Compaction Uniformity Evaluation of Subgrade in Highway Based on Principal Components Analysis and Back Propagation Neural Networks

Abstract

This paper proposes a comprehensive method for the compaction uniformity evaluation of subgrade in highways based on the principle components analysis and BP neural network. A field test on resilient and Young’s moduli of subgrade during compaction is performed on Zun-Qin highway. The moduli representing the compaction uniformity are the key factors in the principal component analysis, and the components are used as input in Back Propagation (BP) neural networks. The degree of variation and synthesis score of moduli in three subgrade sections are discussed, and the results show that the comprehensive method has a good performance in evaluating the compaction uniformity of the subgrade. The insight from this study provides a novel evaluation method and incites a better understanding of the compaction uniformity of subgrade in highways.

1. Introduction

With the rapid development of highways, the compaction quality of subgrade is becoming more and more important in practice. Some of the subgrade during operation in China has occurred severe cracking and large uneven settlement since the compaction of the subgrade is not uniform during construction [1,2], as shown in Figure 1. The riding comfort is thus significantly influenced and the driving speed is limited in these roadways. Therefore, it is essential to evaluate the compaction uniformity of subgrade in highways.

In this paper, the strength is used to characterize the quality of subgrade compaction so as to study the evaluation of the uniformity of subgrade compaction quality. There are some conventional compaction quality detection methods, for instance, the sand cone test [3], the dynamic cone penetration test (DCP) [4], the nuclear gauge test [5,6], and the California bearing ratio test (CBR) [7]. However, these quality detection methods are time-consuming and they rely on very limited testing points. The portable falling weight deflectometer (PFWD) and GeoGauge are two advanced compaction quality testing methods for subgrade during construction [8,9]. Compared with traditional testing methods, the two new methods can achieve non-destructive, fast and muti-points detection.

Although the compaction quality of the subgrade can be controlled, the uniformity of subgrade during compaction on a relatively large scale still cannot be evaluated. There is very limited literature and specification regarding the compaction uniformity evaluation of subgrade in highways. With the development of computational science, the principal components analysis and Back Propagation (BP) neural networks provide novel solutions for evaluating the compaction uniformity of subgrade [10,11,12]. The principal components analysis is a multivariate statistical method, which uses the idea of dimensionality reduction to convert multiple indicators into several comprehensive indicators without losing original information. The principal components are independent of each other, indicating that they can exhibit superior performance than the original parameters [13]. For the compaction quality evaluation of subgrade, the original parameters are normally the resilient and Young’s moduli of subgrade obtained from the PFWD and Geogauge tests and they are dependent on each other. While the principal components are independent and derived from the moduli, and they become the evaluation parameters in the BP neural networks. The BP neural networks have the advantages of simple structure, multiple training algorithms, good manoeuvrability, and so on, making this method can well evaluate the compaction uniformity of subgrade in highways.

Considering the compaction uniformity of subgrade is important in practice, this paper proposes a novel method to evaluate the compaction uniformity of subgrade during construction based on the principal components analysis and BP neural networks. The field test on resilient and Young’s moduli of the subgrade is first carried out, and the compaction uniformity evaluation methods are then introduced. The evaluation results are finally discussed. The insight from this study provides a novel evaluation method and incites a better understanding of the compaction uniformity of subgrade in highways.

At present, there are few pieces of research on the detection and evaluation of subgrade compaction uniformity using single point detection index. There are no more comprehensive regulations and norms for the detection, evaluation and control of freeway subgrade compaction uniformity at home and abroad to guide the construction of freeway subgrade. It is easy to cause uneven construction quality of subgrade and affect the durability of subgrade and pavement structure. In the construction process, in order to speed up the construction process under the premise of ensuring the construction quality, it is necessary to make a rapid detection and objective and reliable overall evaluation of the compaction uniformity of the subgrade, so as to provide great help for the site construction. Therefore, it is necessary to carry out research on the evaluation and control technology of uneven compaction of highway subgrade. The flowchart of the research method of the paper is shown in Figure 2:

2. Field Test on Moduli of Subgrade

The field test on the compaction of the subgrade is carried out on the Zun-Qin highway, which is constructed with six lanes in two directions and has a design speed of 100 km/h. The subgrade is filled with soils and stones. Since the filled height is different from the normal section and transition section of the subgrade, the tests are carried out in three section areas, as shown in Figure 3. Section 1 is the normal subgrade section, and Section 2 is the transition section on the east side of the culvert. Section 3 is the transition section on the west side of the culvert.

The length of the three testing sections of the subgrade is 40 m and the width is 33.5 m. The testing points are distributed uniformly in Section 1 with every 3 m in the longitudinal direction and every 3 m in the wide direction. For sections 2 and 3, the longitudinal space is 1 m and the width is 3 m, as shown in Figure 4. There are 200 testing points. The resilient and Young’s moduli at every testing point are tested using the PFWD and GeoGauge after the compaction of the subgrade.

3. Principal Components Analysis

Since the inevitable repetition of information between the original indicators, i.e., moduli of subgrade, and the original indicators are not conducive to direct calculation, the method of principal component analysis is used to screen the original indicators. Let the data be an N×M-dimensional matrix, where N is the number of samples, and M is the number of evaluation indicators.

3.1. Standardize the Data

The original variable data is processed in a positive and standardized way. Eliminate possible effects due to unit differences. The standardized formula is:

$$X_I^{*}=\frac{X_i-E\left(X_i\right)}{\sqrt{Var\left(X_i\right)}},i=1,2,\dots ,$$

(1)

3.2. Calculate the Sample Correlation Matrix

The correlation coefficient matrix R is calculated according to the standardized data matrix:

$$R={\left(r_{ij}\right)}_{p\times p}=\left[\begin{array}{cccc}{r}_{11}& r_{12}& \cdots & r_{1p}\\ r_{21}& r_{22}& \cdots & r_{2p}\\ ⋮& ⋮& \cdots & ⋮\\ r_{p1}& r_{p2}& \cdots & r_{pp}\end{array}\right]$$

(2)

$r_{ij}\left(i,j=1,2,\dots ,p\right)$ is the correlation coefficient between $X_i$ of the original variable $X_j$, and its calculation formula is:

$$r_{ij}=\frac{{{\displaystyle \sum }}_{k=1}^p(X_{ki}-{\overline{X}}_i)\left(X_{kj}-{\overline{X}}_j\right)}{\sqrt{{{\displaystyle \sum }}_{k=1}^p{(X_{ki}-{\overline{X}}_i)}^2{{\displaystyle \sum }}_{k=1}^p{(X_{kj}-{\overline{X}}_j)}^2}}$$

(3)

3.3. Calculate Eigenvalues and Eigenvectors

Calculate the eigenvalues ${\lambda }_1$, ${\lambda }_2$, …, ${\lambda }_m$(${\lambda }_i\ge 0$) and the corresponding eigenvectors $c_1$, $c_2$, …, $c_m$.

Through the above steps, calculating $y_i={\mu }_i^{\prime }x$ side obtains the main component. The p eigenvalues and corresponding eigenvectors are the initial solution of factor factorization.

3.4. Calculate the Contribution Rate of the Principal Components

The number of principal components is determined by the eigenvalue ${\lambda }_i$ To interpret the analysis results, multiple principal components need to be taken, and there is no clear index. The principal component is usually taken when the eigenvalue ${\lambda }_i$ is greater than 1, or when the cumulative contribution rate of eigenvalue reaches 80%, or when the statistical test level of eigenvalue p < 0.05.

$$v_i=\frac{{\lambda }_i}{{\displaystyle \sum _{i=1}^m{\lambda }_i}}$$

(4)

3.5. Filter Principal Components

Arrange the contribution rates of each principal component from high to low. When the sum of the contribution rates of the current principal components (cumulative contribution rate) meets the requirements of information reflection accuracy, take the principal components ($z_1$, $z_2$, …, $z_s$) as evaluation indicators in the next stage.

3.6. Calculation of Principal Component Factor Score

The eigenvector matrix is respectively multiplied by the standardized variables of the original variables, namely, the function expression of the principal components is as follows: for each sample, the values of the first m principal components are calculated, i.e.,

$$\begin{gathered} Y\left(i,j\right)={\mu }_{g1}Z{X}_{i1}+{\mu }_{g2}Z{X}_{i2}+\cdots +{\mu }_{gp}Z{X}_{ip},i=1,2,\cdots ,n \\ g=1,2,\cdots ,m \end{gathered}$$

(5)

4. Back Propagation Neural Networks

4.1. Design of Network Structure

According to the characteristics of the evaluation object, a three-layered (input layer, middle layer, and output layer) neural network model is adopted. The input is the result of the principal component analysis, and the output is the synthesis score.

It is necessary to follow the five principles of comprehensiveness, scientificity, operability, comparability and relative independence to establish the evaluation index system of highway subgrade compaction. The principle of comprehensiveness requires to consider the factors that affect the compaction uniformity of the highway subgrade on the whole. Scientific principle means that objective facts must be followed and the selected index can scientifically reflect the uniformity of subgrade compaction. Weaken the influence of subjective evaluation factors. Operability means that the evaluation index of compaction uniformity of highway subgrade is measured by quantitative or qualitative methods.

4.2. Determination of Parameters of Each Layer

The input of the input layer is $A_k=\left(\begin{array}{cccc}{a}_1& a_2& \dots & a_s\end{array}\right)$. The desired output of the output layer is $Y_k=\left(\begin{array}{cccc}{y}_1& y_2& \dots & y_q\end{array}\right)$. The input of the middle layer is $S_k=\left(\begin{array}{cccc}{s}_1& s_2& \dots & s_p\end{array}\right)$. The output of the middle layer is $B_k=\left(\begin{array}{cccc}{b}_1& b_2& \dots & b_p\end{array}\right)$. The input of the output layer is $L_k=\left(\begin{array}{cccc}{l}_1& l_2& \dots & l_q\end{array}\right)$. The actual output of the output layer is $C_k=\left(\begin{array}{cccc}{c}_1& c_2& \dots & c_q\end{array}\right)$. The threshold of neurons in the middle layer is ${\theta }_j$, (j = 1, 2, …, p). The threshold of output layer is ${\gamma }_t$, (t = 1, 2, …, q). The connection weight of input layer and middle layer is $W_{ij}$, (i = 1, 2, …, s; j = 1, 2, …, p). The connection weight of the middle layer and the output layer is $V_{jt}$, (j = 1, 2, …, p; t = 1, 2, …, q).

Determine the BP-neural network architecture. The network structure includes input, output and the number of hidden nodes. The number of layers determines the complexity, the ability to deal with problems and the convergence speed of the neural network. The paper selects six statistical analysis variables of the detection index, so the number of neurons in the input layer is 6, that is, the number of indicators for the evaluation index system. The number of neurons in the hidden layer is selected empirically and can be set to 75% of the number of nodes in the input layer in general. The number of neurons in the output layer is determined by the evaluation result of the expected output.

4.3. Processing Steps of the BP Neural Networks

A row of field detection data along the subgrade cross-section is taken as a group, and ten sets of data from the top of the road bed on the back of the platform and the top of the road bed in the general subgrade section are selected, with a total of 20 sets of data. The sample data are detected by the drop hammer bending meter and the stiffness meter respectively.

Step 1: Initialize the $\left\{W_{ij}\right\}$$\left\{V_{jt}\right\}$$\left\{{\theta }_j\right\}$$\left\{{\gamma }_t\right\}$, and assign the random value in the range of (−1, 1).

Step 2: The learning sample mode is randomly selected to operate ($A_k,Y_k$).

Step 3: Calculate the input and output of the middle layer unit,

$$s_j={\displaystyle \sum _{i=1}^s{w}_{ij}\times a_i-{\theta }_j} \mathrm{and} b_f=f(s_j)$$

(6)

where $f(x)=\frac{1}{1+e^{-x}}$.

Step 4: Calculate the input and output of the output layer unit,

$$l_t={\displaystyle \sum _{j=1}^p{v}_{jt}\times b_j-\gamma } (t=1,2,\dots q), c_t=f(l_t^{})$$

(7)

Step 5: Calculate the general error of each unit of the output layer,

$$d_t=(y_t-c_t)f^{\prime }(l_t)$$

(8)

Step 6: Calculate the general error of each unit of the middle layer,

$$e_t={\displaystyle \sum _{t=1}^q\left(\begin{array}{cc}{d}_t& v_{jt}\end{array}\right)f^{\prime }(s_j)}$$

(9)

Step 7: Revision of weight. Define the $\alpha ,\beta$ from (0, 1), and if the error still does not reach the accuracy requirements after some cycles of calculation, change the $\alpha ,\beta$ from small values.

Step 8: Calculate the error.

$$E_k=\frac{1}{2}{\displaystyle \sum _{t=1}^q{(y_t-c_t)}^2}$$

(10)

When $E_k={\displaystyle \sum _{k=1}^m{E}_k}<\epsilon$ ($\epsilon$ is predetermined value), the training is finished. If not, then go back to step 2.

Step 9: The model is determined after determining each weight.

An evaluation model of subgrade compaction uniformity based on principal component analysis and BP-neural network is established. This step includes performing principal component analysis on the test data, obtaining the quantitative values of the indicators and inputting them into the specific BP-neural network model. The uniformity degree is specified as uniform, mild uneven, moderate uneven, and severe uneven, and the values of 0.1, 0.3, 0.6, 0.9 are respectively assigned as the initial weights of the network, until the fitting value of the learning sample is consistent with the actual observation value. The initial value is relative to the fitting value, and the initial value is the result of the principal component analysis comprehensive score. The fitting value is obtained by fitting the initial value, and the effect of modifying the initial value is achieved. The output from BP neural network model is the synthesis score, which represents the uniformity level of the subgrade.

5. Results

Descriptive statistical analysis is performed on the normalized principal component analysis, and the variation of moduli is shown in

Table 1. Variation of moduli.

Indicator Parameters	Mean Value	Standard Deviation	Coefficient of Variation	Degree of Variation
Ea	0.25	0.16	0.66	medium
Eb	0.48	0.28	0.59	medium
Ec	0.26	0.08	0.52	medium
Ed	0.15	0.06	0.42	medium
Ee	0.39	0.24	0.61	medium
Ef	0.45	0.29	0.66	medium

Note: Ea is the resilient modulus of subgrade in Section 1; Eb is Young’s modulus of subgrade in Section 1; Ec is the resilient modulus of subgrade in Section 2; Ed is Young’s modulus of subgrade in Section 2; Ee is the resilient modulus of subgrade in Section 3; Ef is Young’s modulus of subgrade in Section 3.

. Since the coefficient of variation is distributed in the range of 0.42–0.66, the degree of variation of all moduli is medium.

The uniformity of the three sections of the subgrade is shown in Figure 5, Figure 6 and Figure 7. Figure 5, Figure 6 and Figure 7 show the results of the deviation rate, which is used to represent the degree of influence of the uniformity of a single cross-section on the overall compaction uniformity of the subgrade. The greater the deviation of the measured value from the standard value, the greater the deviation rate. It can be seen that most of the measured values are close to the distance of the standard values. In the general subgrade section, due to the existence of half-filled and half-excavated subgrade, the measured value at the filling and excavation junction in Figure 5 is far from the standard value, which is because these test points are located at the junction of filling and excavation. Figure 6 and Figure 7 show the uniformity results of the embankment on both sides of the culvert and the general subgrade section. It can be seen that there is a trend of large fluctuation on both sides and relatively stable in the middle. The evenness of the abutment and the general subgrade section is similar, but there are still different degrees of deviation between the single cross-section and the whole subgrade due to different subgrade filling processes in the road and bridge transition section. The overall uniformity of the ITD is good.

The predicted values and tested values in three sections are shown in Figure 8, Figure 9 and Figure 10. The predicted values are derived from the principal components analysis and B-P neural networks, and these values are in good agreement with the tested values, indicating that the comprehensive method is suitable for the evaluation of compaction uniformity of subgrade in highways.

According to the evaluation results of subgrade compaction uniformity, it can be seen that under the condition of the same other factors, there is uneven subgrade compaction in different subgrade sections, and the uneven degree of different subgrade sections is different, and the uniformity degree of the abutment is better than that of the general subgrade section, which is mainly caused by the difference of subgrade packing and filling process between the abutment and general subgrade section. There is a sudden change of compaction mass in the transitional section of the road and bridge, which results in the worst uniformity of the transitional section. Due to the existence of uneven subgrade compaction and the difference of uneven degrees, under the influence of traffic load and dead weight, the expressway will produce uneven subgrade settlement, bridge jump, pavement cracks and other diseases. Therefore, it is necessary to study, evaluate and control the compaction uniformity of subgrade.

6. Conclusions

This paper proposes a comprehensive method for evaluating the compaction uniformity of subgrade based on principal components analysis and BP neural networks. The following conclusions can be drawn:

(1)

The key factors in principal components analysis and BP neural networks contain prominent information on the evaluation of uniformity by dealing with the resilient and Young’s moduli of subgrade. The weight of different evaluation indicators is also considered, making the evaluation results more accurate, reasonable and credible.

(2)

Through statistical analysis of the field test data, the indicators that can reflect the uniformity of the data are obtained, and these indicators and the uniformity of the subgrade compaction are regarded as variables. However, there are usually complex causal relationships among the six indicators such as mean, standard deviation, mean SE, kurtosis coefficient, skewness coefficient and coefficient of variation, which are largely manifested by interrelated influencing factors, and the relationship between the variables is usually nonlinear. Bp-neural network determines the relationship between variables through an autonomous learning process, and can simultaneously consider the influence of multiple variables on the compaction uniformity of the subbase.

(3)

The subgrade in the normal section exhibits relatively poor uniformity due to the filling and excavation, and the subgrade in the transition section shows better compaction uniformity during construction.

(4)

The comprehensive method considers the rationality of selecting evaluation factors and their influence weights on the compaction uniformity of subgrade, and it is suitable for the evaluation of compaction uniformity of subgrade in highways.