Effective prestress estimation and evaluation characteristic value calculation method for concrete structure

Abstract

An effective prestress estimation and evaluation characteristic value calculation method for a concrete structure, wherein the method comprises: S1, calculating effective prestress probability distribution of prestressed tendons of the concrete structure; S2, establishing Gaussian mixture models of effective prestress of the concrete structure; S3, performing normalization processing and normal significance determination on the Gaussian mixture models; S4, performing sampling estimation on structure's effective prestress probability distribution with theoretical distribution being normal distribution; S5, calculating an effective prestress evaluation characteristic value of a component under the normal distribution; S6, performing grouping processing and normal significance determination on the Gaussian mixture models; S7, performing sampling estimation on structure's effective prestress probability distribution with theoretical distribution being N sub-distribution Gaussian mixture models; and S8, calculating the effective prestress evaluation characteristic value of the component under the N sub-distribution Gaussian mixture models. According to the method, the evaluation characteristic value of effective prestress can be scientifically determined based on the actually measured sampling data of the effective prestress.

Description

Calculation method of characteristic value for estimation and evaluation of effective prestress in concrete structures Technical Field

[0001] The present invention relates to the technical field of prestressed concrete structures, and more particularly to a method for estimating characteristic values ​​of measured evaluation of effective prestress in existing concrete structures based on sequential sampling theory. Background Art

[0002] Prestressed concrete technology, with its low carbon footprint, material-saving properties, and strong spatial spanning capabilities, is a core technology for major civil engineering projects in my country, widely used in infrastructure such as road and rail bridges, specialized structures, and housing. In recent years, with the ongoing advancement of urban renewal, the demand for service performance evaluation has increased significantly. Prestressed concrete structures are subject to multiple factors, including fatigue, overloading, and extended service life. The effective prestressing force in these structures decreases, gradually reducing their service performance, which can lead to serious safety incidents. Therefore, scientifically evaluating the service performance of prestressed concrete structures is crucial for ensuring their long-term, safe service.

[0003] The effective prestress level is a significant factor affecting the service performance of prestressed concrete structures. In actual service, effective prestress is subject to multiple uncertainties and inherent randomness. Affected by design factors such as alignment and reinforcement ratio, the structural prestressing system exhibits significant nonuniformity. Furthermore, due to the concealed nature of prestressed tendons and the difficulty of measurement, efficient and accurate detection technologies have only been gradually implemented in recent years. Limited by structural type and detection efficiency, only a small amount of scattered effective prestress data can be obtained. This suggests that prestressed concrete structures often have a large number of tendons, high uncertainty, uneven distribution, and a prominent discrete effect. However, existing evaluation methods often use the mean of the test data as the characteristic value for evaluating effective prestress, ignoring the uncertainty and nonuniformity of the effective prestress distribution. This makes it impossible to accurately characterize the structural prestressing behavior. Consequently, there is a serious lack of effective prestressing system behavior and structural service performance evaluation methods based on measured data.

[0004] Therefore, how to provide an estimation method for the real-time probability distribution of effective prestress in existing concrete structures based on the measured sampling data of effective prestress, and then scientifically determine the evaluation characteristic value of effective prestress, is an urgent problem that technical personnel in this field need to solve.

[0005] Summary of the Invention

[0006] In view of this, the present invention provides a method for estimating and evaluating characteristic values ​​of effective prestress in concrete structures, aiming to solve the above technical problems.

[0007] In order to achieve the above object, the present invention adopts the following technical solutions:

[0008] A method for estimating and evaluating characteristic values ​​of effective prestress in concrete structures comprises the following steps:

[0009] S1. Calculate the probability distribution of effective prestress in prestressed tendons of concrete structures:

[0010] According to the design and construction drawings, the number of prestressed tendons n in the prestressed concrete structure is counted. Based on the statistical characteristics of each influencing factor, the Monte Carlo method is used to calculate the probability distribution of the effective prestress of the i-th tendon in the prestressed concrete structure (i = 1, 2, ... n). The statistical parameters of influencing factors such as tensioning control stress and friction coefficient and the theoretical calculation formula of prestress loss can be selected by referring to the paper "Prestress distribution characteristics and estimation method of concrete structure based on Gaussian mixture model. Journal of Building Structures, 2022, 43 (10): 60-67". The Monte Carlo method is used to calculate the probability distribution of the effective prestress of the i-th prestressed tendon (i = 1, 2, ... n), and establish the normal distribution probability density curve, as shown in formula (1).

[0011] Where x is the effective prestress, μ i is the design value of the effective prestress of the i-th reinforcement, which is also the mean of the probability distribution, s i is the standard deviation.

[0012] S2. Establish a Gaussian mixture model of effective prestress in concrete structures:

[0013] The set of effective prestress probability distributions of all prestressed tendons (n ​​tendons) in the structural prestressed system is regarded as a population and defined as the effective prestress probability distribution of the structure. The probability density p of the effective prestress of the i-th (i=1,2,...,n) single tendon of the structure is obtained through step S1. i (x), and the probability density is superimposed to form a Gaussian mixture model of the effective prestress of the structure, as shown in formula (2).

[0014] S3. Normalization of Gaussian mixture model and normal significance determination:

[0015] Research shows that the Gaussian mixture model composed of sub-distributions with similar means and variances can be approximated by the normal distribution. In order to reduce the uneven distribution of effective prestress of the structure caused by design factors, the effective prestress of each single reinforcement can be normalized by dividing the random variables of the effective prestress of all single reinforcements by their corresponding design values ​​μ i The normalized structural effective prestress Gaussian mixture model is formed, as shown in formula (3).

[0016] In statistics, the kurtosis and skewness can be used to calculate the normal significance of the model. Referring to the paper "Characteristics and estimation method of prestressed concrete structure based on Gaussian mixture model. Journal of Building Structures, 2022, 43(10): 60-67", the skewness S of the Gaussian mixture model with the same sub-distribution mean is 0, and the kurtosis K can be calculated using formula (4).

[0017] If the kurtosis is in the interval [3,3.5], it is preliminarily considered that the distribution can approximately obey the normal distribution, but further quantitative judgment is needed.

[0018] Based on the mean μ=1 and variance of formula (3) Construct a normal distribution model, as shown in formula (5).

[0019] Establish the difference function D(x)=P(x)-N(x) between the Gaussian mixture model and the normal distribution model. Due to monotonicity, it can be seen that D(x) is the maximum value at x=1. It is only necessary to determine whether this difference is within 5%, as shown in formula (6). It can be determined that the Gaussian mixture model (see formula 3) can be approximately represented by the normal distribution (see formula 5).

[0020] S4. Sampling estimation of probability distribution of effective prestress of structures with normal theoretical distribution:

[0021] In actual evaluation projects, the influence of factors such as construction errors and material properties should be considered, and the effective prestress control parameters of the structure should be corrected through the measured sample distribution to accurately reflect the actual distribution characteristics of the effective prestress of the structure.

[0022] According to step S3, when the probability distribution of effective prestress in the structure is normalized and can pass the normal significance test, the probability distribution of effective prestress in the structure is considered to come from a population with an approximate normal distribution, and its mean and variance determine the probability distribution form of the effective prestress. However, since accurate estimation of the variance requires a large number of samples, the actual measurement cost is high. Therefore, the present invention introduces a 95% guaranteed rate eigenvalue that contains both mean and variance information as a constrained eigenvalue, and based on the idea of ​​sequential sampling, conducts a measured sampling estimate of the effective prestress in the structure:

[0023] ① The effective prestress of the structure is considered to obey the normal distribution, see formula (7). 2 The effective prestress of the prestressed tendons in the structure is randomly sampled and tested to obtain the real-time effective prestress of the prestressed tendons and divide it by the design value of the prestressed tendons. The first batch of samples is k bars in total, forming Initial sample. P(x)=p(x|μ,s 2 ) (7)

[0024] ②Use Bootstrap method to analyze the samples Perform k×B sampling with replacement to form B Bootstrap samples that can reflect the statistical characteristics of the population (see formula 8), where a column of the matrix is ​​a Bootstrap sample, and B is taken at least 100,000 times. Calculate the 95% guaranteed rate eigenvalue of each Bootstrap sample Forming the eigenvalue probability density function Calculate the 95% confidence interval length l F .

[0025] ③If the maximum error It is considered that the sample The estimation accuracy meets the predetermined requirements, where for The mean of , 0.05 and 0.1 are the preset estimation accuracy requirements. When the accuracy requirement is high, 0.05 can be taken, and when the accuracy requirement is not high, 0.1 can be taken. It is considered that the sample If the estimation of the constrained eigenvalue cannot meet the predetermined requirements, it is necessary to continue to extract m samples to increase the total number of samples to k+m to form the measured sample.

[0026] ④ Repeat steps ② to ③ for the sample. When the estimation accuracy requirement is not met, add m samples each time and repeat p-1 times in total until the estimation accuracy requirement is met, that is, the maximum estimation error The final sample

[0027] ⑤Use Bootstrap method to Expand the sample processing to form the probability density function of the mean and variance Calculate 95% confidence intervals for the two parameters separately To ensure the safety of the structure, it is recommended to use the lower limit of the mean and upper bound of variance As a characteristic parameter, substituting it into formula (7) yields the measured probability estimation model of the effective prestress of the structure, as shown in formula (9).

[0028] ⑥ Perform inverse normalization on formula (9) and multiply it by the design value μ of each prestressed reinforcement in the structure i , calculate the measured probability estimation model of the effective prestress of each single reinforcement, see formula (10).

[0029] S5. Calculation of the effective prestress evaluation characteristic value of the component under normal distribution:

[0030] When the measured effective prestress of the structure obeys the normal distribution, the effective prestress of a single tendon also obeys the normal distribution (see formula 10), and the probability distribution of the effective prestress of the component is the superposition of multiple normal distributions. Based on the principle of normal distribution superposition, the probability distribution of the effective prestress of a component equipped with T-root prestressing tendons can be expressed as:

[0031] Among them, μ R 、s R are the measured estimated mean and standard deviation of the effective prestress of each single reinforcement in the component, respectively, obtained by formula (10), R is the number of the prestressing reinforcement in the component (R = 1, 2, ..., T);

[0032] In order to fully consider the discrete characteristics of the effective prestress distribution of the component, the upper and lower limits of the 95% confidence interval are used to evaluate the characteristic values ​​of the effective prestress distribution in the component. As shown in formula (12), when it is favorable to the component, the lower limit of the 95% confidence interval should be used for estimation, and the formula should take a negative sign; when it is unfavorable to the component, the upper limit of the 95% confidence interval should be used for estimation, and the formula should take a positive sign.

[0033] S6. Grouping processing of Gaussian mixture model and normal significance determination:

[0034] When the above normal significance cannot be satisfied, return to step S2, arrange the means and variances of the effective prestressing forces of all single bars in formula (2) according to their size, and divide them into N groups so that the difference in means and variances between each group is minimized. The number of prestressing bars in each group is n. j The Gaussian mixture model of the effective prestress of this structure is shown in formula (13).

[0035] Where j represents the group number (j=1,2…N), n j is the number of prestressed tendons in each group, q represents the number of prestressed tendons in group j (q=1,2,…n j ), μ qj , are the design value (mean) and variance of the qth reinforcement in the jth group respectively.

[0036] Referring to the paper “Distribution characteristics and estimation method of prestressed concrete structures based on Gaussian mixture model. Journal of Building Structures, 2022, 43(10): 60-67”, the kurtosis and skewness of each group of Gaussian mixture models were calculated using the following formula:

[0037] Where: S is skewness, K is kurtosis, E[(x-μ j ) 2 ]、E[(x-μ j ) 3 ]、E[(x-μ j ) 4 ] are the second-order, third-order, and fourth-order center distances of the jth group, μ j is the mean of the design values ​​of the j-th group model,

[0038] If the skewness S of the j-th group of Gaussian mixture models is in the interval [-0.2, 0.2] and the kurtosis K is in the interval [3, 3.5], then the model can be approximately normal distributed.

[0039] If the above N groups of Gaussian mixture models can all pass the above normal significance test, then the structural effective prestress Gaussian mixture model can be approximately transformed into N sub-distribution Gaussian mixture models, so that each sub-distribution can be approximately represented by a normal distribution, as shown in formula (19).

[0040] Where N is the number of groups. In practical projects, it is recommended to start with 2. If the above normal significance criterion cannot be met, increase the number of groups to make N = N + 1, and finally make N sub-distributions pass the normal significance test. For general projects, it is recommended to set 2 ≤ N ≤ 5. j is the weight of the jth group.

[0041] S7. Sampling estimation of the probability distribution of effective prestress of structures with N-subdistribution Gaussian mixture model as the theoretical distribution:

[0042] When the probability distribution of effective prestress of the structure can be processed and tested through step S6, it can be regarded as coming from a population of approximately N sub-distribution Gaussian mixtures, see formula (20).

[0043] The effective prestress of the structure is estimated by sampling with reference to the measured sampling estimation of the normal distribution (step S5). The specific steps are as follows:

[0044] ① The effective prestress of the structure obeys the N-distribution Gaussian mixture model, where ω j , μ j , s j are unknown. Make the first sampling from the population and draw k initial samples

[0045] ②Use Bootstrap method to analyze the samples Perform sampling with replacement k×B times to form B bootstrap samples that can reflect the statistical characteristics of the population.

[0046] ③ Using the EM algorithm, the ω of each Bootstrap sample j , μ j , s j The statistical parameters are estimated and the probability distribution of the 95% guaranteed rate characteristic value is obtained. Calculate the 95% confidence interval length l F .

[0047] ④ If the maximum estimation error It is considered that the sample The estimation accuracy meets the predetermined requirements. It is considered that the sample If the estimation of the constrained eigenvalue cannot meet the predetermined requirements, it is necessary to continue to extract m samples to increase the total number of samples to k+m to form the measured sample.

[0048] ⑤ With Repeat steps ② to ③ for the sample. When the estimation accuracy requirement is not met, add m samples each time and repeat p-1 times in total until the estimation accuracy requirement is met, that is, the maximum estimation error The final sample

[0049] ⑥ Using EM algorithm to The control parameters of the sample are estimated to obtain the weight mean and variance Substituting into formula (20), we can obtain the probability distribution of effective prestress of the structure.

[0050] Calculation of the effective prestress evaluation characteristic value of components under the S8 and N-subdistribution Gaussian mixture models:

[0051] When the measured distribution of effective prestress in the structure approximately follows an N-distribution Gaussian mixture model (N=2, 3, 4, 5), the visual effective prestress of a single reinforcement also follows an N-distribution Gaussian mixture model. Since there are no reports on the probabilistic superposition calculation of Gaussian mixture models, this paper derives the superposition formula of the Gaussian mixture model based on the normal distribution superposition principle.

[0052] When the measured effective prestress of the structure is a twin-distribution Gaussian mixture model, that is, N = 2, see formula (22).

[0053] Based on the probability superposition principle, when there are two reinforcement bars in a component, the Gaussian mixture model superposition can be regarded as the superposition of two normal sub-distributions with weights. For example, the sub-distribution With sub-distribution The probability of simultaneous occurrence and mutual superposition is 2ω1ω2. Because all sub-distributions are normal distributions, the mean and variance of the superposition can be calculated using the normal distribution superposition principle. Therefore, the probability of effective prestressing for a component equipped with two bars is shown in Equation (23). The superposition of the two-sub-distribution model forms a three-sub-distribution Gaussian mixture model.

[0054] By analogy, when the component is equipped with T-bars, it is equivalent to the Gaussian mixture model being superimposed T-1 times, as shown in Equation (24).

[0055] Among them, ω r is the weight coefficient, which is a polynomial (ω1+ω2) T The corresponding single term after factorization is C r is a single coefficient, r=0,1,2,…T, see Table 1.

[0056] Table 1 Weight coefficient calculation

[0057] When the measured effective prestress of the structure obeys the N-sub-distribution Gaussian mixture model (N=3, 4, 5), the effective prestress probability density of the component can be calculated by referring to the superposition principle of the twin-distribution Gaussian mixture model proposed in this invention. It can be determined by integration. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are merely embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on the provided drawings without paying any creative work.

[0059] FIG1 is a flow chart of a method for estimating the probability distribution of effective prestress of a structure and calculating the evaluation characteristic value based on sequential sampling provided by the present invention;

[0060] FIG2 is a diagram of a normalized Gaussian mixture model provided by an embodiment of the present invention;

[0061] FIG3 is a diagram showing the variation of the maximum estimation error of effective prestressing force of a beam with the number of samples according to an embodiment of the present invention;

[0062] FIG4 is a diagram of a twin distribution Gaussian mixture model provided by an embodiment of the present invention;

[0063] FIG5 is a diagram showing how the maximum estimation error of effective prestress of a plate varies with the number of samples provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0064] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0065] 1 , an embodiment of the present invention discloses a method for estimating and evaluating characteristic values ​​of effective prestress in a concrete structure, which specifically includes the following steps:

[0066] S1. Calculate the probability distribution of effective prestress in prestressed tendons of concrete structures;

[0067] S2. Establish a Gaussian mixture model of effective prestress in concrete structures;

[0068] S3, normalization processing of Gaussian mixture model and normal significance determination;

[0069] S4. Sampling estimation of probability distribution of effective prestress of structures whose theoretical distribution is normal distribution;

[0070] S5. Calculation of characteristic values ​​for effective prestress evaluation of components under normal distribution;

[0071] S6. Grouping processing of Gaussian mixture model and normal significance determination;

[0072] S7. Sampling estimation of probability distribution of effective prestress of structure using Gaussian mixture model with N sub-distributions as theoretical distribution;

[0073] S8. Calculation of characteristic values ​​for effective prestress evaluation of components under the N-subdistribution Gaussian mixture model.

[0074] Example:

[0075] A certain project employed unbonded prestressing technology for its beams and slabs. This structure, a nine-story, wide-flat beam frame with unbonded prestressed concrete, boasts a building area of ​​approximately 43,000 square meters. Nineteen years after completion, reinforcement and renovation efforts were planned, necessitating an evaluation of the effective prestress level within the structure. The project employed 1570MPa steel strands for prestressing, with a tensioning control stress of 70% of the tendon's tensile strength. The concrete beam strength was C35.

[0076] Taking into account the significant differences in the design values ​​of effective prestressing of beams and slabs in the structure, the effective prestressing in beams and slabs is classified, tested and evaluated.

[0077] Sampling evaluation of prestressed tendons in beams

[0078] 1. Obtain the design conditions and number of prestressed tendons in the beam (n = 1817) from the design drawings, requiring the maximum estimated error to be controlled within 5%.

[0079] 1. Normalization of theoretical model and normal significance test:

[0080] According to steps S1 and S2, the effective prestressed Gaussian mixture model of the structural beam is calculated, and the effective prestressed tendons in the beam are normalized to form a Gaussian mixture model with a mean of 1 for each sub-distribution. The kurtosis is calculated to be K = 3.08, and the difference between the two models D(x) = 0.023 < 0.05. The results show that the Gaussian mixture model can pass the S3 test, indicating that it can be approximated by the normal distribution for sampling evaluation. The theoretical distribution is shown in Figure 2 and Equation (25). P(x) = p(x|1, 0.067 2 ) (25)

[0081] 2. Take the first batch of prestressed steel samples

[0082] The number of samples for the first batch was 125 according to Article 3.3.10 of the Technical Standard for Inspection of Building Structures (GB / T 50344-2019), and the type of project was taken. The effective prestress of unbonded prestressed tendons was measured on site using the pull-off method. The measured effective prestress value was divided by the design mean value of the prestressed tendon to form the normalized measured data.

[0083] 3. Combine the Bootstrap method to perform replacement sampling on the first batch of samples

[0084] Sampling with replacement is performed 100,000 times to generate the sample space required for eigenvalue estimation Calculate the 95% confidence level of each Bootstrap sample. Forming the eigenvalue probability density function Calculate the length of its 95% confidence interval l F125 .

[0085] After KS test, the 95% assurance rate characteristic value can accept the normality hypothesis, and the characteristic value mean is calculated. Eigenvalue variance Confidence interval length l F125 =3.92×s 125 =0.051, estimated error Unable to meet reservation requirements.

[0086] 4. Supplement sampling samples and calculate eigenvalue estimation error

[0087] Supplement the measured samples to expand the sample information and improve the accuracy of the estimation. Repeat steps (② to ③) in S4 with 10 measured samples each time (m = 10), as shown in Figure 3. When the number of samples increases to 175, the mean of the characteristic value of the 95% guarantee rate of effective prestressing 95% confidence interval length l F175 =3.92*s 175 =0.045, maximum estimation error That is, 175 samples can ensure that the eigenvalue estimation error is controlled within 5%.

[0088] 5. Establish a measured probability estimation model for effective prestress of the structure

[0089] For 175 samples Combined with the Bootstrap method, sampling is performed 100,000 times with replacement, and the mean of each sampling result is calculated. and variance The probability density function, with the lower limit of the mean and the upper limit of the variance as characteristic parameters, estimates the actual probability distribution of the effective prestress of the structure.

[0090] For the probability distribution of effective prestress of a single reinforcement, we can perform inverse normalization and multiply it by the design mean value corresponding to the prestressed reinforcement. The measured probability estimation model is as follows:

[0091] For example, if the effective prestressing design value of the i-th reinforcement is 754 MPa, the probability distribution of the prestressed reinforcement is:

[0092] 6. Calculation of evaluation characteristic value of effective prestress of beam components:

[0093] Taking a typical prestressed concrete beam as an example, the effective prestress evaluation characteristic value of the beam component is carried out. The beam is equipped with 12 prestressed tendons, and the probability distribution of each tendon is calculated by formula (27), which is 6 tendons and 6 roots Based on the normal distribution superposition principle of step S5, the evaluation characteristic value of the effective prestress of the beam component is:

[0094] 2. Sampling evaluation of prestressed tendons in slabs

[0095] The design conditions and number of prestressed tendons in the slab (n=3587) are obtained from the design drawings, and the maximum estimation error is required to be controlled within 5%.

[0096] 1. Normalization of theoretical model and normal significance test:

[0097] According to steps S1 and S2, the effective prestressed Gaussian mixture model of the structural beam is calculated, and the effective prestressed stress of the prestressed tendons in the slab is normalized to form a Gaussian mixture model with a mean of 1 for each sub-distribution. The kurtosis is calculated to be K = 3.69 > 3.5. The results show that this Gaussian mixture model cannot pass the S3 test. It is necessary to return to step S2 and sort its mean and variance from large to small, dividing it into two groups (N = 2), as shown in formula (30).

[0098] The normal significance of each Gaussian mixture model was verified through step S5. The skewness of group 1, S, was calculated to be less than 0.2, and the kurtosis, K, was calculated to be less than 3.12. For group 2, S was calculated to be less than 0.2, and the kurtosis, K, was calculated to be less than 3.5. Both Gaussian mixture models passed the normal significance test, indicating that the Gaussian mixture model in the plate can be approximately transformed into a Gaussian mixture model of a twin distribution, as shown in Figure 4 and Equation (31).

[0099] 2. Take the first batch of prestressed steel samples

[0100] The number of samples for the first batch is 200 according to Article 3.3.10 of the Technical Standard for Building Structure Inspection (GB / T 50344-2019). The effective prestress of unbonded prestressed tendons was measured on site using the pull-off method to obtain the measured data.

[0101] 3. Combine the Bootstrap method to perform replacement sampling on the first batch of samples

[0102] Sampling with replacement is performed 100,000 times to generate the sample space required for eigenvalue estimation The EM algorithm is used to estimate the parameters of each sampling result, and the 95% guaranteed rate characteristic value of each sampling result is calculated according to the S6 method. Form the probability density function of the eigenvalue under the conditions of 200 sampling samples Calculate the length of its 95% confidence interval l F200 .

[0103] After KS test, the 95% assurance rate characteristic value can accept the normality hypothesis, and the characteristic value mean is calculated. Eigenvalue variance Length of 95% confidence interval of eigenvalue l F200=3.92×s 200 =40.49, maximum estimation error Unable to meet reservation requirements.

[0104] 4. Supplement sampling samples and calculate eigenvalue estimation error

[0105] Supplement the measured samples to expand the sample information and improve the accuracy of the estimation. Repeat steps (② to ③) in S7 with 20 measured samples each time (m = 20). As shown in Figure 5, when the number of samples increases to 320, the mean of the characteristic value of the 95% guarantee rate of effective prestressing Length of 95% confidence interval of eigenvalue l F320 =3.92×s 320 =32.49, maximum estimation error That is, 320 samples can ensure that the eigenvalue estimation error is controlled within 5%.

[0106] 5. Establish a measured probability estimation model for effective prestress of the structure

[0107] Using EM algorithm The control parameters of the sample are estimated to obtain the weight mean and variance Substituting into formula (20), the probability distribution of effective prestress of the structure is obtained as follows:

[0108] The effective prestress of a single reinforcement also follows this distribution:

[0109] 6. Calculation of evaluation characteristic values ​​of effective prestress of plate components

[0110] Taking a typical prestressed concrete slab as an example, the effective prestress evaluation characteristic value of the slab component is carried out. The slab is equipped with 124 prestressed tendons. The probability distribution of each tendon is shown in Equation (33). Based on the superposition principle of the twin distribution Gaussian mixture model in step S8, the probability distribution of the effective prestress of the slab component is:

[0111] The evaluation characteristic value of the effective prestress of the plate member can be obtained by integrating formula (34) and calculating the value of x when the area is 95%. This x is the evaluation characteristic value of the effective prestress.

[0112] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. Reference can be made to the common and similar parts between the various embodiments. For the devices disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple, and the relevant parts can be referred to the method description.

[0113] The above description of the disclosed embodiments is intended to enable one skilled in the art to implement or use the present invention. Various modifications to these embodiments will be readily apparent to one skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention is not limited to the embodiments shown herein but is intended to conform to the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for calculating characteristic values ​​of effective prestress estimation and evaluation of concrete structures, characterized in that: The specific steps include: S1. Calculate the probability distribution of effective prestress of prestressed tendons in concrete structures; S2. Establish a Gaussian mixture model of effective prestress in concrete structures; S3, normalization processing of Gaussian mixture model and normal significance determination; S4. Sampling estimation of probability distribution of effective prestress of structures whose theoretical distribution is normal distribution; S5. Calculation of characteristic values ​​for effective prestress evaluation of components under normal distribution; S6. Grouping processing of Gaussian mixture model and determination of normal significance; S7, Sampling estimation of probability distribution of effective prestress of structure with N-subdistribution Gaussian mixture model as theoretical distribution; S8. Calculation of eigenvalues ​​for effective prestress evaluation of components under the N-subdistribution Gaussian mixture model.

2. A method for calculating characteristic values ​​of effective prestress estimation and evaluation of concrete structures according to claim 1, characterized in that: In step S1: the number n of prestressed tendons in the prestressed concrete structure is counted; the Monte Carlo method is used to calculate the probability distribution of the effective prestress of the i-th tendon in the prestressed concrete structure (i=1, 2, ...n); the Monte Carlo method is used to calculate the probability distribution of the effective prestress of the i-th prestressed tendon (i=1, 2, ...n), and a normal distribution probability density curve is established, as shown in formula (1): Where x is the effective prestress, μ i is the design value of the effective prestress of the i-th reinforcement, that is, the mean of the probability distribution, s i is the standard deviation.

3. A method for calculating characteristic values ​​of effective prestress estimation and evaluation of concrete structures according to claim 2, characterized in that: In step S2: the set of probability distributions of effective prestress of all prestressed tendons in the structural prestressing system is regarded as a population and defined as the probability distribution of effective prestress of the structure; the probability density p of the effective prestress of the i-th (i=1,2,...,n) single tendon of the structure is obtained through step S1: i (x), and the probability density is superimposed to form a Gaussian mixture model of effective prestress of the structure, as shown in formula (2).

4. A method for calculating characteristic values ​​of effective prestress estimation and evaluation of concrete structures according to claim 3, characterized in that: In step S3: the random variables of the effective prestress of all single bars are divided by their corresponding design values ​​μ i , forming a normalized structural effective prestress Gaussian mixture model, as shown in formula (3): The skewness S of the Gaussian mixture model with the same sub-distribution mean is 0, and the kurtosis K is calculated using formula (4): If the kurtosis is in the interval [3,3.5], it is preliminarily considered that the distribution can be approximately normal; Based on the mean μ = 1 and variance of formula (3) Construct a normal distribution model, as shown in formula (5): The difference function D(x)=P(x)-N(x) between the Gaussian mixture model and the normal distribution model is established. Due to monotonicity, D(x) is the maximum value at x=1. Therefore, it is only necessary to determine whether the difference is within 5%, as shown in formula (6). It is determined that the Gaussian mixture model can be approximated by the normal distribution.

5. A method for calculating characteristic values ​​of effective prestress estimation and evaluation of concrete structures according to claim 4, characterized in that: In step S4: Based on the idea of ​​sequential sampling, the effective prestress of the structure is estimated by actual sampling: ① The effective prestress of the structure obeys the normal distribution, see formula (7); where μ, s 2 All of them are unknown. Random sampling test is performed on the effective prestress of the prestressed tendons in the structure to obtain the real-time effective prestress of the prestressed tendons and divide it by the design value of the prestressed tendons. The first batch of samples takes k tendons in total to form Initial sample; P(x)=p(x|μ,s 2 ) (7) ②Use Bootstrap method to sample Perform sampling with replacement k×B times to form B Bootstrap samples that can reflect the statistical characteristics of the population, as shown in formula (8), where a column of the matrix is ​​a Bootstrap sample, and B is taken at least 100,000 times; calculate the 95% guaranteed rate eigenvalue of each Bootstrap sample Forming the eigenvalue probability density function Calculate the 95% confidence interval length l F ; ③If the maximum error It is considered that the sample The estimation accuracy meets the predetermined requirements, where for The mean of It is considered that the sample The estimation of the eigenvalue cannot meet the predetermined requirements, and it is necessary to continue to extract m samples to increase the total number of samples to k+m to form the measured sample. ④ Repeat steps ② to ③ for samples. When the estimation accuracy requirement is not met, add m samples each time and repeat p-1 times in total until the estimation accuracy requirement is met, that is, the maximum estimation error The final sample ⑤Use Bootstrap method to Expand the sample and form the probability density function of the mean and variance Calculate the 95% confidence intervals for the two parameters separately To ensure the safety of the structure, the lower limit of the mean and the upper limit of the variance As a characteristic parameter, substituting into formula (7), it is the measured probability estimation model of the effective prestress of the structure, see formula (9); ⑥ Perform inverse normalization on equation (9) and multiply it by the design value μ of each prestressed tendon in the structure i , calculate the measured probability estimation model of the effective prestress of each single reinforcement, see formula (10).

6. A method for calculating characteristic values ​​of effective prestress estimation and evaluation of concrete structures according to claim 5, characterized in that: In step S5: when the measured effective prestress of the structure obeys the normal distribution, the effective prestress of a single tendon also obeys the normal distribution as shown in formula (10), and the probability distribution of the effective prestress of the component is the superposition of multiple normal distributions; based on the principle of normal distribution superposition, the probability distribution of the effective prestress of the component equipped with T-root prestressing tendons is expressed as: Among them, μ R 、s R are the measured estimated mean and standard deviation of the effective prestress of each single reinforcement in the component, respectively, obtained by formula (10), R is the number of the prestressing reinforcement in the component (R = 1, 2, ..., T); The upper and lower limits of the 95% confidence interval are used as the characteristic values ​​for evaluating the effective prestress distribution in the component. As shown in formula (12); when it is favorable to the component, the lower limit of the 95% confidence interval is used for estimation, and the formula takes a negative sign; when it is unfavorable to the component, the upper limit of the 95% confidence interval is used for estimation, and the formula takes a positive sign.

7. A method for calculating characteristic values ​​of effective prestress estimation and evaluation of concrete structures according to claim 6, characterized in that: In step S6: when the above normal significance cannot be satisfied, return to step S2, arrange the means and variances of all single reinforcement effective prestressing in formula (2) according to their size, and divide them into N groups, so that the difference between the means and variances of each group is minimized, and the number of prestressing tendons in each group is n. j ; Then the Gaussian mixture model of the effective prestress of this structure is shown in formula (13); Where j represents the group number (j=1,2…N), n j is the number of prestressed tendons in each group, q represents the number of prestressed tendons in the jth group (q = 1, 2, ... n j ), μ qj , are the design value and variance of the qth reinforcement in the jth group respectively; The following formula is used to calculate the kurtosis and skewness of each group of Gaussian mixture models: Where: S is skewness, K is kurtosis, E[(x-μ j ) 2 ]、E[(x-μ j ) 3 ]、E[(x-μ j ) 4 ] are the second-order, third-order, and fourth-order center distances of the jth group, μ j is the mean of the j-th group of model design values, If the skewness S of the j-th group of Gaussian mixture models is in the interval [-0.2, 0.2], and the kurtosis K is in the interval [3, 3.5], then the model can be approximately normal distributed; If the above N groups of Gaussian mixture models can all pass the above normal significance test, the structural effective prestress Gaussian mixture model can be approximately transformed into N sub-distribution Gaussian mixture models, so that each sub-distribution can be approximately represented by a normal distribution, as shown in formula (19); Where N is the number of groups. In practical projects, N is first set to 2. If it cannot meet the above normal significance criterion, the number of groups is increased to make N = N + 1, and finally all N sub-distributions can pass the normal significance test. For general projects, 2 ≤ N ≤ 5, ω j is the weight of the jth group.

8. A method for calculating characteristic values ​​of effective prestress estimation and evaluation of concrete structures according to claim 7, characterized in that: In step S7: when the effective prestress probability distribution of the structure can pass the processing and verification of step S6, it is considered that the effective prestress probability distribution of the structure comes from a population of an approximate N-subdistribution Gaussian mixture, see formula (20); The measured sampling estimation of the effective prestress of the structure is carried out by referring to the measured sampling estimation of the normal distribution; the specific steps are as follows: ① The effective prestress of the structure obeys the N-distribution Gaussian mixture model, where ω j , μ j ,s j are all unknown; make the first sampling from the population and draw k initial samples ②Use Bootstrap method to sample Perform sampling with replacement k×B times to form B Bootstrap samples that can reflect the statistical characteristics of the population; ③ Through the EM algorithm, the ω of each Bootstrap sample j , μ j ,s j The statistical parameters are estimated and the probability distribution of the 95% guaranteed rate characteristic value is obtained. Calculate the 95% confidence interval length l F ; ④ If the maximum estimation error It is considered that the sample The estimation accuracy meets the predetermined requirements; if It is considered that the sample The estimate of the constraint eigenvalue cannot satisfy To meet the predetermined requirements, m samples need to be drawn to increase the total number of samples to k + m to form the measured samples. ⑤ Repeat steps ② to ③ for samples. When the estimation accuracy requirement is not met, add m samples each time and repeat p-1 times in total until the estimation accuracy requirement is met, that is, the maximum estimation error The final sample ⑥ Using EM algorithm The sample control parameters are estimated to obtain the weights Mean With variance Substituting into formula (20), we can obtain the probability distribution of effective prestress of the structure.

9. A method for calculating characteristic values ​​of effective prestress estimation and evaluation of concrete structures according to claim 8, characterized in that: In step S8: when the measured distribution of the effective prestress of the structure approximately obeys the N-distribution Gaussian mixture model (N=2, 3, 4...), the effective prestress of the single reinforcement also obeys the N-distribution Gaussian mixture model; When the measured effective prestress of the structure is a twin distribution Gaussian mixture model, that is, N = 2, see formula (22); The probability superposition of effective prestress of a component equipped with two bars is shown in formula (23). The superposition of the two-distribution model forms a three-distribution Gaussian mixture model; By analogy, when the component is equipped with T-root bars, it is equivalent to superposition of Gaussian mixture model T-1 times, as shown in formula (24); Among them, ω r is the weight coefficient, which is a polynomial (ω1+ω2) T The corresponding single term after factorization is C r is a single coefficient, r=0,1,2,…,T; When the measured effective prestress of the structure obeys the N-distribution Gaussian mixture model (N=3, 4, 5), the probability density of the effective prestress of the component is calculated by referring to the superposition principle of the twin distribution Gaussian mixture model, and the evaluation characteristic value of the effective prestress of the component is It is determined by integration.

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