#### Abstract

In order to predict the creep settlement of high-fill embankments, the time-dependent viscoelastic model of Poynting–Thomson (the standard linear solid) has been chosen to represent the creep behavior of soils. In the present study, the hereditary integral was applied to calculate the strain while the load increase is varied with time. Calculation expressions of the creep settlement of an embankment during and after construction were obtained under one-dimensional compression conditions. Using this approach, the three parameters of every layer can be determined and adjusted to accommodate in situ monitoring data. The calculated results agreed well with those from the field, which imply that the method proposed in this paper can give a precise prediction of creep settlement of high-fill embankments.

#### 1. Introduction

As an illustration of the creep settlement performance of a high-filled embankment, the case of the dam-like embankment of the Lanzhou–Yongjing highway was investigated using the proposed new method.

#### 2. Theory

For the model of Poynting–Thomson (Figure 1),where

The constitutive equation of the Maxwell model in Figure 1 iswhere

Substituting equations (5) and (6) into equation (4) yields

The calculation expression of is obtained from equation (7):and substituting the expressions of and equation (3) into equation (1) yields

The constitutive equation of the Poynting–Thomson model is

In a creep test, the pressure maintains a constant value when ; equation (10) becomes

The Poynting–Thomson model (Figure 1) first produces the instantaneous elastic deformation under the constant pressure . As time increases, the deformation continues to increase such that when , , and the solution of equation (11) is

Which otherwise can be written aswhere

R is called the creep factor.

is called the time factor.

And equation (13) can also be written aswhere

The mechanical meaning of is the strain under unit force, which can easily be measured in practical tests. is called the creep compliance. After applying the stress when , the strain of the model is . Then, when another stress is applied when , there is another strain , which is added to when , which is proportional to . This strain is also a function of T, but only adds when and is still dependent upon the same creep compliance:

If is applied when , then should increase . The effect of each on can be superposed in this way, which is called the Boltzmann superposition principle [27]. If changes when , the relationship between applied stress and the time factor will be replaced by a stepped polyline, as shown in Figure 2. When , the sum becomes integral:

If the applied load varies with time, according to Figure 3, because , the strain at time factor T will be given by the hereditary integral:

The solution of the hereditary integral iswhere

The strain from T to (Figure 3) can be described as

is obtained from equation (21):

And when , the strain from to is (Figure 3)

From equation (21), it is supposed that the stress maintains a constant value when (Figure 3):

#### 3. Application to a High-Filled Embankment

If it is assumed that the height of an embankment varies with time according to Figure 4, the stress at any point depends on the height of fill above it.

##### 3.1. Settlement of the Embankment during Construction
###### 3.1.1. Degree of Settlement of the Embankment at Height during Construction

The degree of settlement of the embankment at height during construction can be described as follows:

can be obtained from equation (21):where

The integrals listed above must be calculated by numerical methods. In order to do this, the loading curve is divided into N intervals; from up to , the curve is divided into intervals, and from up to , the curve is divided into intervals.

Substituting equation (28) into equation (27), the settlement of a point at height during construction is obtained fromwhere

If it is assumed that the height of the embankment increases linearly with time, and in order to simplify the deduction process, we assumed that the slope of the loading curve is 1, and then equations (33) and (35) will become

Substituting equations (36) and (37) into equation (31) gives

###### 3.1.2. Settlement of a Finite Layer () during Construction

The strain from time factor up to is

Substituting by and equation (39) into equation (21) gives

The compression of a finite layer is

Substituting equation (40) into equation (41) giveswhere

##### 3.2. Settlement of the Embankment after Construction
###### 3.2.1. Settlement of an Embankment at Height after Construction

The settlement of an embankment at height after construction can be described as follows:

Substituting equation (26) into equation (48) giveswhere the value of is equal to equation (32).

At the top of the embankment,

If the height of the embankment varies linearly with time, with a slope of 1, then equations (49) and (51) will become

###### 3.2.2. Settlement of a Finite Layer after Construction

The strain of a finite layer after construction is

The settlement of a finite layer after construction iswherewhere the value of is equal to equation (44).

##### 3.3. Parameter Determination

The three parameters of the model are the elastic modulus , the creep ratio R, and the time factor .

If settlement is measured from different heights of the embankment, from equation (55), the time factor is obtained fromwhere is the time increment after construction.

The solution of equation (57) with different time increments will permit an adjustment for .

Substituting equation (55) into equation (42), the values of from different layers are obtained:

Substituting the value of and into equation (41) gives

#### 4. Practical Verification

For the Lanzhou–Yongjing highway, a section from 24 + 160 km to 25 + 838 km was completed with a high-fill subgrade. The settlement meters were set up from 24 + 400 km to 24 + 405 km, which was semifilling and semiexcavating subgrades (Figure 5(a)). On the right side of the subgrade, a steep slope was there. However, flood drainage was on the left side of the subgrade. The materials excavated from the slope were used to fill the subgrade. The maximum fill height of this section was 18.3 m. From 0 to 15 m height, the subgrade was filled with loess. From 15–16.1 m height, it was filled with a sandy pebble. From 16.1–18.3 m height, it was filled with gravel. The physical and mechanical properties of the subsoils are shown in Table 1. Single-point settlement meters were adopted to monitor the long-term settlement of the subgrade [29]. It can be seen that for both the road shoulder and driveway, 6 layers of settlement meters were embedded. The layout of settlement meters is shown in Figure 5, which were 3 m, 6 m, 9 m, 12 m, 15 m, and 18.3 m. And the embedment depth is reflected in Figure 5(b). Almost two-year settlement data are obtained from the embankment after construction. And the field-measurement settlement data during construction is obtained from [28], which monitored the settlement of each layer of the filling during the construction, and the relationship between filling height and settlement of the filling during construction is represented in Figure 6. And from the monitored data during construction in Figure 6, firstly, the value of is obtained from equation (57), secondly, substituting the value of and all the monitored parameters into equation (58) to calculate the value of , thirdly, the values of and are substituted into equation (59) to get the value of , and finally, the creep settlement of every layer of the embankment is calculated from equations (42) and (55), respectively, which are presented in Figures 7 and 8. From the field-measured data, the , , and parameters for every layer were established as indicated in Table 2.

Settlement values during construction from field measurement and settlement values calculated from equation (31) are presented in Figure 6, which is basically consistent with the experimental finding [30].

The measured and calculated settlements after construction obtained with equation (49) are presented in Figure 7.

The agreement between the measured and calculated settlements is excellent, although the measured settlement values are more irregular because this area is in a soil region that is seasonally frozen, and during the winter, soil freezing leads to the ground heave [31, 32].

#### 5. Conclusions

To describe the time-dependent behavior of clay inside a high-filled embankment, the viscoelastic model of Poynting–Thomson was applied in this study in order to calculate the creep settlement of high-filled embankments. And the stress history is reflected by the hereditary integral. One-dimensional compression was assumed in this investigation. The settlement values at a point and of a finite layer during construction and after construction are obtained. The three parameters of this model, , , and , can be adjusted for the in situ monitoring data from every layer. By comparison of calculated results obtained by the proposed method with the field monitoring data, it was concluded that the proposed method can give a precise prediction of the creep settlement of high-filled embankments. In the future work, the other viscoelastic models which can better reflect the time-dependent creep behavior of soils will apply to amplify understanding of settlement behavior in more complex soil conditions.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest.

#### Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (Grant no. 51568044).