The resilient modulus of subgrade is a design parameter of the pavement structure, which is significantly affected by the overlying load and traffic load. It is important to calculate the equivalent resilient modulus of the top surface of subgrade based on the nonuniform distribution of resilient modulus in subgrade. This paper takes the fully weathered granite soil as the research object. Firstly, the soil density of different layers of the subgrade structure is calculated by the degree of compaction of different subgrade layers. Secondly, the overlying load of each point in the subgrade is determined based on the quality of subgrade. Thirdly, the subprogram of the finite element software is compiled and redeveloped based on the elastic constitutive model, and the calculation method for the resilient modulus of each point in the subgrade under the traffic load is proposed when the convergence criterion is set up. Finally, according to the deflection equivalence of the elastic double layer and elastic half-space, the calculation and control methods for equivalent resilient modulus of the top surface of subgrade under nonuniform stress distribution are put forward, and the field verification tests are carried out. The results show that the error range between numerical calculation and field measurement of equivalent resilient modulus of subgrade is 10%. It shows that the calculation method for equivalent resilient modulus of subgrade proposed in this study is reasonable and effective. The equivalent resilient modulus of subgrade decreases with the increase of traffic load. And the resilient modulus of subgrade soil increases with the increase of subgrade depth. The resilient modulus of subgrade soil has a significant impact on the equivalent resilient modulus of subgrade after the overlaying gravel layer. The equivalent resilient modulus of the subgrade with the gravel layer increases with the increase of the resilient modulus of the subgrade soil.
The resilient modulus of all pavement layers, including subgrades, is one of the primary material properties as it is used in mechanistic pavement thickness design [
Subgrade soil mainly withstands the impact of traffic load transferred from the road surface. As traffic load is transferred from top to bottom, stress decreases gradually [
The resilient modulus of subgrade is an important parameter to characterize the performance of subgrade [
The value of the
To sum up, the resilient modulus of subgrade soil strongly depends on stress; however, the spatial distribution of the resilient modulus inside the subgrade is nonuniform because of the nonuniform distribution of static load and dynamic stress inside the subgrade; few researchers focus their research on this topic, which leads to the lack of a calculation method for the equivalent resilient modulus of subgrade based on nonuniform spatial distribution of stress, let alone a regulation and control method in this regard. Therefore, taking into account the nonuniform stress field in the subgrade, it is essential for the pavement design to calculate accurately the equivalent resilient modulus of the subgrade. Meanwhile, it is reasonable that increasing the stiffness of the roadbed can effectively control the deformation and fatigue life of the pavement structure.
The dynamic load and overlying load on the subgrade soil vary among different depths within the subgrade work area. Therefore, the state of confining pressure (overlying load) and the state of dynamic load (traffic load) vary among different points inside the subgrade. These two stress states exert impacts on the resilient modulus of the subgrade soil, which is also the main reason for the difference of resilient modulus among various points inside the subgrade soil and results in the nonuniform distribution of the resilient modulus inside the subgrade soil. Before the calculation of the equivalent resilient modulus of subgrade, it is necessary to first accurately calculate the resilient modulus of various points inside the subgrade under the nonuniform distribution state and then calculate the equivalent resilient modulus of the top surface of subgrade.
Before the determination of the overlying load at various points of the subgrade, it is necessary to calculate the soil density at different layers of the subgrade. Generally speaking, subgrade soil is filled at certain compactness while the optimum water content of the soil is reached [
Roadbed:
Upper embankment:
Lower embankment:
To determine the overlying load on various nodes inside the subgrade is a relatively complex process. Formulas for calculating the overlying load on various nodes of the roadbed, upper embankment, and lower embankment are as follows.
Roadbed:
Upper embankment:
Lower embankment:
A subgrade structure model is created through the ABAQUS software. Subgrade soil is subject to the isotropic linearity-elasticity constitutive relation, with the stiffness matrix shown as
As the subgrade withstands traffic load transferred to the road surface, the dynamic load varies among soil of different depths, resulting in the nonuniform distribution of the modulus field inside the subgrade. For finite element calculation, each grid cell can be regarded as an elastomer; that is to say, the resilient modulus of each grid cell is a fixed value. To determine the resilient modulus of each grid cell, it is necessary to use the UMAT program for secondary development in ABAQUS. The specific method is given below.
The initial modulus (taken as 60 MPa herein) is set for each grid cell in the subgrade structure, the standard axle load is given, and then the first trial calculation is undertaken, whereby a dynamic stress
Many researches have studied the nonlinear convergence of the elastic modulus. The tangent stiffness method was applied to solve the nonlinear stress-strain behaviour by updating the tangent stiffness at each iteration until the stress increment converges [
Flow chart for the calculation of the nonuniform modulus field.
The modulus of each node inside the subgrade varies under different dynamic stresses. When the error in the modulus between two calculations for a definite node is within 0.5%, and the accumulated error in the modulus between two calculations for all the nodes of the subgrade structure is within 5%, the accuracy of the modulus of each grid cell inside the subgrade is considered to meet the project requirements. The finite element calculation is concluded at this point. The convergence criteria for the modulus field inside the subgrade are as follows:
The flow chart for the calculation of the nonuniform modulus field inside the subgrade is shown in Figure
For the calculation of the deflection of the top surface of subgrade under the uniform distribution of resilient modulus of subgrade, the above-mentioned finite element calculation model and its relevant parameters are adopted. The resilient modulus field of subgrade does not vary with the stress state in this model. To numerically calculate the deflection (deformation) of the top surface of the subgrade, the dynamic loading applied on the top surface of the subgrade is half-sinusoid cyclic loading with a stress amplitude of 0.7 MPa, a loading cycle of 0.2 s, an interval time of 0.8 s, and a loading area of 30 cm diameter circle. The cycle period is one second. The loading conditions mentioned above are used to calculate the resilient modulus of subgrade soil at 20∼120 MPa, and the deflection at the central point of stress applied on the top surface of the subgrade can be obtained. The relation between the resilient modulus of subgrade soil and the deflection is shown in Figure
Deflection results of subgrade under dynamic loading. (a) Calculation deflection in the FEM. (b) Relationship between the resilient modulus and the deflection of subgrade.
The site verification with a portable falling weight deflectometer (PFWD) is carried out on the top surface of the upper embankment (subsoil) of K116 + 980∼K117 + 180 (left part) in Lot C of Guangzhou–Foshan–Zhaoqing Highway. The PFWD site tests are also conducted on the top surface of unscreened gravel layers with different thicknesses laid on the upper embankment (subsoil).
In this study, the PFWD site test section is 200 m in length. Firstly, two testing lines are placed on the top surface of the upper embankment along the vehicle wheelmark of the carriageway, and the horizontal distance between the two test lines is 2.0 m. The PFWD site tests are conducted on each testing line at a spacing of 20 m. The deflection of each test point on the top surface of the subgrade can be obtained by the PFWD. There are 24 test points in total. Secondly, after each PFWD test on the top surface of the upper embankment (subsoil) was carried out, the unscreened gravel layer of 10 cm thickness is laid on the upper embankment. And the PFWD site tests are also done in the same positions on the surface of the unscreened gravel layer of 10 cm thickness. Thirdly, the above experimental methods had been reused on the top surface of the unscreened gravel layer of 20 cm thickness and 30 cm thickness. There are 72 test points in total, and the stress-time curves of the dynamic loading can be obtained. And also the measured deflection of each test point in each layer can be gained.
The filling subgrade soil is granite eluvial soil (sandy clay with a low liquid limit). After the PFWD tests on the top surface of the subgrade (subsoil) are implemented, the unscreened gravel is placed on the top surface of the subgrade and compacted to the layer with a thickness of 10 cm, and the compactness of the unscreened gravel layer is not less than 96%. After that, the test points are located by the electronic total station. The site tests of the PFWD are conducted in the same positions when the unscreened gravel layers are laid on the top surface of subgrade of 10 cm thickness, 20 cm thickness, and 30 cm thickness. More than 5 PFWD tests are carried out at each test point, and the average value of tests is taken as the measured value for every test point.
Finite element model of subgrade is undertaken according to the size and parameters of the test section in Figure
To study the impact of unscreened gravel on the equivalent resilient modulus of subgrade for different thicknesses of unscreened gravel layers, finite element models with different thicknesses of unscreened gravel layers are created. The method mentioned above for the finite element calculation of equivalent resilient modulus of subgrade is used for numerical calculation of the standard half-sinusoid loading curve with a stress amplitude of 100 kPa and a dynamic loading cycle of 0.2 s.
To investigate the difference between the calculated values and the PFWD-measured values of the deflection of the top surface of subgrade, a half-sinusoid loading curve under standard axle load is obtained according to the above-mentioned method. The calculated deflection value of the top surface of the subgrade is 152.4 (0.01 mm), measured by the PFWD tests. The test results are shown in Tables
Results of the numerical value and PFWD-measured value of deflection at the center belt.
Mileage pile number | Deflection at the top of subgrade (0.01 mm) | Deflection on a 10 cm thick layer (0.01 mm) | Deflection on a 20 cm thick layer (0.01 mm) | Deflection on a 30 cm thick layer (0.01 mm) | ||||
---|---|---|---|---|---|---|---|---|
Measured value | Numerical value | Measured value | Numerical value | Measured value | Numerical value | Measured value | Numerical value | |
K116 + 980 | 162.6 | 152.4 | 144.2 | 133.0 | 132.0 | 122.0 | 123.3 | 113.6 |
K117 + 000 | 163.1 | 144.1 | 132.6 | 123.4 | ||||
K117 + 020 | 163.8 | 144.2 | 132.8 | 123.2 | ||||
K117 + 040 | 165.5 | 144.6 | 132.7 | 123.9 | ||||
K117 + 060 | 165.1 | 140.5 | 132.8 | 123.7 | ||||
K117 + 080 | 165.5 | 146.8 | 132.9 | 123.8 | ||||
K117 + 100 | 166.4 | 144.8 | 132.8 | 123.2 | ||||
K117 + 120 | 164.4 | 144.7 | 132.3 | 123.4 | ||||
K117 + 140 | 163.6 | 144.8 | 132.4 | 124.1 | ||||
K117 + 160 | 165.3 | 144.9 | 132.3 | 123.8 | ||||
K117 + 180 | 164.8 | 144.0 | 132.1 | 124.0 | ||||
Mean of deflection | 164.6 | 152.4 | 144.3 | 133.0 | 132.5 | 122.0 | 123.6 | 113.6 |
Error of deflection | 12.2 | 11.3 | 10.5 | 10.0 | ||||
Equivalent |
63.45 | 68.72 | 73.03 | 79.63 | 79.95 | 87.28 | 86.08 | 94.15 |
Error of the |
5.27 | 6.60 | 7.33 | 8.07 |
Results of the numerical value and PFWD-measured value of deflection at the wheelmark belt.
Mileage pile number | Deflection at the top of subgrade (0.01 mm) | Deflection on a 10 cm thick layer (0.01 mm) | Deflection on a 20 cm thick layer (0.01 mm) | Deflection on a 30 cm thick layer (0.01 mm) | ||||
---|---|---|---|---|---|---|---|---|
Measured value | Numerical value | Measured value | Numerical value | Measured value | Numerical value | Measured value | Numerical value | |
K116 + 980 | 164.3 | 152.4 | 141.4 | 133.0 | 132.0 | 122.0 | 123.2 | 113.6 |
K117 + 000 | 162.5 | 143.2 | 132.1 | 123.6 | ||||
K117 + 020 | 161.8 | 145.7 | 131.7 | 123.2 | ||||
K117 + 040 | 164.4 | 138.6 | 131.3 | 123.4 | ||||
K117 + 060 | 163.1 | 144.2 | 131.9 | 123.3 | ||||
K117 + 080 | 164.7 | 147.1 | 131.8 | 123.5 | ||||
K117 + 100 | 164.3 | 143.3 | 131.7 | 123.4 | ||||
K117 + 120 | 161.5 | 143.8 | 131.5 | 123.6 | ||||
K117 + 140 | 163.9 | 143.2 | 132.1 | 123.6 | ||||
K117 + 160 | 161.8 | 143.6 | 131.5 | 123.0 | ||||
K117 + 180 | 161.6 | 143.2 | 132.1 | 123.1 | ||||
Mean of deflection | 163.1 | 152.4 | 143.4 | 133.0 | 131.8 | 122.0 | 123.4 | 113.6 |
Error of deflection | 11.3 | 10.4 | 9.8 | 9.8 | ||||
Equivalent |
64.12 | 68.72 | 73.51 | 79.63 | 80.41 | 87.28 | 86.23 | 94.15 |
Error of the |
4.60 | 6.12 | 6.87 | 7.92 |
After the deflection of the top surface of subgrade soil in the test section is measured, unscreened gravel is placed on the top surface layer by layer; there are three layers in total, and each layer is 10 cm thick. The PFWD test is conducted on each layer. The coordinates of the test point correspond to those of the test point on the top surface of the subsoil.
When each unscreened gravel layer (10 cm, 20 cm, and 30 cm in thickness) is placed on the top surface of the subgrade, the deflection of the top surface of the subgrade is calculated. The material resilient modulus of the unscreened gravel layers is 200 MPa, which is obtained through the laboratory test [
It can be clearly found after comprehensive analysis of Tables
From Tables
Considering that many previous researchers paid more attention to the resilient modulus of the overlay or calculated the equivalent resilient modulus of the gravel layers under static loading, and the research results on the equivalent modulus of the gravel layer top under dynamic loading are few, especially considering the nonuniform distribution of the resilient modulus of subgrade soil, there are few related in-depth studies. One of the purposes of this paper is to provide a quantitative variation rule of equivalent resilient modulus for road engineering researchers under the condition of nonuniform distribution of resilient modulus in subgrade.
To study the impact of different material modulus values and layer thicknesses on the equivalent resilient modulus of subgrade under different subsoil modulus conditions, a finite element calculation model is created for different material modulus values and layer thicknesses, based on the cross-sectional size and subsoil material parameters of the test section of Nanchang–Zhangshu Highway. The working conditions for calculation are shown in Table
Calculation conditions under different modulus values of subgrade soil and modulus values and thicknesses of overlays.
Calculation condition | Parameter values |
---|---|
Modulus of subgrade soil (MPa) | 20, 40, 60, 80, 100, 120 |
Modulus of overlays (MPa) | 50, 100, 200, 300, 400, 500 |
Thickness of overlays (cm) | 10, 20, 30, 40, 50, 60, 70, 80 |
Equivalent resilient modulus with different
According to Figure
To study the influence of different material resilient modulus values and layer thicknesses on the equivalent resilient modulus of subgrade under different subgrade soil modulus conditions, the same calculation model and parameters in Table
Equivalent resilient modulus with different
According to Figure
In conclusion, layers with a certain thickness and higher material modulus on the top surface of subgrade (top surface of subsoil) can effectively increase the overall equivalent resilient modulus of the subgrade. Though layers cannot increase the equivalent resilient modulus of the subsoil, we can design corresponding layers according to the attenuation of the subsoil modulus during road operation, propose a regulation and control method for the equivalent resilient modulus of the subgrade, increase the overall equivalent resilient modulus of the subgrade, and reduce the attenuation rate of the structural modulus of the subgrade during its service life so as to ensure the stability of the structural modulus of the subgrade throughout the road operation period.
Based on the comparison between the PFWD-measured deflection and the calculated deflection of the top surface of the subgrade, the error values between calculated and measured results of the equivalent resilient modulus of the subgrade are within 10%. This demonstrates that the calculation method for the structural modulus of subgrade proposed in this study is reasonable and effective. The equivalent resilient modulus of the subgrade reduces as the traffic load increases; the material modulus of subgrade soil inside the subgrade increases as the subgrade depth increases. When the load is less than 2.0 times the standard load, the equivalent resilient modulus of the subgrade reduces more significantly as the load increases; within the vertical depth of 1.5 m, the resilient modulus of the material of the subgrade soil increases more significantly as the depth increases, at about 60%. The resilient modulus of subgrade soil has important impacts on the overall equivalent resilient modulus of the subgrade provided by laid layers. In response to the fact that the resilient modulus of subgrade gradually reduces with climatic changes, we have proposed a control method for the equivalent resilient modulus of the subgrade; that is, the gravel layers with a certain thickness, higher modulus, and good water stability are placed on the top surface of the subgrade to increase the equivalent resilient modulus of the top surface of the subgrade. And it can ensure the stability of the resilient modulus of subgrade throughout the road operation period.
The data used to support the findings of this study are available from the corresponding author upon request.
The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.
The authors declare no conflicts of interest.
The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (51878078, 51608057, and 51908562), Hunan Provincial Innovation Foundation for Postgraduate (CX2018B521), Open Research Fund of Science and Technology Innovation Platform of State Engineering Laboratory of Highway Maintenance Technology, and Changsha University of Science & Technology (kfj150103). This study was also supported by the State Bureau of Forestry 948 Project (2015-4-38), the Youth Scientific Research Foundation, Central South University of Forestry and Technology (QJ2018008B), and the Open Fund of Engineering Research Center of Catastrophic Prophylaxis and Treatment of Road & Traffic Safety of Ministry of Education (Changsha University of Science & Technology) (kfj170405).