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This paper establishes the equivalent relationships between the half-sinusoidal load, triangular load, vertical stepwise load, and moving traffic load. The governing equation was established for analyzing the dynamic responses of pavement, and half-sinusoidal load, triangular load, and vertical stepwise load functions were transformed into Fourier series expressions. The partial differential governing equations were simplified as ordinary differential equations and the analytical solutions were obtained. Further, the solutions were validated through comparing the theoretical results with numerical simulated results. Calculation results revealed that, for unchanged load periods, increasing the amplitudes of the three loads by 1.06, 1.31, and 1.35 times can better simulate the moving traffic loads. For unchanged load function amplitudes, increasing the function periods by 1.07, 2.23, and 2.1 times (for half-sinusoidal, triangular, and vertical stepwise loads, resp.) can improve the simulation performance. The fatigue life of asphalt pavements under the moving traffic load agrees with that of the three load simulations, indicating that the fatigue life of asphalt pavements is only associated with the load amplitude but not the load patterns.

Due to the excellent suitability for vehicles, asphalt pavements have become the preferred pavement structure type in highway construction in many countries. Owing to increasing traffic volume, vehicle loads, and vehicle speed on highways in recent years, the requirements for pavement structure reliability and durability have increased. Nevertheless, the mismatch between the growing high-speed heavy-load highway transportation demands and the insufficient lifespan of pavement restricts highway development [

The current focuses in road research have become studying the kinetic behavior of pavement structures under traffic load, discovering the pavement damage mechanism, and how to switch from a static pavement design to a dynamic one. Over recent years, road researchers have conducted many indoor and outdoor studies on the dynamic response of asphalt pavements and formulated various methodologies to refine the design of asphalt pavements [

It is well known that many literatures analyzed the traffic load with a fixed location of load application, but a changing load size and half-sinusoidal load functions can better simulate actual traffic loads. Many researchers often replaced the actual traffic load with the impulsive load and input the latter in finite calculation models for the dynamic response of asphalt pavements [

Therefore, by analyzing the research results from previous studies, this paper aims at establishing the governing equations of pavement dynamic responses based on the characteristics of semirigid base asphalt pavements, the elastic layer system theory, and soil constitutive equations. Half-sinusoidal load, triangular load, and vertical stepwise load functions were transformed into Fourier calculation expressions and introduced into the governing equations. The governing equation will be solved analytically and verified by comparing the analytic results with the numerical simulated ones. Based on the methodology, the load equivalence will be established in terms of rutting and fatigue life.

There are mainly three loading patterns currently employed in pavement dynamic response analyses: half-sinusoidal impulsive load [

Three typical loading patterns.

The sinusoidal impulsive load

The triangular impulsive load

The vertical stepwise load

The expressions for the three different impulsive load functions are as follows.

The sinusoidal impulsive load function can be expressed as

The triangular impulsive load function can be expressed as

The vertical stepwise load function can be expressed as

The layout of soil foundation and pavement system in this paper was a two-dimensional and eight-layered model, including upper, middle, and lower pavement surface layers, upper, middle, and lower pavement base layers, and upper and lower foundation layers, as shown in Figure

Schematic of the roadbed pavement system under loading.

The subject of this study is asphalt concrete pavement, which in general will be regarded as a continuous medium. Different materials have different properties, for instance, asphalt concrete has a higher porosity and a greater overall material strength than soil. Therefore, corresponding to the actual situation, a rutting model suitable for asphalt concrete pavement is established in this study to analyze the rutting formation. The following assumptions were made in setting up the mathematical model.

The deformation of the asphalt concrete is very small.

The aggregates in asphalt concrete are incompressible.

The displacement and stress between the structural layers are continuous.

No consideration will be made of the gradual compaction process of the pavement and the shrinkage process of the pores. Without taking water into account, the following governing equations are obtained [

In (

For the two-dimensional planar strain problem, the following expressions are obtained from Hooke’s principle of stress and strain with damping property of the material:

On the right side of (

For the pavement layers, the base layers, and the subgrade, the dynamic governing equations are shown in (

In reality, when a vehicle passes through the pavement surface, the load changes in magnitude and location. The dynamic load acting on the pavement structure has two components:

In two-dimensional situations, the load applied can be expanded with the Fourier series. After the load passes through a certain point, the dynamic response at that location will gradually diminish until next load application. Assuming the time for the load to be applied once as_{0} (_{0}=2_{1} (_{1}=2

The three different load functions can be expressed with the Fourier series as follows.

In the equation

According to the Fourier transformation,

According to the Fourier transformation,

According to the Fourier transformation,

For a linear system, an arbitrary function_{m}(_{m} is a function of the single independent variable

Accordingly, the governing equation of the entire system can be written in the following form:

The general solution of the governing equation may be expressed as follows:_{ij}(

The solutions (roots) can be obtained through numerical methods (e.g., MATLAB program).

In order to solve the governing equation, it is necessary to obtain the integration constant_{ijm} (

At the top of the upper surface layer (_{1xz}=_{1z}=

At the bottom of the upper surface layer (_{1}), the boundary condition of the displacement is_{1x}=_{1} and_{1z}=_{2}. Then, the group of equations for the boundary condition can be written as

The boundary conditions for the middle surface layer, the lower surface layer, the upper base layer, the middle base layer, the lower base layer, and the upper subgrade (

At the interface between the lower subgrade and upper subgrade (_{7}), the boundary conditions for the displacements are_{x}=_{71} and_{z}=_{72}.

At the interface between the lower subgrade and upper subgrade (_{7}), the boundary conditions for the stress are_{xz}=_{H7} and_{z}(_{4})=_{H7}.

At the bottom of the lower subgrade (_{8}), the boundary conditions of the displacements are_{x}=0 and_{z}=0. Then, a group of equations for the corresponding boundary conditions can be written as

For all the groups of equations of the boundary conditions shown above, the numerical methods can be used to obtain the displacement (_{ix} and_{iy}), the positive stress (_{Hi}), and the shear stress (_{Hi}). These solutions can then be substituted into the general expressions for the corresponding dynamic governing equations of the subgrade pavement system under a moving load. The solutions for the displacements can be expressed of the form

In addition, the expressions of the shear stress and the positive stress are

The expressions for the vertical normal stress and vertical normal strain acting on the pavement system under the half-sinusoidal impulsive load are

The expressions for the vertical normal stress and vertical normal strain acting on the pavement system under the triangular load are

The expressions for the vertical normal stress and vertical normal strain acting on the pavement system under the vertical stepwise load are

There were four loading patterns in this paper (i.e., half-sinusoidal load, triangular load, vertical stepwise load, and moving load (in two-dimensional situations, moving load could be simplified into strip load, which was shown in Figure

Asphalt mixture is a temperature-sensitive material, and its road performance is closely related to its temperature sensitivity. Rutting usually occurs easily under high temperatures and changes with temperatures. Being a composite material, asphalt mixture is a classic complex of elasticity, viscosity, and plasticity. On the one hand, under low temperatures and a small deformation range, its behavior is close to a linear elastic body; on the other hand, under high temperatures and a wide deformation range, it behaves as a viscoplastic body. In the transition zone under normal temperatures, it is a normal viscoelastic body. Owing to vehicle loading, the properties of the asphalt mixture become highly complex. In the actual application situations, it is usually an inelastic body, with irreversible deformation after unloading. With reference to the pavement design guidelines in AASHTO [

RD is the rutting depth on asphalt-concrete pavement surface layer (units: m),

∆

Overloading or insufficient structural strength of the surface layer usually causes an enormous tensile strain. The following can be obtained using the pavement design guidelines in AASHTO [

This paper compared the theoretical calculation results to those from numerical simulations to configure the application periods of the three load conditions (half-sinusoidal load, triangular load, and vertical stepwise load). Their respective amplitudes were obtained through the comparison between the rutting under the three load conditions and the rutting under the moving traffic load. With the new load amplitudes and application periods, mutual conversion relationships were established to confirm that the three load conditions can provide a better simulation of the dynamic responses of asphalt pavements under the moving traffic load. The structure of the asphalt pavement adopted in the analysis is shown in Figure

Elasticity parameters of the asphalt mixture.

Asphaltic layers | Temperature (°C) | Elastic modulus (MPa) | Poisson ratio |
---|---|---|---|

SMA-13 | 20 | 870 | 0.25 |

30 | 620 | 0.30 | |

40 | 554 | 0.35 | |

50 | 530 | 0.40 | |

60 | 526 | 0.45 | |

Sup20 | 20 | 910 | 0.25 |

30 | 752 | 0.30 | |

40 | 600 | 0.35 | |

50 | 440 | 0.40 | |

60 | 380 | 0.45 | |

Sup25 | 20 | 1031 | 0.25 |

30 | 900 | 0.30 | |

40 | 710 | 0.35 | |

50 | 500 | 0.40 | |

60 | 390 | 0.45 |

Elasticity parameters of materials.

Materials | Elastic modulus (MPa) | Poisson ratio | Density (kg/m^{3}) |
---|---|---|---|

Asphalt mixture | See Table | See Table | 2300 |

Cement stabilized gravel | 1200 | 0.20 | 2200 |

Lime soil | 300 | 0.30 | 2100 |

Soil base | 45 | 0.40 | 1800 |

Creep parameters of the asphalt mixture.

Asphaltic layers | Temperature(°C) | | | | ^{2} |
---|---|---|---|---|---|

SMA-13 | 20 | 6.536E-11 | 0.937 | -0.592 | 0.9326 |

30 | 3.325E-9 | 0.862 | -0.587 | 0.9459 | |

40 | 1.446E-8 | 0.792 | -0.577 | 0.9420 | |

50 | 1.39E-6 | 0.414 | -0.525 | 0.9244 | |

60 | 1.464E-5 | 0.336 | -0.502 | 0.9049 | |

Sup20 | 20 | 4.58E-11 | 0.944 | -0.596 | 0.9264 |

30 | 2.461E-9 | 0.796 | -0.585 | 0.9227 | |

40 | 3.673E-8 | 0.773 | -0.570 | 0.9364 | |

50 | 4.802E-6 | 0.595 | -0.532 | 0.8494 | |

60 | 7.778E-5 | 0.384 | -0.441 | 0.9138 | |

Sup25 | 20 | 4.59E-11 | 0.922 | -0.581 | 0.9377 |

30 | 3.461E-9 | 0.859 | -0.576 | 0.9208 | |

40 | 1.956E-8 | 0.830 | -0.562 | 0.9063 | |

50 | 1.200E-6 | 0.322 | -0.522 | 0.8015 | |

60 | 3.755E-5 | 0.210 | -0.418 | 0.8994 |

Structural patterns of semirigid base asphalt pavements.

The pavement structure [

Three-dimensional model of asphalt pavement structure.

Three-dimensional finite element meshing of asphalt pavement structure.

For a scenario with a vehicle speed c=20m/s, a tire pressure

Comparison of rutting depths of asphalt pavement under different load conditions.

Comparison of rutting depths of asphalt pavement under different load conditions.

As shown in Figures

As shown in Figures

Relationships between rutting of asphalt pavement under the moving traffic load and the three load simulation conditions.

As all the three load conditions resulted in errors in simulating the moving traffic load, this paper improved the following two aspects of the three load conditions such that they can well simulate the moving traffic load. The first method to improve the load conditions was altering the amplitudes of the three load functions whilst keeping their periods unchanged to simulate the moving traffic load. The second method was to change the periods of the three load functions with their amplitudes unchanged

Rutting of asphalt pavement after varying the amplitudes of the three load conditions.

As shown in Figure

(

Rutting of asphalt pavement under sinusoidal load functions with different periods.

Rutting of asphalt pavement under triangular load functions with different periods.

Rutting of asphalt pavement under vertical stepwise load functions with different periods.

As shown in Figure

As shown in Figure

As shown in Figure

As shown by the above two methods, regardless of whether the load amplitudes or the load periods were altered, the moving traffic load was well simulated.

Calculation parameters for asphaltic layers

Asphaltic layers | Elastic modulus (MPa) | Density (kg/m^{3}) | Poisson ratio |
---|---|---|---|

AK-13A | 1200 | 2500 | 0.25 |

AC-20C | 1835 | 2500 | 0.25 |

AC-20F | 1800 | 2500 | 0.25 |

Cement-stabilized gravel | 1500 | 2400 | 0.25 |

Lime-fly-ash gravel | 1400 | 2000 | 0.25 |

Lime soil | 550 | 1930 | 0.35 |

Soil base | 48 | 1900 | 0.40 |

Asphalt pavement structure with multiasphaltic layers.

For a scenario with a vehicle speed c=20m/s, a tire pressure

Comparison of rutting depth of asphalt pavement under different load conditions.

Comparing Figures

Relationship between rutting of asphalt pavement under the moving traffic load and that under the three load functions.

Comparison between rutting under the moving traffic load and that under the three load functions after load amplitude modification.

Comparison between rutting under the moving traffic load and that under the three load functions after load period modification.

From (

Relationship between the pavement fatigue life and the tensile strain at the layer bottom.

where

For a scenario with a vehicle speed c=20m/s, a tire pressure

Calculated tensile strains under the different load functions.

Load types | Tension strain( |
---|---|

Moving traffic load | 8.28E-5 |

Half-sinusoidal load | 7.53E-5 |

Triangular load | 7.53e-5 |

Vertical stepwise load | 7.53e-5 |

As shown in Table

As aforementioned, when the rutting of pavement was considered as the control index, for the unchanged load periods for the four load conditions, the amplitude of the moving traffic load was 1.06, 1.31, and 1.35 times of the amplitude of the sinusoidal, triangular, and vertical stepwise load functions. When the load amplitudes were kept unchanged, the period of the moving traffic load was 1.07, 2.1, and 2.23 times the half-sinusoidal, triangular, and vertical stepwise load functions, respectively. On the other hand, when the fatigue life was employed as the control index, only the load amplitudes of the four load conditions were relevant, but not the load periods. Under the same peak loads, the tensile strains at the pavement base were identical for the sinusoidal, triangular, and vertical stepwise load functions. However, the pavement base tensile strain under the moving traffic load was twice that under the three load functions. Thus, in order to ensure the equivalent rutting and fatigue life under the four load conditions simultaneously, the relationship between the peak load values and the load periods was obtained through comparison during application and is shown in Table

Loading equivalence between various loading types in terms of the peak load values and the load periods.

Load types | The peak load values | Load periods | Indexes | |
---|---|---|---|---|

Moving traffic load | P | T | Rutting | Fatigue life |

Half-sinusoidal load | 1.1P | 1.03T | Rutting | Fatigue life |

Triangular load | 1.1P | 2.50T | Rutting | Fatigue life |

Vertical stepwise load | 1.1P | 2.74T | Rutting | Fatigue life |

This paper derived the explicit analytic solutions to dynamic response of asphalt pavements under three impulsive load conditions and compared the results with the calculated results by the finite software ABAQUS to verify the validity of the explicit analytical solutions derived. This can provide theoretical supports for the treatment of rutting of asphalt pavement with theoretical bases, which is of practical importance to asphalt pavement projects. In order to establish the equivalence between four loading types (i.e., half-sinusoidal load, triangular load, vertical stepwise load, and moving load), the rutting and fatigue life of asphalt pavement were calculated in terms of keeping load amplitude and load period unchanged, respectively. Through comparison analysis, some conclusions can be drawn as follows.

Although the analysis in this paper is lack of the experimental result to conduct comparison to validate the analytical and numerical solution, it provides an insight into understanding the effect and equivalence of different loads applied on the asphalt pavement, which can be applied in the field of simulation and calculation related to the dynamical analysis, rutting, and fatigue life evaluation.

The data used to support the findings of this study are available from the corresponding author upon request.

We declare that there are no conflicts of interest regarding the publication of this paper.

This research has been supported by the National Natural Science Foundation (Grant No. 51308554) to the first author. The author also appreciates the Guizhou Transportation Science and Technology Foundation (Grant Nos. 2013-121-013 and 2019-122-006), the Hunan Transportation Science and Technology Foundation (Grant Nos. 201237 and 201622), and the Central South University Graduate Student Innovation Fund Project (2015zzts062).