Concrete rebound method total probability transformation model based on Copula function

Abstract

The invention relates to the technical field of concrete field detection, in particular to a concrete rebound method total probability transformation model based on a Copula function. According to the method, based on a Copula function method, a mean value conversion model, a variable coefficient conversion model and a Copula related structure between the concrete rebound value and the compressive strength are constructed respectively, and the concrete rebound value and compressive strength edge probability distribution are updated by adopting a Bayesian method. Compared with the prior art, in the process of establishing the conversion model, the total probability information of the concrete rebound value and the compressive strength is adopted, the model accuracy is higher, and the total probability information of the concrete compressive strength can be finally given.

Description

Technical Field

[0001] The invention relates to the technical field of concrete on-site detection, and in particular to a concrete rebound method full probability conversion model based on Copula function. Background Art

[0002] The rebound method is one of the most widely used concrete on-site testing technologies. Its basic principle is: when a steel hammer hits the concrete surface, energy proportional to the hardness of the concrete surface is transmitted to the steel hammer, causing it to rebound to a certain height. The compressive strength of the concrete is calculated based on the relationship between the rebound height and the hardness of the concrete surface, as well as the relationship between hardness and strength. Since the result of the rebound method is not the compressive strength, a conversion model is required to convert the test results into the compressive strength of concrete. This model is usually obtained from the test data using regression analysis methods. In addition, in order to improve the accuracy of the test results, a certain number of concrete core samples will be taken from the structure in actual applications. According to the rebound value and compressive strength of the core samples, the conversion model is corrected using a calibration method.

[0003] However, the rebound value of concrete is not only related to the concrete strength, but also affected by many factors. Therefore, the biggest challenge of the rebound method is to build a conversion model between the rebound value and the compressive strength. However, the regression method and the calibration method do not fully utilize the full probability information of the concrete rebound value and the compressive strength, and these two methods can only give an estimate of the mean value of the concrete compressive strength, but cannot give the full probability information. Summary of the invention

[0004] In order to solve the problem of conversion model accuracy in the process of concrete rebound detection, the present invention proposes a full probability conversion model of concrete rebound method based on Copula function, which uses the full probability information of concrete rebound value and compressive strength to fit the conversion model, improves the accuracy of the conversion result, and can give the probability density function of concrete compressive strength during application.

[0005] To achieve the above object, the present invention adopts the following technical solution:

[0006] A full probability conversion model of concrete rebound method based on Copula function specifically includes the following steps:

[0007] Step 1: Establishment of conversion model

[0008] (1) Mean Shift Model

[0009] Based on a large number of different batches of concrete rebound value and compressive strength data, a conversion model between the concrete rebound value and the mean compressive strength is established using the nonlinear regression method:

[0010] f cm =f 1 (Rm ,α) (1)

[0011] In the formula, f cm is the mean compressive strength, R m is the mean of the rebound value, α is the regression coefficient;

[0012] (2) Coefficient of variation conversion model

[0013] Based on a large number of different batches of concrete rebound value and compressive strength data, a conversion model between concrete rebound value and compressive strength coefficient of variation was established using nonlinear regression method:

[0014]

[0015] In the formula, is the coefficient of variation of compressive strength, δ R is the coefficient of variation of rebound value, β is the regression coefficient;

[0016] (3) Copula related structure

[0017] The empirical probability distribution function values ​​of the rebound value and compressive strength of different batches of concrete are calculated respectively, and the maximum likelihood method is used to estimate the parameters of the Copula function. The likelihood function is:

[0018]

[0019] Where c[·,·;θ] is the Copula density function, θ is the parameter of the Copula function, and T(·) and S(·) are the empirical probability distribution functions of rebound value and compressive strength.

[0020] (4) Full probability description

[0021] Combining the mean conversion model, the coefficient of variation conversion model and the Copula correlation structure, the joint probability density function between the concrete rebound value and the compressive strength is:

[0022]

[0023] Where G(·) and g(·) are the probability distribution function and probability density function of normal distribution; f c is the compressive strength of concrete; R is the rebound value of concrete.

[0024] Step 2: Use of conversion model

[0025] (1) Mean and coefficient of variation

[0026] Given n rebound values ​​(r 1 ,r 2 …,r n ), first calculate the mean value R of the rebound valuem and coefficient of variation δ R :

[0027]

[0028] Combined with the mean and coefficient of variation conversion model, the mean and coefficient of variation of concrete compressive strength are calculated:

[0029] f cm =f 1 (R m ,α) (7)

[0030]

[0031] (2) Probability density function

[0032] The probability density function of the compressive strength of the concrete to be tested is obtained by superimposing the conditional probability density function:

[0033]

[0034] (3) Bayesian updating

[0035] When the rebound value and compressive strength data of a small number of concrete core samples are obtained, the Bayesian method is used to update the marginal probability distribution of concrete rebound value and compressive strength; after the update, the probability density function of concrete compressive strength is:

[0036]

[0037] Where T(·) and t(·) are the updated probability distribution function and probability density function.

[0038] Compared with the prior art, the present invention has the following beneficial effects:

[0039] Based on the Copula function method, the present invention respectively constructs a mean conversion model between concrete rebound value and compressive strength, a coefficient of variation conversion model and a Copula correlation structure, and uses the Bayesian method to update the marginal probability distribution of concrete rebound value and compressive strength. Compared with the traditional technology, the present invention uses the full probability information of concrete rebound value and compressive strength in the process of establishing the conversion model, and the model is more accurate, and can finally provide the full probability information of concrete compressive strength. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 It is a scatter diagram of the rebound value and the compressive strength of the present invention.

[0041] Figure 2 It is a diagram of the mean and coefficient of variation conversion model of the present invention.

[0042] Figure 3It is the probability distribution diagram of the compressive strength of concrete of the present invention. DETAILED DESCRIPTION

[0043] In order to further explain the technical means and effects adopted by the present invention to achieve the predetermined invention purpose, the specific implementation mode, structure, characteristics and effects of the present invention are described in detail below in combination with the accompanying drawings and preferred embodiments.

[0044] Embodiment: A full probability conversion model of concrete rebound method based on Copula function specifically includes the following steps:

[0045] Step 1: Establishment of conversion model

[0046] (1) Mean Shift Model

[0047] Based on a large number of different batches of concrete rebound value and compressive strength data, a conversion model between the concrete rebound value and the mean compressive strength is established using the nonlinear regression method:

[0048] f cm =f 1 (R m ,α) (1)

[0049] In the formula, f cm is the mean compressive strength, R m is the mean rebound value, and α is the regression coefficient.

[0050] (2) Coefficient of variation conversion model

[0051] Based on a large number of different batches of concrete rebound value and compressive strength data, a conversion model between concrete rebound value and compressive strength coefficient of variation was established using nonlinear regression method:

[0052]

[0053] In the formula, is the coefficient of variation of compressive strength, δ R is the coefficient of variation of the rebound value, and β is the regression coefficient.

[0054] (3) Copula related structure

[0055] The empirical distribution function values ​​of the rebound value and compressive strength of different batches of concrete are calculated respectively, and the parameters of the Copula function are estimated by the maximum likelihood method. The likelihood function is:

[0056]

[0057] Where c[·,·;θ] is the Copula density function, θ is the parameter of the Copula function, and T(·) and S(·) are the empirical probability distribution functions of rebound value and compressive strength.

[0058] (4) Full probability description

[0059] Combining the mean conversion model, the coefficient of variation conversion model and the Copula correlation structure, the joint probability density function between the concrete rebound value and the compressive strength is:

[0060]

[0061] Where G(·) and g(·) are the probability distribution function and probability density function of the normal distribution.

[0062] Step 2: Use of conversion model

[0063] (1) Mean and coefficient of variation

[0064] Given n rebound values ​​(r 1 ,r 2 …,r n ), first calculate the mean value R of the rebound value m and coefficient of variation δ R :

[0065]

[0066] Combined with the mean and coefficient of variation conversion model, the mean and coefficient of variation of concrete compressive strength are calculated:

[0067] f cm =f 1 (R m ,α) (7)

[0068]

[0069] (2) Probability density function

[0070] The probability density function of the compressive strength of the concrete to be tested is obtained by superimposing the conditional probability density function:

[0071]

[0072] (3) Bayesian updating

[0073] When the rebound value and compressive strength data of a small number of concrete core samples are obtained, the Bayesian method is used to update the marginal probability distribution of concrete rebound value and compressive strength. After the update, the probability density function of concrete compressive strength is:

[0074]

[0075] Where T(·) and t(·) are the updated probability distribution function and probability density function.

[0076] Experimental data:

[0077] To verify this patent, 20 groups of 1838 pairs of concrete rebound value and compressive strength data were collected. The scatter plot of rebound value and compressive strength is shown in the figure below. Figure 1 shown.

[0078] Firstly, the power-rate model was used to establish the conversion model between the concrete rebound value and the compressive strength, and the logarithmic model was used to establish the conversion model between the coefficient of variation of the rebound value and the compressive strength, as shown in Figure 2 The model parameters are shown in Table 1.

[0079] Table 1 Conversion model parameters

[0080]

[0081] The two-dimensional Student Copula function is used to fit the correlation structure between the concrete rebound value and the compressive strength, and the parameter values ​​are 0.838 and 3.94. The Copula density function of the two-dimensional Student Copula function is:

[0082]

[0083] Where θ and υ are the parameters of the Student Copula function, t υ (·) is the probability density function of the t distribution with degrees of freedom υ, T υ (·) is the probability distribution function of the t distribution with υ degrees of freedom.

[0084] Taking a group in the data set as an example, the estimated values ​​obtained by the regression method, calibration method and the full probability conversion model of the present invention are shown in Table 2. The probability density function of the concrete compressive strength after the full probability model of the present invention and Bayesian update is as follows: Figure 3 As shown in the figure, it can be seen that compared with the regression method, the prediction accuracy of the full probability model is improved, and the full probability information of the compressive strength of concrete can be given. After updating the marginal probability distribution of the rebound value and the compressive strength by the Bayesian method, the prediction accuracy is also significantly improved, which is much higher than the calibration method, and the full probability information of the compressive strength of concrete can be given at the same time.

[0085] Table 2 Prediction results and errors

[0086]

[0087]

[0088] The above description is only a preferred embodiment of the present invention and does not limit the present invention in any form. Although the present invention has been disclosed as a preferred embodiment as above, it is not used to limit the present invention. Any technical personnel in this field can make some changes or modify the technical contents disclosed above into equivalent embodiments without departing from the scope of the technical solution of the present invention. However, any brief modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solution of the present invention are still within the scope of the technical solution of the present invention.

Claims

1. A full probability conversion model of concrete rebound method based on Copula function, characterized by: The specific steps include: Step 1: Establishment of conversion model (1) Mean Shift Model Based on a large number of different batches of concrete rebound value and compressive strength data, a conversion model between the concrete rebound value and the mean compressive strength is established using the nonlinear regression method: f cm =f1(R m ,α) (1) In the formula, f cm is the mean compressive strength, R m is the mean of the rebound value, α is the regression coefficient; (2) Coefficient of variation conversion model Based on a large number of different batches of concrete rebound value and compressive strength data, a conversion model between concrete rebound value and compressive strength coefficient of variation was established using nonlinear regression method: In the formula, is the coefficient of variation of compressive strength, δ R is the coefficient of variation of rebound value, β is the regression coefficient; (3) Copula related structure The empirical probability distribution function values ​​of the rebound value and compressive strength of different batches of concrete are calculated respectively, and the maximum likelihood method is used to estimate the parameters of the Copula function. The likelihood function is: Where c[·,·;θ] is the Copula density function, θ is the parameter of the Copula function, T(·) and S(·) are the empirical probability distribution functions of rebound value and compressive strength; (4) Full probability description Combining the mean conversion model, the coefficient of variation conversion model and the Copula correlation structure, the joint probability density function between the concrete rebound value and the compressive strength is: Where G(·) and g(·) are the probability distribution function and probability density function of normal distribution; Step 2: Use of conversion model (1) Mean and coefficient of variation Given n rebound values ​​r1, r2…, r n , first calculate the mean value R of the rebound value m and coefficient of variation δ R : Combined with the mean and coefficient of variation conversion model, the mean and coefficient of variation of concrete compressive strength are calculated respectively: f cm =f1(R m ,α) (7) (2) Probability density function The probability density function of the compressive strength of the concrete to be tested is obtained by superimposing the conditional probability density function: (3) Bayesian updating When the rebound value and compressive strength data of a small number of concrete core samples are obtained, the Bayesian method is used to update the marginal probability distribution of concrete rebound value and compressive strength; after the update, the probability density function of concrete compressive strength is: Where T(·) and t(·) are the updated probability distribution function and probability density function.

Images