This work presents the dynamic response of a pavement plate resting on a soil whose inertia is taken into account in the design of pavements by rational methods. Thus, the pavement is modeled as a thin plate with finite dimensions, supported longitudinally by dowels and laterally by tie bars. The subgrade is modeled via PasternakVlasov type (threeparameter type) foundation models and the moving traffic load is expressed as a concentrated dynamic load of harmonically varying magnitude, moving straight along the plate with a constant acceleration. The governing equation of the problem is solved using the modified Bolotin method for determining the natural frequencies and the wavenumbers of the system. The orthogonal properties of eigenfunctions are used to find the general solution of the problem. Considering the load over the center of the plate, the results showed that the deflections of the plate are maximum about the middle of the plate but are not null at its edges. It is therefore observed that the deflection decreased 18.33 percent when the inertia of the soil is taken into account. This result shows the possible economic gain when taking into account the inertia of soil in pavement dynamic design.
Pavements are an essential feature of the urban communication system and provide an efficient means of transportation of goods and services. Depending on its rigidity compared to the subsoil, pavements are classified as flexible, rigid, and semiflexible [
In the case of rigid pavements, the most used models are the multilayer elastic model of Burmister [
In most models used previously, the dynamic effect is taken into account only by the inertia of the plate [
This work investigates dynamic response of a rigid pavement resting on an inertial soil. For that, the rigid pavement is modeled as a thin plate with dowels and tie bars in its edges.
In order to take into account its inertia, the soil is modeled as a threeparameter type (
In this research work, an isotropic homogeneous elastic rectangular plate resting on an elastic threeparameter soil is considered to model a pavement. The adjacent plates are supposed to be joined by dowels and tie bars. Based on the work of Asik [
According to the classic theory of thin plates and if taking into account the reduced mass of soil, the transverse deflection of the Kirchhoff plate satisfies the following partial differential equation:
The boundary conditions (Figure
Modeling of doweled rigid pavement under moving load [
(i) The restriction of the elastic vertical translation is characterized by the four equations [
(ii) The restriction of the elastic rotation is characterized by the following four equations [
The initial conditions (
In order to solve governing equation (
Equation (
This equation is independent of time as the function
Here,
To obtain the mode numbers
Substituting (
The solutions of the characteristic equation of (
For
Equation (
Boundary conditions along
In order to obtain no trivial solution, it is necessary to propose that the determinant of (
Substituting (
The solutions of the characteristic equation of (
Equation (
Boundary conditions along
In order to obtain no trivial solution, it is necessary to propose that the determinant of (
To obtain the couples
The natural mode of the plate is therefore given by
Suppose the solution of governing equation (
Thus, for
The corresponding homogeneous solutions of (
Using the procedure described above, a rigid roadway pavement subjected to a dynamic traffic load is analyzed. In this work, a finite rectangular plate doweled along its edges is considered as shown in Figure
Figure
Variation of the deflection directly under load versus time for different types of load (
Figure
Variation of the deflection at the fixed point of coordinates (
Based on the data listed above, the first five mode numbers of the plate modeling the pavement were determined in the
Figure
Variation of the deflection in the center of the plate (
It is deduced from these observations that the dynamically activated soil depth greatly influences the response of the plate. We have chosen in this study the depth maximizing the threeparameter soil type;
Figure
Variation of the deflection along the central axis of the plate (
Figure
Variation of the deflection as a function of plate magnitude, for different values of dynamically activated soil depth, at time
Table
Displacement versus soils parameters for different values of








3.77 × 10^{−4} m  3.78 × 10^{−4} m  5.73 × 10^{−4} m 









5.27 × 10^{−4} m  5.50 × 10^{−4} m  7.08 × 10^{−4} m 









5.36 × 10^{−4} m  5.8 × 10^{−4} m  7.81 × 10^{−4} m 









5.48 × 10^{−4} m  5.73 × 10^{−4} m  8.48 × 10^{−4} m 









5.13 × 10^{−4} m  5.77 × 10^{−4} m  9.37 × 10^{−4} m 



This paper dealt with some significant results from a study of the dynamic analysis of rigid pavements. The soil models used in this work are the wellknown Pasternak model, the PasternakVlasov model which takes into account the interaction between soil layers, and the improved threeparameter model considering the inertia of the soil. The main conclusions of this study are the following:
The soil inertia influences the pavement response at the middle of the plate when the load evolving along its centerline arrived at the center. This indicates a possible overdesign of pavements when using the twoparameter soil model.
The effect of dynamically activated depth of PasternakVlasov soil and threeparameter soil on the response is found to be significant for both soils types but more for threeparameter type than the PasternakVlasov type.
Before the stationary domain of oscillations, a transient response of the plate during
This study only covers plates pavement interconnected by dowels and tie bars. So, we could extend it to continuous pavements plates.
This study does not take into account the cyclic effect of the load. So, the fatigue response of the studied system could be further analyzed.
The resonance is not studied in this work despite the effect that soil inertia can have in that, so we intend to study it in further work.
The authors declare that there is no conflict of interests regarding the publication of this paper.