4, perform actual pre-peak shear curve simulation.

[0039] Statistical damage methods, due to their convenience and intuitiveness, are widely used to study the constitutive response of quasi-brittle materials and are considered a very attractive tool for studying the deformation and failure mechanisms of concrete. According to statistical damage theory, this method first assumes that the damage variable obeys a certain distribution function. Commonly used distribution functions include the Weibull function, the modified Harris function, the normal distribution function, and the Haldane distribution.

[0040] Among the most commonly used distribution functions, the Weibull function plays an absolute dominant role in constitutive relationship modeling, and its expression is: Where p is the variable parameter; p0 and m are the Weibull distribution parameters, but the physical meaning and mathematical definition of these two parameters are not clear.

[0041] Although studies have shown that the results obtained by Weibull distribution are statistically acceptable, it is not feasible to only focus on statistical indicators (such as R 2 and RMSE) may mask the shortcomings of this distribution, and blindly using this method may still cause large errors. In addition, some researchers have found that there is not enough evidence to show that the Weibull distribution is always preferred over other distributions, and seriously question whether the Weibull distribution is the most suitable statistical distribution function for concrete materials. In fact, it is not convenient to use the Weibull distribution to create a constitutive model because it requires determining at least three unknown parameters (p, m and p0); therefore, at least three data points are required, which is more difficult in actual work. Therefore, this application uses the Haldane distribution for simulation.

[0042] The expression of Haldane distribution is When x = 0, F(x) = 0 (corresponding to the characteristic that the shear constitutive curve must pass through the origin). Moreover, when x is greater than 0, F(x) is always greater than 0, which also conforms to the characteristics of the shear curve (shear stress and displacement are always greater than 0). Therefore, it is possible to consider using the Haldane distribution to simulate the pre-peak section of the shear curve.

[0043] The effectiveness of the method of the present invention is illustrated below in conjunction with specific experiments:

[0044] In the verification test, the results of the direct shear test of the plain concrete interface were used. Three groups of concrete specimens with different strengths (numbered as group a, group b, and group c) were made according to different mass mix ratios (water: sand: cement: water reducer: silica fume). During the test, the normal stress was set to 0.2MPa, 0.5MPa, and 1MPa, respectively. The specimen number is XY MPa-Z°; X represents the group (abc), Y represents the normal stress, and Z represents the shear direction (0°, 90°, 180°, 270°). The anisotropic shear test results of each specimen under different normal stresses are shown in Figure 2. Figure 3 shown.

[0045] To facilitate comparison of model results, the present model is compared with the models of Ban et al., Bandis et al., and Nassir et al. The expressions of each model are as follows:

[0046] The expression of Ban et al.'s model is:

[0047]

[0048] In the formula, parameters a and b are model parameters, k i is the initial shear stiffness; these three parameters can be obtained by fitting the test data;

[0049] The model of Bandis et al. is expressed as:

[0050]

[0051] Where m is the inverse of the initial shear stiffness, and n is the inverse of the horizontal asymptote of the hyperbola; both m and n are greater than 0;

[0052] The expression of the model of Nassir et al. is:

[0053]

[0054] The parameter δ is Where η is the expansion cracking coefficient (range 0 to 1), is the basic friction angle.

[0055] like Figure 4 As shown, the above four models are substituted into the experimental results in turn to solve. This application only gives the comparison chart of the results of test b-0.5MPa-0°. The R of each theoretical curve fitted by the proposed model and other models is 2 Calculated and presented in Table 1:

[0056] Table 1 Determination coefficient R of each model and experimental results 2 Schematic table

[0057]

[0058]

[0059]

[0060] As can be seen from Table 1, the satisfactory R 2 Value (all R 2 ≥0.95) shows the effectiveness of the model of the application. In addition, the model of the application only contains one unknown parameter r, while the other three models all contain multiple unknown parameters. The advantage of the model of the application is that it has fewer parameters and is easy to apply.

[0061] During the solution process, it was found that the models of Ban et al., Bandis et al., and Nassir et al. did not describe the initial loading of some test results well, which were significantly higher than the test results. However, this application model does not have this problem. Taking the test data of a-0.5MPa-0° and a-1MPa-0 as examples, the application of the above four models is given respectively, as shown in the following figure: Figure 5 As shown. It can be found that the model of Ban et al., the model of Bandis et al. and the model of Nassir et al. are unable to describe the compaction part in the initial loading stage. Figure 6 The enlarged area shown is used to calculate the R of each model in this area. 2 , the results are as follows Figure 6As shown. It can be seen that the fitting effect of the model of this application is still good (R 2 >0.95), while the fitting effects of the other three models were not ideal (R 2 The range is between 0.539 and 0.834. Furthermore, the model of the present application guarantees that the model curve will pass through the peak point; this is not the case with the models of Ban et al. and Bandis et al. These factors demonstrate the superiority of the model of the present application.

Claims

1. A method for simulating the pre-peak shear curve of a concrete interface based on Haldane distribution, comprising the following steps: S1. Obtaining the distribution function of the Haldane distribution; specifically comprising the following steps: The distribution function F(x) of the Haldane distribution is obtained as Where x is the independent variable of the distribution function; S2. Applying the distribution function obtained in step S1 to the pre-peak segment of the concrete interface shear curve to obtain an initial simulation function of the pre-peak shear curve; specifically comprising the following steps: The obtained distribution function F(x) is applied to the pre-peak section of the concrete interface shear curve, and the initial simulation function τ(u) of the pre-peak shear curve is obtained as Where α is the first parameter; u is the shear displacement; S3. Substitute the coordinates of the peak point of the pre-peak shear curve into the initial simulation function of the pre-peak shear curve obtained in step S2 to obtain the parameter values ​​of the initial simulation function of the pre-peak shear curve; specifically comprising the following steps: The peak point coordinates of the pre-peak shear curve (u p ,τ p ) is substituted into the initial simulation function τ(u) of the pre-peak shear curve, and the first parameter α is calculated as S4. Substituting the parameter values ​​obtained in step S3 into the initial simulation function of the pre-peak shear curve, and correcting it using an exponential function to obtain the final pre-peak shear curve simulation function; specifically comprising the following steps: A. Substitute the parameter values ​​obtained in step S3 into the initial simulation function of the pre-peak shear curve to obtain the function τ(u): B. Correction coefficient using exponential function The function τ(u) obtained in step A is corrected to obtain the final pre-peak shear curve simulation function τ: Where γ is the second parameter, which is determined by the statistical method of successive approximation in practical applications; S5. Using the pre-peak shear curve simulation function obtained in step S4, perform actual pre-peak shear curve simulation.

Images