International Scholarly Research Notices

Volume 2016, Article ID 4975345, 9 pages

http://dx.doi.org/10.1155/2016/4975345

## Dynamic Response of a Rigid Pavement Plate Based on an Inertial Soil

University of Abomey-Calavi, 01 BP 2009 Cotonou, Benin

Received 7 November 2015; Accepted 15 December 2015

Academic Editor: Ömer Cívalek

Copyright © 2016 Mohamed Gibigaye et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 Pasternak-Vlasov type (three-parameter 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.

#### 1. Introduction

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 [1]. To design these classes of pavements, the rational methods models are often used. The backbone of a method of roadway design is the mechanical model used to define the structure [2].

In the case of rigid pavements, the most used models are the multilayer elastic model of Burmister [3] and the Westergaard model, assuming pavement as a plate resting on the Winkler soil type [2, 4–6]. The differences between the pavement and the soil rigidities were conducted by Ullidtz [7], to deduce that Burmister model is generally not considered as an appropriate tool for the analysis of a rigid pavement response. Then, the large used design model of existing rigid pavement is Westergaard’s, using the Winkler soil type. Although Winkler model leads to relatively simplified results, it has serious limitations [1]. Firstly, there is the deflection discontinuity between the charged and the uncharged part of pavement plate. Secondly, both models consider loads usually as a static one applied on a plate [2–4, 8]. According to Sun and Greenberg (2000) cited by St-Laurent [9], traffic loads on the pavement induce inertial effects that must be supported by foresaid pavement. Thus, the static load model does not reflect accurately the actual conditions of load on the pavement [10]. During the last decade, many researchers have examined the problem assuming the loads as a dynamic one [1, 8, 10]. In the studies mentioned above, the soil used in the structure modeling are Winkler soil type. But, according to the design guide of United States National Cooperative Highway Research Program (NCHRP) [11], the two-parameter Pasternak model is designated in 1998 as the best pavement option to model the foundation of pavements. From that, many researchers base theirs works on using this type of soil [12, 13]. Alisjahbana and Wangsadinata [14] looked at the dynamic analysis of a rigid pavement under mobile load resting on Pasternak soil type. Based on this model, the determination of soil parameters is only based on the elasticity modulus and Poisson’s ratio. On the other hand, the Pasternak-Vlasov model that takes into account the logarithmic decrement of soil was used by Rahman and Anam [1] using the finite element method. The study of Rahman also showed that the Pasternak-Vlasov model is more economical than that of Winkler and cannot arbitrarily set the values of the intrinsic characteristics of the soil.

In most models used previously, the dynamic effect is taken into account only by the inertia of the plate [11, 15]. Concerning the soil, inertia is neglected in dynamic modeling of pavement structure. However, Civalek [16] took into account the inertia of the soil but had defined as constant independents values the intrinsic parameters of soil. He concluded that the effect of foundation inertia on the central deflection of the finite plate is not considerable. But according to the results of Pan and Atluri’s work [17], in engineering practice, this is still not always the case, and this factor may have significant effects on the dynamic response of the plate modeling the pavement. For these reasons, several studies have tried to modify the Pasternak-Vlasov soil introducing the inertia of the soil to a depth of soil susceptible to dynamic forces applied to the structure. Gibigaye in his work used an inertia soil model foundation for the study of the behavior of shells modeling underground shells [18]. He concluded that it is important to take into account the inertia of the foundation soil on the dynamic response of civil engineering structures. Besides, Dimitrovová [19] noted that the classical formula which predicts a critical velocity of load significantly is overestimated compared to the one experienced in reality. She indicated that this formula should be revised by introducing two important notions: the effective finite depth of the foundation that is dynamically activated and the inertial effect of this activated foundation layer.

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 three-parameter type (, , and , resp., integral characteristics in compression and in shearing and reduced mass of the foundation soil). The boundary conditions of these plates are modeled by two linear relationships between strains and stresses at the plate edges. The homogeneous solution of the problem is achieved by the method of separation of variables, so that the superposition gives a solution satisfying the boundary conditions. Since the deformation is expressed as eigenfunctions products, the solution of the dynamic problem is obtained based on the orthogonality properties of eigenfunctions. The general solution of the problem is obtained from the specific properties of the Dirac delta function. This paper provides an overview of the dynamic analysis of rigid pavements response as described above.

#### 2. Materials and Methods

##### 2.1. Governing Equation

In this research work, an isotropic homogeneous elastic rectangular plate resting on an elastic three-parameter 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 [20] and Gibigaye work in the cylindrical axis system [18], the soil response according to the deflection at a given point inside the soil layer is equivalent towhere , , , and are space-time coordinates of the soil studying point; is the deflection inside the soil layer defined as ; and is a vertical decay function of soil that must verify and ; , , and are, respectively, integral characteristics in compression and in shearing and linear reduced mass of the foundation soil, supposed to be homogeneous and monolayer. They are expressed as follows [21]:where is the effective finite depth of the foundation that is dynamically activated; is the density of the subgrade; is a constant, named logarithmic decrement of the soil, which determinates the rate of decrease of the deflections depending on the depth; is Young’s Modulus; is Poisson’s ratio.

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: is the deflection of Kirchhoff plate which is equal to the deflection of the plate/soil interface.

is the load transmitted to the pavement [14]. Here ; are the geometrical position of load at the time ; is the magnitude of the moving wheel load; is the acceleration of the load; is the angular frequency of the applied load; is the Dirac function; , , and are the dimensions of the finite plate and is the flexural stiffness of the plate.

The boundary conditions (Figure 1) are modeled as follows: