Method for calculating variance of initial elastic modulus and bulk modulus parameters of Duncan-Zhang EB model

Abstract

The invention belongs to the field of geotechnical engineering, and particularly relates to a variance calculation method for initial elastic modulus and bulk modulus parameters of a Duncan-Zhang EB model. Index values of initial elastic modulus parameters K and n and bulk modulus parameters Kb and m in a Duncan-Zhang EB model are mostly obtained by adopting a grouping test and arrangement method at present, but the mean value error and the standard deviation of the parameters are relatively large. If all test points are placed in the same coordinate system for regression calculation, the problems of small sample capacity and large parameter error can be well solved. However, the Duncan-Zhang EB model parameters are regressed by the least square method, regression residuals have square difference or correlation or both, so that the effectiveness of the parameters is reduced, and the variance is no longer minimum variance estimation. According to the method, through variable replacement, the residual error of a new regression equation has variance homogeneity and irrelevance, the method is applied to a triaxial test for regression of 12 groups of rockfill materials of a certain height earth and rockfill dam, and the result shows that the standard deviation of parameters is minimum by the generalized least square method provided by the invention.

Description

Technical Field

[0001] The invention belongs to the field of geotechnical engineering, and specifically relates to a method for calculating the variance of initial elastic modulus and bulk modulus parameters of a Duncan-Zhang EB model. Background Art

[0002] The Duncan-Zhang EB model is a constitutive model widely used at home and abroad to describe the nonlinear stress-strain behavior of various soils. Its model parameters are key parameters for stress-deformation calculation of geotechnical structures, which will directly affect the safety evaluation of geotechnical structures. The variance of deformation parameters is a key parameter for stress-deformation sensitivity analysis and reliability analysis of important geotechnical structures.

[0003] The "Design Specifications for Rolled Earth-Rock Dams (SL274-2020, NB / T10872-2021)" stipulates that for high dams, it is advisable to combine engineering characteristics, dam material test results and engineering experience to carry out sensitivity analysis of stress-deformation parameters. In order to study whether plastic zones, tensile stress zones, cracks and hydraulic fracturing are generated in the dam body. To conduct sensitivity analysis of stress and deformation of high earth-rock dams, it is necessary to understand the statistical characteristics of stress-deformation parameters, that is, the mean and variance of deformation parameters. Due to the limited number of test groups for each dam material, the variance samples for determining deformation parameters are small and the accuracy is low. Even for ultra-high earth-rock dams, 11 groups of stress-deformation parameter tests are carried out for each material, and a combination of small value average, average and large value average is generally used to determine the calculation scheme for the stress-deformation sensitivity of the dam body. Table 1 gives the calculation scheme for stress-deformation sensitivity analysis of a high core wall rockfill dam. The quantiles of the small value average, average and large value average of the parameters are 25%, 50% and 75% respectively.

[0004] Table 1 Parameter scheme for stress-deformation sensitivity analysis of a high core rockfill dam

[0005]

[0006] It can be seen that such a combination and its parameter sensitivity analysis scheme are relatively rough. A more scientific method is to determine the variance of the deformation parameters to develop a parameter combination scheme at any quantile between different dam materials, and perform finite element calculations on this basis to more reasonably judge the safety of the dam body.

[0007] At present, the constitutive relationship of earth-rock materials in stress-deformation analysis of earth-rock dams mostly adopts the nonlinear elastic model represented by the Duncan-Zhang hyperbola, namely the Duncan-Zhang EB model. Its parameters are determined by group test sorting and statistical method. The samples of each group of tests are small and the parameter errors are large. If statistics are performed on this basis, the variance of the parameters obtained is too large and does not reflect the actual situation. Some scholars also take the earth-rock dam materials of several projects as a whole and statistically analyze the parameters of the Duncan-Zhang EB model. However, due to the different dam materials of different projects, the mean and variance of their parameters are only of reference significance and cannot be directly applied to the calculation and analysis of a certain project. The least squares method is to put all the test points of each group in the same coordinate system for regression calculation, which not only solves the sample capacity problem well, but also eliminates the errors contained in each group of test parameters. However, the study found that when the least squares method is applied to the regression of the initial elastic modulus and bulk modulus parameters of the Duncan-Zhang EB model, the residuals have variance heterogeneity or correlation or both, which violates the conditions for the use of the least squares method. Although we can still obtain an unbiased estimate of the parameter itself, the variance of the parameter is a biased estimator. Summary of the invention

[0008] In order to eliminate the variance heterogeneity and correlation of the residuals of the initial elastic modulus and bulk modulus parameters of the Duncan-Zhang EB model, the present invention squares the covariance matrix of the original regression residuals to obtain its square root matrix, multiplies the explanatory variables, dependent variables and residual vectors of the regression equation by the square root matrix of the covariance, and performs variable replacement, so that the new regression equation has the homogeneity of variance and irrelevance of the residuals. Regression calculations are performed on all test data, and the mean and variance of the stress-deformation parameters are obtained by the regression parameters. The variance of the deformation parameters obtained by the new method is smaller than that of the least squares method and the traditional grouping and arranging parameter statistics. It can be seen that the generalized least squares method that eliminates variance heterogeneity and correlation improves the estimation accuracy of the initial elastic modulus and bulk modulus parameters of the Duncan-Zhang EB model.

[0009] The technical solution of the present invention:

[0010] The method for calculating the variance of the initial elastic modulus and bulk modulus parameters of the Duncan-Zhang EB model includes the following steps:

[0011] Step 1: Select the regression expression of the initial elastic modulus and bulk modulus of the Duncan-Zhang EB model and organize the regression data.

[0012] According to the Duncan-Zhang EB model, the initial elastic modulus expression E i =KP a (σ3 / P a ) n , where K and n are the initial elastic modulus coefficient and index, P a It is expressed as an atmospheric pressure value, σ3 represents the confining pressure; the bulk modulus expression is B = Kb P a (σ3 / P a ) m , where K b , m is the bulk modulus coefficient and index; both sides of the formula are divided by P a And taking the logarithm, we get

[0013] lg(E i / P a )=lg(K)+nlg(σ3 / P a )(2)

[0014] lg(B / P a )=lg(K b )+mlg(σ3 / P a )(3)

[0015] Let lg(σ3 / P a )=x, when returning to the initial elastic modulus, let lg(E i / P a )=y,lg(K)=β0,n=β1;When regressing the bulk modulus, let lg(B / P a )=y,lg(K b )=β0,m=β1.

[0016] The regression equations of the two can be expressed as

[0017] y=β0+β1x+e(4)

[0018] Where e is the regression residual.

[0019] By conducting several triaxial tests on a soil material, several pairs of initial elastic modulus and several pairs of bulk modulus regression data equal to the number of tests can be determined, which are [lg(σ3 / P a )lg(E i / P a )], [lg(σ3 / P a )lg(B i / P a )];

[0020] Step 2: Least squares regression and homogeneity of variance and correlation test of its residuals

[0021] The matrix expression of the regression equation is

[0022] y=Xβ+e, E(e)=0, Cov(e)=σ 2 I (5) In the formula, X is the independent variable, y is the dependent variable, and e is the regression residual term, which represents the influence of other factors on y besides X and the experimental measurement error. y=[y1 y2…yi …y n ] T ,y i =lg(E i / P a ) or y i =lg(B i / P a ), T is the matrix transpose. β=[β0β1] T , β0 and β1 are unknown parameters to be estimated. E i , B i are the confining pressure, initial elastic modulus and bulk modulus of the ith test, respectively.

[0023] In formula (5), formula (1) is the regression equation, and formulas (2) and (3) are the prerequisites for the application of the least squares method.

[0024] The n groups of observation data (x i ,y i ) into the equation, we get an estimate of the regression parameter β

[0025]

[0026] The covariance matrix of the regression parameters is

[0027]

[0028] Where V is the covariance matrix of the residual vector under different confining pressures, σ 2 is the variance of the entire residual vector, i.e. Cov(e) = σ 2 V. If the regression residual satisfies the third formula of formula (5), then V = I is the unit matrix.

[0029] With the regression parameters of the initial elastic modulus and bulk modulus, the regression prediction value of y can be calculated. The difference between the regression prediction value and the measured value under various confining pressures is the regression residual e i (i=1,2,…,k), k represents the confining pressure level. The covariance matrix of the regression residual vector under each confining pressure is calculated as follows:

[0030]

[0031] In the formula, D(e1) and D(e k ) is the variance of the regression residual vector under the first and kth level confining pressure, Cov(e1,e k ) is the covariance of the regression residual vector under the 1st and kth level confining pressure.

[0032] (1) KS test of normal distribution of regression residuals under various confining pressures

[0033] The null hypothesis and alternative hypothesis are H0: F(x) = F0(x), H1: F(x) ≠ F0(x), where F0(x) is the normal distribution function. The test statistic is D n =sup|F n (x)-F0(x)|,F n (x) is the sample distribution function. At the significance level α, Probability Find the quantile t α , then the rejection region is [t α , +∞).

[0034] The test statistic is calculated based on the residuals under various confining pressures, its maximum value is determined, and it is judged whether to accept the null hypothesis.

[0035] (2) Homogeneity test of regression residuals under different confining pressures

[0036] The F test is used to determine whether the residuals have homogeneity of variance under different confining pressures. The test statistic is are the sample variances of the regression residual vectors under two confining pressures, n1 and n2 are the number of samples. The null hypothesis and alternative hypothesis are H0: H1: Not all equal, The residual variance under the i-th level of confining pressure. At the significance level α, make P(F≤F 1-α / 2 (n1-1,n2-1))=α / 2, P(F≥F α / 2 (n1-1,n2-1))=α / 2. Calculate the variance of the residual under each confining pressure With the statistic F, look up the table to determine the critical value and decide whether to accept the null hypothesis.

[0037] (3) Independence test of regression residuals under different confining pressures

[0038] The independence test can be determined by testing the correlation coefficient ρ. The null hypothesis and alternative hypothesis are H0: ρ = 0, H1: ρ ≠ 0. Test statistic It follows a t distribution with n-2 degrees of freedom. X i , Y i , are the residuals and their means under two confining pressures respectively. Given a significance level α, let P(|t|≥t α / 2 (n-2)) = α, if the observed value of t satisfies |t| ≥ t α / 2 (n-2), the null hypothesis H0 is rejected. The point estimate R and t of the correlation coefficient are calculated based on the regression residuals under each confining pressure to determine whether to accept the null hypothesis.

[0039] If the regression residuals have homogeneous variance and are uncorrelated, skip the third to fourth steps, and use the regression parameter estimates obtained by the least squares method and their covariance matrix to obtain the mean and variance of the initial elastic modulus and bulk modulus parameters; if the regression residuals have heterogeneous variance or correlation, or both, perform the third step and re-estimate the regression parameters and covariance matrix;

[0040] Step 3: Generalized least squares regression to eliminate variance and correlation

[0041] The covariance matrix V of the regression residuals is

[0042] Cov(e)=σ 2 V (8)

[0043] Let K T K=K -1 K=V, multiply both sides of the regression equation y=Xβ+ε by K -1 ,have

[0044] K -1 y=K -1 Xβ+K -1 e

[0045] Let z = K -1 y,B=K -1 X,g=K -1 e, then

[0046] z=Bβ+g (10)

[0047] After the transformation, the g elements have zero mean, constant variance and uncorrelated variances, satisfying the application conditions of least squares estimation.

[0048] An estimate of the solution to the equation is

[0049]

[0050] The covariance matrix of

[0051]

[0052] The fourth step is to obtain the mean and covariance matrix of the initial elastic modulus and bulk modulus parameters based on the regression parameters and their covariance matrix.

[0053] (1) Calculation of initial elastic modulus coefficient and index

[0054] Since lg(K)=β0,n=β1,there is

[0055]

[0056] Taylor expansion of the above formula, ignoring the second-order and above infinitesimals, and applying the variance formula of probability theory, we get

[0057]

[0058] In the formula is the variance of the initial elastic modulus regression parameters β0 and β1, is the variance of the initial elastic modulus coefficient K and the exponent n, Cov(K,n) is the covariance of K and n, and Cov(β0,β1) is the covariance of β0 and β1.

[0059] (2) Calculation of bulk modulus coefficient and index

[0060] Because lg(K b )=β0,m=β1,there are

[0061]

[0062] Taylor expansion of the above formula, ignoring the second-order and above infinitesimals, and applying the variance formula of probability theory, we get

[0063]

[0064] In the formula, is the variance of the bulk modulus regression parameters β0 and β1, K is the bulk modulus coefficient b The variance of index m, Cov(K b ,m) is K b The covariance with m, Cov(β0,β1) is the covariance of β0 and β1.

[0065] The beneficial effects of the present invention are:

[0066] The present invention provides a variance calculation method for the initial elastic modulus parameters and bulk modulus parameters of the Duncan-Zhang EB model of soil. Based on this, the initial elastic modulus parameters and bulk modulus parameters of the Duncan-Zhang EB model of any quantile can be determined, which improves the flexibility of the combination of different dam material parameter schemes for stress-deformation sensitivity analysis of earth-rock dams and provides key parameters for reliability analysis of earth-rock dams. For example, the stress-deformation sensitivity analysis of high earth-rock dams is no longer limited to the combination of the small average value, average value, and large average value of deformation parameters, and combinations of parameters of any quantile of different dam materials can be performed. Compared with the traditional method of grouping tests and sorting parameters and then statistically analyzing them, the generalized least squares method proposed by the present invention is used to regress the initial elastic modulus and bulk modulus parameters, and the parameter mean obtained is unbiased; compared with the classical least squares method, the parameter variance obtained by the new method is smaller. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1It is the generalized least square regression diagram of the initial elastic modulus of the rockfill material in zone I in the embodiment.

[0068] Figure 2 It is the generalized least squares regression diagram of the bulk modulus of the rockfill soil material in Zone I in the embodiment. DETAILED DESCRIPTION

[0069] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings and technical solutions.

[0070] The method for calculating the variance of the initial elastic modulus and bulk modulus parameters of the Duncan-Zhang EB model includes the following steps:

[0071] Step 1: Select the regression equation expression

[0072] The Duncan-Zhang EB stress-strain model is

[0073] E t =E i (1-R f S) 2

[0074] In the formula, the destruction ratio R f =(σ1-σ3) f / (σ1-σ3) ult , stress level

[0075] For dam rockfill, c = 0, represents the friction angle under the confining pressure of 1 atmosphere, It indicates the reduction of friction angle for every 10-fold increase in confining pressure. σ1 and σ3 are the maximum and minimum principal pressures, respectively. S is the stress level. P a is an atmospheric pressure value. Initial elastic modulus E i The expression of bulk modulus B is as follows

[0076] E i =KP a (σ3 / P a ) n , B=K b P a (σ3 / P a ) m (1)

[0077] Where, K and n are the initial elastic modulus coefficient and index; K b , m is the bulk modulus coefficient and exponent.

[0078] Divide P on both sides of the two equations in formula (1) a Then take the logarithm and get

[0079] lg(E i / P a )=lg(K)+nlg(σ3 / P a )(2)

[0080] lg(B / P a )=lg(K b )+mlg(σ3 / P a )(3)

[0081] Let lg(σ3 / P a )=x, when returning to the initial elastic modulus, let lg(E i / P a )=y,lg(K)=β0,n=β1;When regressing the bulk modulus, let lg(B / P a )=y,lg(K b )=β0,m=β1.

[0082] The regression equations of the two can be expressed as

[0083] y=β0+β1x+e(4)

[0084] In the formula, e is the residual term.

[0085] For n triaxial tests of a soil material, n pairs of initial elastic modulus and n pairs of bulk modulus regression data are determined. They are [lg(σ3 / P a )lg(E i / P a )],[lg(σ3 / P a )lg(B i / P a )]. i and B i Determined according to the method specified in the specification. i Determined by connecting the two points of stress level 70% and 95% on the stress-strain relationship curve; Bulk modulus ε v It represents volumetric strain. When the strain strain curve is dilatant, the peak point is taken. If dilatation does not occur before the point corresponding to the 70% stress level, the point on the strain strain curve corresponding to the 70% stress level is taken.

[0086] The second step is to determine the regression parameters β0 and β1 using the least squares method.

[0087] The matrix expression of the regression equation is

[0088] y=Xβ+e, E(e)=0, Cov(e)=σ 2 I (5)

[0089] In the formula, X is the independent variable, y is the dependent variable, and e is the residual term, which represents the influence of other factors on y except X and the experimental measurement error. y=[y1 y2 …y i … y n ] T ,y i =lg(E i / P a ) or y i =lg(B i / P a ), T is the matrix transpose; E i , B i are the confining pressure, initial elastic modulus and bulk modulus of the ith test respectively. β=[β0 β1] T , β0 and β1 are unknown parameters to be estimated.

[0090] In formula (5), formula (1) is the regression equation, and formulas (2) and (3) are the prerequisites for the application of the least squares method.

[0091] The least squares method finds an estimate of β such that the residual vector e = the sum of the squares of the lengths of y-Xβ‖y-Xβ‖ 2 Minimum.

[0092] Q(β)=‖y-Xβ‖ 2 =(y-Xβ) T (y-Xβ)

[0093] Taking the partial derivative of β and setting it to zero, we get the canonical equation X T Xβ=X T y, and the solution is

[0094] β=(X T X) -1 X T y

[0095] The n groups of observation data (x i ,y i ) into the equation, we get an estimate of β

[0096]

[0097] The covariance matrix of the regression parameters is

[0098]

[0099] Where V is the covariance matrix of the residual vector under different confining pressures. 2 is the variance of the entire residual vector. If the third equation in equation (5) holds, then V = I is the unit matrix.

[0100] The third step is to test the homogeneity of variance and correlation of the residuals of the classical least squares regression

[0101] The predicted value of y is calculated based on the estimated value of the regression parameters, and the difference between the measured value and the predicted value is the regression residual. After obtaining the regression residual vector under each confining pressure, the regression residual covariance matrix is ​​calculated, which is expressed as follows:

[0102]

[0103] In the formula, D(e1), D(e k ) is the variance of the regression residual vector under the 1st and kth level confining pressure, Cov(e1,e k ) is the covariance of the regression residual vector under the 1st and kth levels of confining pressure, and k is the number of confining pressure levels.

[0104] If the main diagonal elements of the covariance matrix are equal, the residuals under each confining pressure have the same variance; if all non-main diagonal elements are zero, the residuals under each confining pressure are uncorrelated. If the main diagonal elements are not all equal and the non-diagonal elements are not all zero, the normal distribution test, variance homogeneity test and independence test of the regression residuals under each confining pressure are performed.

[0105] 1. Test the variance and correlation of the residuals of the initial elastic modulus regression parameters

[0106] Substituting the initial elastic modulus of 12 groups of triaxial tests of granite rockfill materials in area I of a high dam into equations (6) and (7), the regression equation and covariance matrix of the initial elastic modulus are obtained, as shown in Table 1.

[0107] Table 1 Calculation results of initial elastic modulus by classical least squares regression

[0108]

[0109] The covariance matrix of the residuals of the initial elastic modulus of the granite rockfill in area I is:

[0110]

[0111] It can be seen that the main diagonal elements of the regression residual covariance matrix are not all equal, and the non-diagonal elements are not all zero. It is necessary to determine from a statistical perspective whether the initial elastic modulus regression residuals satisfy the hypothesis test of variance homogeneity and uncorrelation. The specific steps are to first use the KS method to test whether the residuals under each confining pressure satisfy the normal distribution; then perform an F test on the ratio of the sample variances of the regression residuals of each confining pressure to test whether their variances are homogeneous; finally, test the correlation of the regression residuals under each confining pressure.

[0112] Table 4 shows the test table of whether the initial elastic modulus regression residuals under various confining pressures obey the normal distribution. It can be seen from the table that under the condition of a given significance level of α = 0.05, the regression residuals of various confining pressures obey the normal distribution.

[0113] Table 4 Log under various confining pressures 10 (E i / P a ) Regression residual normal distribution test (significance level α = 0.05)

[0114]

[0115] Table 5 shows the results of the variance homogeneity test of the initial elastic modulus regression residuals under various confining pressures. It can be seen from the table that under the condition of a given significance level of α = 0.05, there are 5 data points that do not meet the variance homogeneity assumption.

[0116] Table 5 Ratio of variance of least squares regression residuals under different confining pressures (significance level α=0.05)

[0117]

[0118] Note: = means accepting the null hypothesis, and × means not accepting the null hypothesis.

[0119] Table 6 shows the independence test table of regression residuals. It can be seen from the table that the residuals under each confining pressure meet the independence hypothesis at a given significance level of α = 0.05.

[0120] Table 6 t value and regression residual independence test table (significance level α=0.05)

[0121]

[0122] Note: = means accepting the null hypothesis, and × means not accepting the null hypothesis.

[0123] Therefore, the initial elastic modulus of the Duncan-Zhang EB model of granite rockfill material in area I of a high dam is regressed by classical least squares. The regression residuals under various confining pressures obey the normal distribution and are independent, but do not meet the homogeneity of variance.

[0124] 2. Test the variance and correlation of bulk modulus regression parameter residuals

[0125] Substituting the bulk modulus results of 12 triaxial tests on granite rockfill materials in Area I into equations (6) and (7), the regression equation and covariance matrix of the bulk modulus are obtained. See Table 2.

[0126] Table 7 Classical least squares regression bulk modulus calculation results

[0127]

[0128] When regressing the bulk modulus of granite rockfill in zone I, the covariance matrix of the residuals under various confining pressures is as follows:

[0129]

[0130] It can be seen that the main diagonal elements of the regression residual covariance matrix are not all equal, and the non-diagonal elements are not all zero. Similarly, to determine whether the bulk modulus regression residuals satisfy the variance homogeneity and irrelevance, hypothesis testing is required, and the steps are the same as before.

[0131] Table 8 shows the test table of whether the bulk modulus regression residuals under various confining pressures obey the normal distribution. It can be seen from the table that when the significance level α=0.05, the regression residuals of various confining pressures obey the normal distribution.

[0132] Table 8 Log under various confining pressures 10 (B / Pa) regression residual normal distribution test (significance level α=0.05)

[0133]

[0134] Table 9 shows the results of the test of homogeneity of variance of bulk modulus regression residuals under various confining pressures. It can be seen from the table that when the significance level α = 0.05, the hypothesis of homogeneity of variance under various confining pressures is accepted.

[0135] Table 9 Ratio of variance of least squares regression bulk modulus residuals under various confining pressures (significance level α=0.05)

[0136]

[0137]

[0138] Note: = means accepting the null hypothesis, and × means not accepting the null hypothesis.

[0139] Table 10 shows the results of the residual independence test of bulk modulus regression under various confining pressures. It can be seen from the table that the residual independence hypothesis is not accepted under the condition of a given significance level of α = 0.05.

[0140] Table 10 t-values ​​of the residuals of the least squares regression bulk modulus under various confining pressures and their independence test table (significance level α=0.05)

[0141]

[0142] Note: = means accepting the null hypothesis, and × means not accepting the null hypothesis.

[0143] Therefore, the classical least squares regression of the Duncan-Zhang EB model bulk modulus of granite rockfill material in area I of a high dam shows that the regression residuals under various confining pressures obey the normal distribution and have the same variance, but do not meet the independence assumption.

[0144] The fourth step is to use the generalized least squares method to eliminate the residual variance and correlation.

[0145] If the covariance matrix V of the regression residual is

[0146] Cov(e)=σ 2 V (8)

[0147] Let K'K = K -1 K=V, multiply both sides of the regression equation y=Xβ+ε by K -1 ,have

[0148] K -1 y=K -1 Xβ+K -1 e (9)

[0149] Let z = K -1 y,B=K -1 X,g=K -1 e, then

[0150] z=Bβ+g (10)

[0151] After the transformation, the g elements have zero mean, constant variance and uncorrelated variances, satisfying the application conditions of least squares estimation.

[0152] An estimate of the solution to the equation is

[0153]

[0154] The covariance matrix of

[0155]

[0156] Step 5: Generalized least squares regression calculation to eliminate variance and correlation of regression residuals

[0157] The regression equation and covariance matrix of the initial elastic modulus were calculated based on the triaxial test results of 12 groups of granite rockfill materials in Area I according to equations (11) and (12), as shown in Tables 11 and Figure 1 ; The regression equation and covariance matrix for calculating the bulk modulus are shown in Table 12 and Figure 2 .

[0158] Table 11 Calculation results of the initial elastic modulus using the generalized least squares method

[0159]

[0160] Table 12 Calculation results of generalized least squares regression bulk modulus

[0161]

[0162] The sixth step is to obtain the mean and variance of the initial elastic modulus and bulk modulus parameters of the Duncan-Zhang EB model based on the regression parameters and their covariance matrix.

[0163] 1. Calculation of initial elastic modulus coefficient K and index n

[0164] Since lg(K)=β0,n=β1,there is

[0165]

[0166] Taylor expansion of the above formula, ignoring the second-order and above infinitesimals, and applying the variance formula of probability theory, we get

[0167]

[0168] In the formula is the variance of the initial elastic modulus regression parameters β0 and β1, is the variance of the initial elastic modulus coefficient K and the exponent n, Cov(K,n) is the covariance of K and n, and Cov(β0,β1) is the covariance of β0 and β1.

[0169] According to the regression parameters β0 and β1 of the triaxial test results of the granite rockfill materials in Area I, the mean, variance and standard deviation of the initial elastic modulus K and n were calculated (see Table 13).

[0170] Table 13 Mean, variance and standard deviation of initial elastic modulus parameters of granite rockfill material test in Area I

[0171]

[0172]

[0173] 2. Bulk modulus coefficient K b Calculation with exponent m

[0174] Because lg(K b )=β0,m=β1,there are

[0175]

[0176] Taylor expansion of the above formula, ignoring the second-order and above infinitesimals, and applying the variance formula of probability theory, we get

[0177]

[0178] In the formula is the variance of the bulk modulus regression parameters β0 and β1, K is the bulk modulus coefficient b The variance of index m, Cov(K b ,m) is K b The covariance with m, Cov(β0,β1) is the covariance of β0 and β1.

[0179] The bulk modulus K is calculated based on the regression bulk modulus parameters β0 and β1 of the triaxial test results of granite rockfill in Area I. b The mean, variance and standard deviation of , m are shown in Table 14.

[0180] Table 14 Mean, variance and standard deviation of bulk modulus parameters Kb and m of granite rockfill materials in area I in triaxial tests

[0181]

[0182] Step 7: Compare the standard deviation of the parameters with those of the traditional grouping

[0183] The means and standard deviations of the initial elastic modulus and bulk modulus parameters of the traditional grouping Duncan-Zhang EB model are shown in Table 15, and the parameters and standard deviations obtained by the classical and generalized least squares methods are also listed in Table 15. It can be seen that in terms of parameter standard deviations, the traditional grouping method is greater than (equal to) the classical least squares method, and the classical least squares method is greater than the generalized least squares method.

[0184] Table 15 Comparison of the mean and standard deviation of initial shear modulus and bulk modulus parameters obtained by different methods

[0185]

[0186] Therefore, the method for arranging the initial elastic modulus and bulk modulus parameters of the Duncan-Zhang EB model proposed in the present invention has the characteristic of minimum standard deviation, thereby improving the estimation accuracy of key parameters of the Duncan-Zhang EB model.

Claims

1. A method for calculating the variance of initial elastic modulus and bulk modulus parameters of the Duncan-Zhang EB model, characterized in that: The steps include: Step 1: Select the regression expression of the initial elastic modulus and bulk modulus of the Duncan-Zhang EB model and sort out the regression data; According to the Duncan-Zhang EB model, the initial elastic modulus expression E i =KP a (σ3 / P a ) n , where K and n are the initial elastic modulus coefficient and index, P a It is expressed as an atmospheric pressure value, σ3 represents the confining pressure; the bulk modulus expression is B = K b P a (σ3 / P a ) m , where K b , m is the bulk modulus coefficient and index; both sides of the formula are divided by P a And taking the logarithm, we get lg(E i / P a )=lg(K)+nlg(σ3 / P a ) (2) lg(B / P a )=lg(K b )+mlg(σ3 / P a ) (3) Let lg(σ3 / P a )=x, when returning to the initial elastic modulus, let lg(E i / P a )=y,lg(K)=β0,n=β1;When regressing the bulk modulus, let lg(B / P a )=y,lg(K b )=β0,m=β1; The regression equations of the two can be expressed as y=β0+β1x+e (4) Where, e is the regression residual; By conducting several triaxial tests on a soil material, several pairs of initial elastic modulus and several pairs of bulk modulus regression data equal to the number of tests can be determined, which are [lg(σ3 / P a )lg(E i / P a )], [lg(σ3 / P a )lg(B i / P a )]; Step 2: Least squares regression and homogeneity of variance and correlation test of its residuals The matrix expression of the regression equation is y6Xβ+e,E(e)0,Cov(e)0σ 2 I (5) In the formula, X is the independent variable, y is the dependent variable; e is the regression residual term, which represents the influence of other factors on y except X and the experimental measurement error; y=[y1 y2…y i …y n ] T ,y i =lg(E i / P a ) or y i =lg(B i / P a ), T is the matrix transpose; β = [β0β1] T , β0 and β1 are unknown parameters to be estimated; E i , B i are the confining pressure, initial elastic modulus and bulk modulus of the ith test respectively; In formula (5), formula 1 is the regression equation, and formulas 2 and 3 are the prerequisites for the application of the least squares method; The n groups of observation data (x i ,y i ) into the equation, we get an estimate of the regression parameter β The covariance matrix of the regression parameters is Where V is the covariance matrix of the residual vector under different confining pressures, σ 2 is the variance of the entire residual vector, i.e. Cov(e) = σ 2 V; if the regression residual satisfies the third formula of formula (5), then V = I is the unit matrix; With the regression parameters of the initial elastic modulus and bulk modulus, the regression prediction value of y can be calculated; the difference between the regression prediction value and the measured value under various levels of confining pressure is the regression residual e i (i=1,2,…,k), k represents the confining pressure level; calculate the covariance matrix of the regression residual vector under each confining pressure, the formula is as follows In the formula, D(e1) and D(e k ) is the variance of the regression residual vector under the first and kth level confining pressure, Cov(e1,e k ) is the covariance of the regression residual vector under the 1st and kth level confining pressure; (1) KS test of normal distribution of regression residuals under various confining pressures The null hypothesis and alternative hypothesis are H0: F(x) = F0(x), H1: F(x) ≠ F0(x), where F0(x) is the normal distribution function; the test statistic is D n =sup|F n (x)-F0(x)|,F n (x) is the sample distribution function; at the significance level α, Probability Find the quantile t α , then the rejection region is [t α , +∞); Calculate the test statistic based on the residuals under each confining pressure, determine its maximum value, and judge whether to accept the null hypothesis; (2) Homogeneity test of regression residuals under different confining pressures The F test is used to determine whether the residuals have homogeneity of variance under different confining pressures. The test statistic is They represent the sample variance of the residual vector under two confining pressures, n1 and n2 are the corresponding sample numbers; the null hypothesis and the alternative hypothesis are H0: H1: Not all equal, The residual variance under the i-th level of confining pressure; at the significance level α, make P(F≤F 1-α / 2 (n1-1,n2-1))=α / 2, P(F≥F α / 2 (n1-1,n2-1)) = α / 2; calculate the variance of the residual under each confining pressure With the statistic F, look up the table to determine the critical value and decide whether to accept the null hypothesis; (3) Independence test of regression residuals under different confining pressures The independence test can be determined by testing the correlation coefficient ρ; the null hypothesis and alternative hypothesis are H0: ρ = 0, H1: ρ ≠ 0, and the test statistic is It follows a t distribution with n-2 degrees of freedom; X i , Y i , are the residuals and their means under two confining pressures respectively; given a significance level α, let P(|t|≥t α / 2 (n-2)) = α, if the observed value of t satisfies |t| ≥ t α / 2 (n-2), the null hypothesis H0 is rejected; the point estimate R and t of the correlation coefficient are calculated according to the regression residuals under each confining pressure to determine whether to accept the null hypothesis; If the regression residuals have homogeneous variance and are uncorrelated, skip the third to fourth steps, and use the regression parameter estimates obtained by the least squares method and their covariance matrix to obtain the mean and variance of the initial elastic modulus and bulk modulus parameters; if the regression residuals have heterogeneous variance or correlation, or both, perform the third step and re-estimate the regression parameters and covariance matrix; Step 3: Generalized least squares regression to eliminate variance and correlation The covariance matrix V of the regression residuals is Cov(e)=σ 2 V (8) Let K T K=K -1 K=V, multiply both sides of the regression equation y=Xβ+ε by K -1 ,have K -1 y=K -1 Xβ+K -1 e Let z = K -1 y,B=K -1 X,g=K -1 e, then z=Bβ+g (10) After the transformation, the g elements have zero mean, constant variance and uncorrelated variances, satisfying the application conditions of least squares estimation; An estimate of the solution to the equation is The covariance matrix of The fourth step is to obtain the mean and covariance matrix of the initial elastic modulus and bulk modulus parameters based on the regression parameters and their covariance matrix. (1) Calculation of initial elastic modulus coefficient and index Since lg(K)=β0,n=β1,there is Taylor expansion of the above formula, ignoring the second-order and above infinitesimals, and applying the variance formula of probability theory, we get In the formula is the variance of the initial elastic modulus regression parameters β0 and β1, is the variance of the initial elastic modulus coefficient K and the exponent n, Cov(K,n) is the covariance of K and n, and Cov(β0,β1) is the covariance of β0 and β1; (2) Calculation of bulk modulus coefficient and index Because lg(K b )=β0,m=β1,there are Taylor expansion of the above formula, ignoring the second-order and above infinitesimals, and applying the variance formula of probability theory, we get In the formula, is the variance of the bulk modulus regression parameters β0 and β1, K is the bulk modulus coefficient b The variance of index m, Cov(K b ,m) is K b The covariance with m, Cov(β0,β1) is the covariance of β0 and β1.

Images