1. Introduction

2. Governing equations and results

2.1. Both upper and lower parts of the plate are thin plates

2.1.1. Basic equations and boundary conditions

Governing differential equations of the upper and lower plates are given as

(1a)

(1b)

where $D_{x_i}$ and $D_{y_i}$ are bending stiffness, $w_i$ is deflection, $F(x_1,y_1)$ is interaction force between the upper and lower plates, $Q(x_2,y_2)$ is subgrade reaction, $H_i=D_{x_i}{v}_{y_i}+2D_{x_i{y}_i}$ is equivalent stiffness, $v_{x_i}$ and $v_{y_i}$ are Poisson ratio, $D_{x_i{y}_i}$ is torsional stiffness. The upper and lower plates are numbered $i=1$ and $i=2$, respectively.

The internal force of the upper and lower plates could be written in terms of deflection functions:

(2)

in which $M_{x_i}$ is bending moment, $M_{x_i{y}_i}$ is twisting moment, $Q_{x_i}$ is shear force.

Boundary restrictions are given as

(3a)

(3b)

(3c)

2.1.2. Coordination equation and analytical solution

The deflections of the upper and lower plates can be expressed as double cosine series with supplementary terms:

(4)

where ${\mu }_{2m_i{x}_i}=\frac{H_i+2D_{x_i{y}_i}}{D_{x_i}}$, ${\mu }_{2m_i{y}_i}=\frac{H_i+2D_{x_i{y}_i}}{D_{y_i}}$, $w_{m_i{n}_i}$, $C_{m_i}$, $D_{m_i}$, $G_{n_i}$, $H_{n_i}$ are undetermined parameters.

Eq. (4) has four steps’ derivation for rectangular plate with four free edges, which could satisfy the boundary conditions, such as the corner condition and shear force at the boundary. If the plate is made of isotropic material, the Eq. (4) can degenerate into an expression of an isotropic rectangular plate.

Based on Eq. (1), it could be found that $F(x_1,y_1)$ is related to the control differential equations of the upper and lower plates, so $F(x_1,y_1)$ can be expanded into double cosine series represented by $x_1$, $y_1$ and $x_2$, $y_2$.

(5)

where

(6a)

(6b)

(6c)

$x_0$, $y_0$ represent the relationship between $x_1{y}_1$ and $x_2{y}_2$ coordinate systems, and substituting $x_2=x_1+x_0$, $y_2=y_1+y_0$ into the integral part of the Eq. (6c):

Hence, Eq. (6) is rewritten as

$q(x_1,y_1)$ is expanded into double cosine series represented by $x_1$ and $y_1$.

(7)

where

Subgrade reaction can be expressed in terms of double cosine series as

where

Substituting Eqs. (1a), (5) and (7) into Eqs. (3)~(4), and then expanding the polynomial of the supplementary terms in the formulas to cosine series. Comparing the coefficients of the corresponding items on both sides of the Eqs. (3)~(4), we could obtain the expressions as

(8)

(9)

where ${\alpha }_{m_i}=\frac{m_i\pi }{a_i}$, ${\beta }_{m_i}=\frac{n_i\pi }{b_i}$ and

Considering the boundary conditions of bending moment, we could obtain:

(1) When $x_i=0$, $M_{x_i}=0$

(10)

(2) When $x_i=a_i$, $M_{x_i}=0$

(11)

(3) When $y_i=0$, $M_{y_i}=0$

(12)

(4) When $y_i=b_i$, $M_{y_i}=0$

(13)

The deflection of the upper and lower plate could be expressed by formula [18]

where

Expanding the deflections of the lower plate into double cosine series at the contacting position with the upper plate

(14)

Substituting $x_2=x_1+x_0$, $y_2=y_1+y_0$ into Eq. (14), we obtain

Eq. (8) can be rewritten as

Taking into account that the deflections of upper and lower plates at the contact position are the same, the deformation equation of compatibility is expressed as

(15)

The double Fourier transform of the ground reaction force is expressed as

(16)

Expanding $w\left|{}_{z_2=0}\right.$ into double cosine series form

(17)

in which

(18)

In view of the research contents in the literature [19], $w\left|{}_{z_2{m}_2{n}_2}\right.$ could be expressed as

(19)

where

Eq. (4) could be rewritten in double cosine series form, and considering that the plate and surface of the elastic foundation have the same vertical displacement, thus the coefficients of corresponding items are also the same. The deformation equation of compatibility is given as

(20)

According to Eqs. (8)~(13) and (15)~(20), the undetermined coefficients $w_{m_1{n}_1}$, $F_{m_1{n}_1}$, $C_{m_1}$, $D_{m_1}$, $G_{n_1}$, $H_{n_1}$, $w_{m_2{n}_2}$, $Q_{m_2{n}_2}$, $C_{m_2}$, $D_{m_2}$, $G_{n_2}$, $H_{n_2}$ could be solved. Substituting the solved coefficients into Eq. (1) and Eq. (4), the subgrade reaction, deflection and internal force of the plate could be obtained.

2.1.3. Example

We consider a stepped rectangular plate resting on the surface of an elastic half space foundation. The dimensions of the upper and lower plate are 4.0m×0.2m (side length × thickness) and 4m×0m (side length × thickness) respectively, and the uniform load $q(x_1,y_1)$ on the plate is 0.98MPa. In this case, the stepped rectangular plate degenerates into a rectangular plate. The performance parameters of plate and foundation are given in Table 1.

The subgrade reaction, bending moment and deflection of the plate could be obtained by the solution, as shown in Figure 2.

Table 2 shows that the calculated results are consistent with the results in [19]. This comparison proves the effectiveness of the theory proposed in this paper.

2.2. Both upper and lower parts of the plate are moderately thick plates

2.2.1. Basic equations and boundary conditions

The governing differential equations of moderately thick plate are determined by formulas [20]

(21)

where ${\Phi }_{x_i}$, ${\Phi }_{y_i}$, $w_i$, $F$, $Q$ are unknown coefficient.

Boundary restrictions are given as

2.2.2. Coordination equation and analytical solution

Expanding ${\Phi }_{x_i}$, ${\Phi }_{y_i}$, $w_i$, $F$, $Q$, $q$ into Fourier series

in which ${\Phi }_{m{}_{i}n{}_i}$, ${\psi }_{m_i{n}_i}$, $w_{m_i{n}_i}$, $F_{m_2{n}_2}$, $q_{m_1{n}_1}$ and $Q_{m_i{n}_i}$ are unknown coefficients.

Taking into account the continuous differentiability of the formula on the boundary of the plate, we can write

where $a_{n_i}$, $b_{m_i}$, $c_{n_i}$, $d_{m_i}$, $e_{n_i}$, $f_{m_i}$, $g_{n_i}$ and $h_{m_i}$ are unknown coefficients. According to boundary conditions, we obtain the expressions

(1) Derived from $M_{x_i}=0$ on the edge $x_i=0$, we obtain the expression

(2) Derived from $M_{x_i}=0$ on the edge $x_i=a_i$, we obtain the expression

(3) Derived from $M_{y_i}=0$ on the edge $y_i=0$, we obtain the expression

(4) Derived from $M_{y_i}=0$ on the edge $y_i=b_i$, we obtain the expression

Substituting the unfolded Fourier form of ${\Phi }_{x_i}$, ${\Phi }_{y_i}$ and $w$ into the Eq. (17), the expressions are given as

(22a)

(22b)

(22c)

(22d)

(22e)

(22f)

Considering the same displacement of upper and lower plate at the contacting position, thus the deflection equations of the upper and lower plate could be expanded into double cosine series as

(22)

The deformation coordination equation between the upper and the lower plates is expressed as

The deformation coordination equation between the lower plate and the foundation is given as

(23)

Through the Eqs. (21), (22), (22) and (23), the undetermined coefficients $m_i$, $a_{n_i}$, $b_{m_1}$, $e_{n_1}$, $f_{m_1}$, ${\phi }_{m_i{n}_i}$, ${\psi }_{m_i{n}_i}$, $w_{m_i{n}_i}$, $F_{m_i{n}_i}$, $Q_{m_2{n}_2}$ could be simultaneously solved. Substituting the solved coefficients into related formulas, the subgrade reaction, deflection and internal force of the plate could be obtained.

2.2.3. Example

Recalculating the example in 2.1.3 according to the theory mentioned in this section, the results are given in Figure 3.

The calculated results in this paper are consistent with the results in [19], as shown in Table 3. This comparison proves the effectiveness of the theory proposed in this paper.

When the stepped rectangular plate is simultaneously analyzed using moderately thick plate theory and thin plate theory, the same method (as shown in 2.1 and 2.2) could be used to solve the bending moment and deflection formula of the plate.

3. Discussions

4. Conclusions

References

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