Displacement and Stress Analysis of Thin Plate for Cement Concrete Pavement

In order to analyze the displacement and stress of thin plate for cement concrete pavement with rectangle shape and resting on Winkler soil foundation, an analysis model was set up based on the theory of elastic thin plate on Winkler foundation. According to elasticity Kirchhoff theory of thin plates and Winkler soil foundation model, the expressions of displacement and stress were yielded by using the inverse Hankel transform. Numerical method was employed to calculate displacement and stress of the thin plate. Results showed that displacement and stress corresponded with using the method from Code for Highway Design-Pavement. Results showed that the method can provide a new computation method for displacement and stress of the thin plate about cement concrete pavement.


Introduction
The model of a thin plate on elastic foundation was mainly used in structural engineering in last years.Currently, thin plates or films of metal, ceramic, or synthetic materials were bonded in the surface of machine structural parts or electronic devices to improve their mechanical, thermal, electronic, or tribological properties [1].At these applications, the subgrade of the thin plate can be simulated as a Winkler foundation, which reacted with pressure proportional to the deflection of the plate at each point.The plate in the above applications was loaded by vertical loads or bending moments.For cement concrete pavement, the applied load is mainly from the vehicles, which is vertically applied on pavement.So that the width and thickness of the pavement are usually determined by the traffic volume.And the thickness will increase with the designed traffic volume.A large number of research works have been published to solve problem of a classical thin plate [2][3][4] or a thin plate on an elastic foundation [5][6][7][8][9].Fisher investigated the elastic impact of a sphere on a thin plate which is continuous contact with a foundation [10].Jędrysiak investigated the dynamic response of thin elastic plates in Winkler foundation [11].Some researches introduced an efficient boundary element approach for the analysis of thin plate resting on an elastic Winkler foundation with different boundary conditions [12][13][14][15].Shao combined the Fourier spectral method and differential quadrature method for solving problems for thin plates resting on Winkler foundations with irregular domains [16].
However, few research works have been published concerning the stress and displacement of a thin plate on Winkler foundation.The present work is aimed to obtain an analytical solution of a finite rectangle thin plate on an elastic foundation under vertical forces.For this purpose, both the Hankel transform and inverse Hankel transform were adopted.

Basic Assumption and Governing Equations
According to the theory of elastic thin plate on Winkler soil foundation, a rectangle thin plate with thickness h was considered to rest on a Winkler soil foundation (Figure 1), the thickness of plate was less than one-fifth the size of plane, the plate was loaded by the vertical pressure (, ) and the foundation reaction pressure p assumed to be proportional to the vertical deflection (, ) of the plate [17], and the rectangular coordinate system was placed on midplane.
Kirchhoff proposed the basic assumption of the thin plate as follows [17].
(1) Length of normal line of the plate is unchanged before and after deformation, that is, where  z is normal strain.
(2) Before deformation, the normal line of the plate is perpendicular to midplane; after deformation, the normal line is still perpendicular to midplane, that is, where  zx ,  zy are shear strain.
(3) The plane of parallel to midplane does not squeeze, that is, where  z is stress.
(4) Horizontal displacement of the midplane does not occur during the process thin plate bending, which are where  0 , V 0 are displacement.According to the hypothesis (2), it yields  zx = 0, namely, / + / = 0,  zy = 0, namely, / + V/ = 0. Thus, According to the hypothesis (4), the displacement of thin plate can be obtained as follows: According to (6), the strain of thin plate can be obtained as follows: According to general Hook's law, we can obtain Substituting ( 7) into ( 8), the stress of thin plate can be obtained as follows: where E is elasticity modulus,  is Poisson's ratio, u and v are displacement,  is vertical deflection of the plate, and  = (, ).

Stress Fields of the Elastic Thin Plate
A stress field is a region in a body for which the stress is defined at every point [18].Stress fields were widely used in materials science.The stress field, which was composed of these generalized stresses, was called the generalized stress field.
A hexahedron element with thickness h was taken out from the thin plate (Figure 2); stress components acting on the four sides of hexahedron element were M x dy, M y dx, M xy dy, M yx dx, Q x dy, and Q y dx, where M x , M y , M xy , M yx , Q x , and Q y are called the generalized stress.Generalized stress can be expressed as follows: Substituting ( 9) into (10) yields where ), and D is relative stiffness of the plate.Combining ( 9) and (11) gives Under the condition of axisymmetric loading, stress of the plane can be obtained

Bending Equations of the Thin Plate
As illustrated in Figure 3, uniform load (, ) is applied on hexahedron element, while   ,   ,   ,   ,   , and   are generalized stress on the four sides of the element.The loading added up to zero in the direction of the z axis, and solving bending moments of parallel to the x axis and the y axis yields Substituting ( 14) and ( 15) into (16) yields Substituting ( 11) into (17) yields where ∇ 2 =  2 / 2 +  2 / 2 , ∇ 2 is the Laplace operator.
According to the view of Winkler [1], a rectangle thin plate was considered to rest on a Winkler soil foundation, and the foundation reaction pressure p was only related to the vertical deflection  of the plate.

𝑝 = 𝑘𝜔 (19)
where k is reaction modulus of the foundation.The plate was loaded as follows: A finite elastic thin plate on Winkler soil foundation; the bending equation was obtained as follows:

The Equations of Stress and Displacement
Under the axisymmetric loading, the bending equation of the thin plate on Winkler soil foundation in cylindrical can be obtained as follows: where The application of the first class of zero order Hankel transform to equation ( 22) yields where where  = ℎ 3 /12(1 −  2 ).
Set  4 = /;  is the relative stiffness radius of the plate on Winkler soil foundation.The application of Hankel transform to (24) and expression of deflection can be obtained as follows.
Substituting (25) into (13) yields The maximum radial stress  r and tangential stress   can be obtained as follows: On the position r=0, (25)-( 27)can be transformed as follows: Equations ( 30)-(32) can be solved by numerical method.Combining equations ( 31) and (32) gives The thin plate for cement concrete pavement, , was less than 1.Therefore,   was greater than   on the position  = 0.The maximum stress   was calculated in the following example.

Calculation Example
In order to check the validity of the method presented in the paper, an example of thin plate for cement concrete pavement was given.The material parameters were chosen as follows: h = 0.18 m and h = 0.22 m, E = 11000 MPa,  = 0.15, k = 1.4×10 7 N/m 3 ,  = 0.151 m q(x,y) = 0.7 MPa.We selected the integral interval changed from 0-10, 0-20, 0-30, 0-40, 0-50, and 0-1000, respectively, the displacement and stress were calculated, and the results were summarized in Tables 1, 2, 3, and 4.
According to the result of Table 1, integral interval tends to 0-20; when the thickness h of the thin plate is from 0.18 m to h = 0.22 m, displacement of the thin plate on the position r = 0 was 0.615 mm and 0.595 mm, respectively.According to the results of Table 2, integral interval tends to 0-40; when the thickness h of the thin plate is from 0.18 m to h = 0.22 m, stress of the thin plate on the position r = 0 was 1.53 MPa and 1.41 MPa, respectively.

Validation of the Method
The material parameters were chosen as the calculation example; according to the literature [19], calculation procedure was as follows.
Relative stiffness radius l of the pavement is Based on this, we can obtain coefficient : The deflection of the thin plate on the position r = 0 can be found in the table [19].The bending moment coefficient on the position r = 0 can be found in the table [19].From the above analysis, the stress and deflection based on the method were equal to using the method from Code for Highway Design-Pavement [19].It also indicated that the present method was reasonable in the paper.

Conclusions
(1) The expressions of displacement and stress for cement concrete pavement on elastic foundation were derived.
(2) In order to verify the theoretical results, an example of thin plate for cement concrete pavement was given; the stress and deflection correspond with using the method from Code for Highway Design-Pavement.
(3) Future experiment research is needed to be carried out to verify the theoretical results present in this paper.

Figure 1 :
Figure 1: Model of the thin plate.

Figure 2 :
Figure 2: Stress of the plane.

Figure 3 :
Figure 3: Internal force of the plane.

Table 1 :
Displacement of plate in the middle position (h = 0.18 m).

Table 2 :
Stresses of plate in the middle position (h = 0.18 m).

Table 3 :
Displacement of plate in the middle position (h = 0.22 m).

Table 4 :
Stresses of plate in the middle position (h = 0.22 m).