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
Creep settlement is a main research issue for highway deterioration and other high-fill projects [1–5]. The analysis of embankment creep settlements is the basis for the correct understanding and evaluation of the subgrade stability. In consequence, more research should be carried out into the calculation of creep settlement of embankment. Existing methods related to creep settlement research can be divided into three groups: (1) Empirical models, for example, the Asaoka method [6, 7], the hyperbolic method [8], the parabola method [9], the exponential method, and in situ tests [10, 11], in those methods, the monitored creep settlement of the embankment after construction was adopted to predict the creep settlement in the future. (2) Constitutive models: Bjerrum [12] pointed out that the influence of creep on clays was equivalent to make the clay overconsolidated, which was obtained through a large number of consolidation tests and engineering settlement data analysis, and Yao et al. [13] established the unified hardening (UH) model (a model for over-consolidated clays), which considered the time effects and analyzed the influence of creep on over-consolidated clays. Hashiguchi [14] put forward the concept of a subloading yield surface. Asaoka et al. [15] suggested the concept of superloading and subloading yield surfaces for over-consolidated soils. And Wang et al. [16] proposed an advanced elastic-viscoplastic constitutive model to predict time-dependent behavior of the over-consolidated clay based on the subloading surface model and relative overstress relation. (3) Numerical methods: Loganathan et al. [17] put forward a calculation method named “field deformation analysis” (FDA), which separates and quantifies settlement components. Li et al. [18] used the modified secant modulus method to calculate settlements, an approach that is characterized by the fact that it is not influenced by the initial void ratio. But the empirical models pay less attention on the creep mechanism of clays, and most of the constitutive models are elastoplastic constitutive models, which cannot describe the time-dependent deformation characteristic of over-consolidated clays. Experimental results indicate that the creep deformation of high-filled embankment is affected by both time and stress history. The rheological model can reflect the time-dependent stress-strain behavior of clays, and in order to simulate the layered filling process of the high-filled embankment during construction, the stress applied on the rheological model is according to the loading curve, so the rheological model used in the creep settlement calculation of the embankment can not only reflect the time-dependent stress-strain behavior but also the stress history of the embankment. The application of some relatively simple rheological models has been studied in some cases [19–22]. Morro [23] concluded that a spring-dashpot model (i.e., the Maxwell model in parallel with a dashpot) showed a constant strain rate during the secondary stage. Justo and Durand [24] obtained settlement expressions with the standard linear solid rheological model for one-dimensional settlements of an embankment during and after construction. In this paper, the Poynting–Thomson model (a linear spring in parallel with a Maxwell model) was applied to represent the time-dependent deformation of soil, and the theory of the hereditary integral used to calculate the cumulative strain is explained in detail, which reflects the strain at any time factor and depends upon the entire stress history [25, 26].
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.
(a)
(b)
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).