Revision as of 04:53, 23 February 2024

Abstract: Based on the Pasternak two-parameter foundation model, this study establishes a dynamic model for analyzing the deflection of slabs under moving loads. The variational method and reciprocal equal work theorem are employed to obtain the deflection solution, while the Matlab program is utilized with practical examples to calculate the foundation response modulus considering transfer shear force, bending moment, and their combined effects. By fitting measured dynamic deflections with theoretical predictions using the least square method, the response modulus K of the foundation is determined, demonstrating that our proposed model effectively captures the realistic behavior of rigid pavements.

Keywords: Pasternak model; Least squares method; Foundation reaction modulus

1 Introduction

With the rapid development of China's civil aviation industry, higher performance and safety requirements for airport pavement are inevitable. Currently, the commonly used foundation models include the E. Winkler foundation model and the elastic half-space foundation model. Although the Winkler foundation model theory is simple and intuitive, requiring only one elastic parameter to express ideal elastic foundation characteristics and relating displacement of any point solely to applied stress at that point, it fails to consider soil-ground continuity. In contrast, the elastic half-space foundation model overemphasizes stress diffusion between soil and ground, making calculations more cumbersome and impractical for engineering applications. Consequently, mechanics experts have developed several more reasonable foundation calculation models through extensive experimentation.Zhang Weigang and Chen Xiangming et al [1,2] utilized the Kelvin viscoelastic model under moving load conditions to obtain a relatively accurate foundation reaction modulus.The viscoelastic Winkler dynamic model verified by Xing Yaozhong[3] better reflects actual airport operations.V.Patil[4] employed Pasternak's two-parameter soil medium as a model to investigate material nonlinearity effects on pavement response in supporting soil medium.Yakshansh Kumar[5] applied a novel finite element-based cyclic response model for rigid pavement design while establishing the period range of moving load within rigid pavement parameters. Airport pavements consist of multiple concrete panels working collectively. This paper selects the Pasternak two-parameter foundation model based on independent parameters: foundation reaction modulus K and shear modulus G; thereby establishing a dynamic rigid pavement model considering joint load transfer capacity.

2 The formulation of dual-panel motion equations

2.1 The formulation of the equation of motion

Figure 1 The schematic of a two-parameter foundation model

Figure 2 The forces on the shear layer

Based on the two-parameter foundation, a dynamic model of rigid thin plate pavement on the two-parameter foundation is established, considering boundary shear and bending under moving load [6]. The dynamic equation of the two-parameter foundation incorporates two independent parameters: reaction modulus K and shear modulus G. Figure 1 illustrates the schematic diagram of the two-parameter foundation, while Figure 2 depicts the forces acting on the shear layer [7].

According to the Kirchhoff thin plate theory [8] and the theory of elastic mechanics, we establish the motion equation for plates subjected to a two-parameter foundation under dynamic loading as follows:

D — the flexural stiffness of the plate

µ —Poisson's ratio

${\textstyle M=\rho h}$ —The areal density

${\textstyle {\nabla }^{2}={\frac {{\partial }^{2}}{\partial {x}^{2}}}+{\frac {{\partial }^{2}}{\partial {y}^{2}}}}$ —The Laplace operator in two dimensions

K—Foundation reaction modulus

G—Foundation shear modulus

The aforementioned equation represents the differential equation of motion for the track panel system subjected to a general moving load, denoted as ${\textstyle P\left(x,y,t\right)}$ . In the context of the investigated form of moving load in this study, the equality ${\textstyle P\left(x,y,t\right)}$ can be expressed as follows:

The function ${\textstyle \delta \,}$ is represented by ${\textstyle \delta \left(.\right)}$ , and the uniform gliding speed of the aircraft is denoted by ${\textstyle {v}_{x}}$ .

2.2 Boundary conditions (shear and bending)

The plate dimensions are defined as length a and width b, with simultaneous transfer of shear force and bending moment between the plates in the actual working state. The boundary shear force is generated by the spring support force, where the elastic support rigidity coefficient ${\textstyle {k}_{1}\,}$ represents the boundary condition parameter. Similarly, the bending moment is generated by the spring-supporting moment, with the elastic supporting rigidity coefficient ${\textstyle {k}_{2}}$ representing another boundary condition parameter. Assuming that ${\textstyle Y=}$ $0,b$ corresponds to the side where the seam is located, we can express its general formula for boundary conditions as follows:

3 The solution to the equation of motion

3.1 Solving time factors

The variational formulation of equation (1) for plate motion is as follows:

The lateral load concentration stands out among them. ${\textstyle P=}$ $P\left(x,y,t\right)$ ， ${\textstyle w=w\left(x,y,t\right)}$

Select a function in the specified format as the solution for the variational equation mentioned above.

The mode function of the plate is represented by ${\textstyle W\left(x,y\right)}$ , and the coefficients of the deflection function ${\textstyle w\left(x,y,t\right)}$ and load ${\textstyle P\left(x,y,t\right)}$ are denoted as ${\textstyle T\left(t\right)}$ and ${\textstyle B\left(t\right)}$ respectively. The variation corresponding to the deflection function ${\textstyle w\left(x,y,t\right)}$ is as follows:

Substituting equations (5) and (6) into (4) gives:

In the formula:

The computable expression for T(t) is as follows:

In the equation, ${\textstyle {w}_{mn}}$ represents the natural frequency of undamped free vibration for a thin plate. The first term in equation (8) denotes the thin plate's free vibration, while the second term represents its forced vibration.

Among them, c1 and c2 are constants that depend on the initial conditions of motion, while B(t) is dependent on the load characteristics. In this study, the load is considered as a concentrated load P0, neglecting its mass. Consequently, B(t) can be expressed as follows:

According to formulas (8) and (9), the dynamic deflection of the plate under the influence of a moving concentrated load P, with zero initial conditions (c₁ =c₂), can be determined as follows:

3.2 Solving coordinate factors

In order to address the analytical solution for the bending behavior of diverse boundary plates subjected to moving loads, a four-sided simply supported rectangular plate with dimensions, geometry, and material properties that precisely match those of the actual system is considered as the fundamental model [9], as depicted in the figure 3 below.

Figure 3 Basic system of board

The plate is subjected to a unit concentration force at the flow coordinate ${\textstyle \left(\zeta ,\eta \right)}$ , and the displacement is solved using a trigonometric series.

{\textstyle {w}_{1}={\frac {4}{abD}}\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {\mathrm {sin} \,{k}_{m}\zeta \mathrm {sin} \,{k}_{n}\eta }{({{k}_{m}}^{2}+{{k}_{n}}^{2}{)}^{2}}}\mathrm {sin} \,{k}_{m}x\mathrm {sin} \,{k}_{n}y\quad \quad \quad \,(11)}

In the formula： ${\textstyle {k}_{m}={\frac {m\pi }{a}},\quad {k}_{n}={\frac {n\pi }{b}}}$

The solution ${\textstyle {w}_{1}}$ is referred to as the fundamental solution corresponding to the rectangular plate system.

The boundary condition can be simplified as a simply supported boundary by incorporating the corresponding support settlement and constraint moment under actual boundary conditions. The additional support settlement is assumed to be:

The additional constraint torque shall be determined as follows:

The reciprocal theorem of work between the fundamental system and the real system can be utilized to derive:

Set:

Substituting equations (12), (13) and (15) into (14) yields:

{A}_{mn}={\frac {{\frac {2}{a}}{k}_{m}\left\{D\left[{{k}_{m}}^{2}+\left(2-\mu \right){{k}_{n}}^{2}\right]\left({A}_{n}-{B}_{n}\mathrm {cos} \,m\pi \right)+\left({E}_{n}-{F}_{n}\mathrm {cos} \,m\pi \right)\right\}}{D{\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)}^{2}-\left[M{{w}_{mn}}^{2}-K-G\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)\right]}}

+{\frac {{\frac {2}{b}}{k}_{n}\left\{D\left[{{k}_{n}}^{2}+\left(2-\mu \right){{k}_{m}}^{2}\right]\left({C}_{m}-{D}_{m}\mathrm {cos} \,n\pi \right)+\left({G}_{m}-{H}_{m}\mathrm {cos} \,n\pi \right)\right\}}{D{\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)}^{2}-\left[M{{w}_{mn}}^{2}-K-G\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)\right]}}

Substituting ${\textstyle {A}_{mn}}$ into ${\textstyle W\left(x,y\right)}$ can obtain the mode function expression of the triangular series type:

W\left(x,y\right)=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {{A}_{n}\left[{{k}_{m}}^{2}+\left(2-\mu \right){{k}_{n}}^{2}\right]{\frac {2}{a}}{k}_{m}D}{D{\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)}^{2}-\left[M{{w}_{mn}}^{2}-K-G\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)\right]}}\mathrm {sin} \,{k}_{m}x\mathrm {sin} \,{k}_{n}y

\quad \quad \quad +\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {-{B}_{n}\mathrm {cos} \,m\pi \left[{{k}_{m}}^{2}+\left(2-\mu \right){{k}_{n}}^{2}\right]{\frac {2}{a}}{k}_{m}D}{D{\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)}^{2}-\left[M{{w}_{mn}}^{2}-K-G\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)\right]}}\mathrm {sin} \,{k}_{m}x\mathrm {sin} \,{k}_{n}y

\quad \quad \quad +\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {{C}_{m}\left[{{k}_{m}}^{2}+\left(2-\mu \right){{k}_{n}}^{2}\right]{\frac {2}{b}}{k}_{n}D}{D{\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)}^{2}-\left[M{{w}_{mn}}^{2}-K-G\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)\right]}}\mathrm {sin} \,{k}_{m}x\mathrm {sin} \,{k}_{n}y

\quad \quad \quad +\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {-{D}_{m}\mathrm {cos} \,n\pi \left[{{k}_{n}}^{2}+\left(2-\mu \right){{k}_{m}}^{2}\right]{\frac {2}{b}}{k}_{n}D}{D{\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)}^{2}-\left[M{{w}_{mn}}^{2}-K-G\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)\right]}}\mathrm {sin} \,{k}_{m}x\mathrm {sin} \,{k}_{n}y

\quad \quad \quad +\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {{E}_{n}{\frac {2}{a}}{k}_{m}}{D{\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)}^{2}-\left[M{{w}_{mn}}^{2}-K-G\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)\right]}}\mathrm {sin} \,{k}_{m}x\mathrm {sin} \,{k}_{n}y

\quad \quad \quad +\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {-{F}_{n}{\frac {2}{a}}{k}_{m}\mathrm {cos} \,m\pi }{D{\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)}^{2}-\left[M{{w}_{mn}}^{2}-K-G\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)\right]}}\mathrm {sin} \,{k}_{m}x\mathrm {sin} \,{k}_{n}y

\quad \quad \quad +\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {{G}_{m}{\frac {2}{b}}{k}_{n}}{D{\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)}^{2}-\left[M{{w}_{mn}}^{2}-K-G\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)\right]}}\mathrm {sin} \,{k}_{m}x\mathrm {sin} \,{k}_{n}y

\quad \quad \quad +\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {-{H}_{m}{\frac {2}{b}}{k}_{n}\mathrm {cos} \,n\pi }{D{\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)}^{2}-\left[M{{w}_{mn}}^{2}-K-G\left({{k}_{m}}^{2}+{{k}_{n}}^{2}\right)\right]}}\mathrm {sin} \,{k}_{m}x\mathrm {sin} \,{k}_{n}y

Set:

The mode function of the triangular series can only satisfy the condition that the boundary deflection is zero while ensuring homogeneity in the corresponding boundary shear force and bending moment. Therefore, ${\textstyle {w}_{1}}$ , ${\textstyle {w}_{2}}$ , ${\textstyle {w}_{3}}$ and ${\textstyle {w}_{4}}$ undergo hyperbolic transformations in one of the following directions. Similarly, ${\textstyle {w}_{5}}$ , ${\textstyle {w}_{6}}$ , ${\textstyle {w}_{7}}$ and ${\textstyle {w}_{8}}$ are transformed accordingly, We can get:

{w}_{1}=\sum _{n=1}^{\infty }{\frac {{A}_{n}}{\left({{\alpha }_{n}}^{2}-{{\beta }_{n}}^{2}\right)}}\left\{{\begin{matrix}\quad \left[{\frac {{{\alpha }_{n}}^{2}sh{\alpha }_{n}\left(a-x\right)}{sh{\alpha }_{n}a}}-{\frac {{{\beta }_{n}}^{2}sh{\beta }_{n}\left(a-x\right)}{sh{\beta }_{n}a}}\right]\\\ +\left(2-\mu \right){{k}_{n}}^{2}\left[-{\frac {sh{\alpha }_{n}\left(a-x\right)}{sh{\alpha }_{n}a}}+{\frac {sh{\beta }_{n}\left(a-x\right)}{sh{\beta }_{n}a}}\right]\end{matrix}}\right\}\mathrm {sin} \,{k}_{n}y.

{\textstyle {w}_{2}=\sum _{n=1}^{\infty }{\frac {{B}_{n}}{\left({{\alpha }_{n}}^{2}-{{\beta }_{n}}^{2}\right)}}\left\{{\begin{matrix}\quad \left[{\frac {{{\alpha }_{n}}^{2}sh{\alpha }_{n}x}{sh{\alpha }_{n}a}}-{\frac {{{\beta }_{n}}^{2}sh{\beta }_{n}x}{sh{\beta }_{n}a}}\right]\\\ +\left(2-\mu \right){{k}_{n}}^{2}\left[-{\frac {sh{\alpha }_{n}x}{sh{\alpha }_{n}a}}+{\frac {sh{\beta }_{n}x}{sh{\beta }_{n}a}}\right]\end{matrix}}\right\}\mathrm {sin} \,{k}_{n}y.}

{w}_{3}=\sum _{m=1}^{\infty }{\frac {{C}_{m}}{\left({{\alpha }_{m}}^{2}-{{\beta }_{m}}^{2}\right)}}\left\{{\begin{matrix}\quad \left[{\frac {{{\alpha }_{m}}^{2}sh{\alpha }_{m}\left(b-y\right)}{sh{\alpha }_{m}b}}-{\frac {{{\beta }_{m}}^{2}sh{\beta }_{m}\left(b-y\right)}{sh{\beta }_{m}b}}\right]\\\ +\left(2-\mu \right){{k}_{m}}^{2}\left[-{\frac {sh{\alpha }_{m}\left(b-y\right)}{sh{\alpha }_{m}b}}+{\frac {sh{\beta }_{m}\left(b-y\right)}{sh{\beta }_{m}b}}\right]\end{matrix}}\right\}\mathrm {sin} \,{k}_{m}x.

{w}_{4}=\sum _{m=1}^{\infty }{\frac {{D}_{m}}{\left({{\alpha }_{m}}^{2}-{{\beta }_{m}}^{2}\right)}}\left\{{\begin{matrix}\quad \left[{\frac {{{\alpha }_{m}}^{2}sh{\alpha }_{m}y}{sh{\alpha }_{m}b}}-{\frac {{{\beta }_{m}}^{2}sh{\beta }_{m}y}{sh{\beta }_{m}b}}\right]\\\ +\left(2-\mu \right){{k}_{m}}^{2}\left[-{\frac {sh{\alpha }_{m}y}{sh{\alpha }_{m}b}}+{\frac {sh{\beta }_{m}y}{sh{\beta }_{m}b}}\right]\end{matrix}}\right\}\mathrm {sin} \,{k}_{m}x.\quad \quad \quad

The coordinate factor after the aforementioned transformation not only satisfies the condition of zero boundary deflection but also fulfills equations (12) and (13). In this scenario, the actual system must adhere to boundary conditions (3). The substitution of ${\textstyle {B}_{n}}$ , ${\textstyle {C}_{m}}$ , ${\textstyle {D}_{m}}$ , ${\textstyle {E}_{n}}$ , ${\textstyle {F}_{n}}$ , ${\textstyle {G}_{m}}$ and ${\textstyle {H}_{m}}$ into equation (3) yields eight equations, the solutions of which provide the corresponding values. Substituting these values into equation (17) determines the coordinate factor ${\textstyle W\left(x,y\right)}$

By substituting the coordinate factor ${\textstyle W\left(x,y\right)}$ into equation (10), one can derive the expression for ${\textstyle W\left(x,y,t\right)}$ and obtain an analytical formula for the deflection of the pavement panel.

4 The identification of foundation modulus.

According to the analytical expression of deflection, the deflection value of the plate can be determined when the reaction modulus K of the foundation is fixed. Conversely, if the measured deflection at each measuring point is known, a fitting process can be conducted by establishing an objective function to compare and match the measured dynamic deflection with theoretical dynamic deflection. The program [10] is implemented using Matlab for identifying the response modulus of the foundation. During parameter optimization, an objective function is formulated based on the least squares criterion:

Among them, ${\textstyle \mathrm {min} \,\epsilon (K)}$ For the objective function, ${\textstyle {w}_{i}}$ and ${\textstyle {\overline {{w}_{i}}}}$ represents the theoretically calculated and measured deflection values of the measurement points, respectively, and n represents the number of measurement points on the pavement. Adjust K one by one to minimize the objective function, and the K value at this point is the identified foundation response modulus.

5 Verification of the case

The model and inversion recognition method presented in this paper was utilized to compute the pavement surface of an airport located in Nanjing, with geometric dimensions of a=6m, b=4m, elastic modulus ${\textstyle E=}$ $3.5\times 1{0}^{4\,}MPa$ , Poisson ratio ${\textstyle \mu =}$ $0.167$ , density ${\textstyle \rho =2500\,{\frac {kN}{{m}^{2}}}}$ , thickness ${\textstyle h=}$ $15.75mm$ . Additionally considered were foundation shear modulus ${\textstyle G=}$ $35.17{\frac {N}{c{m}^{3}}}$ , rigidity coefficients ${\textstyle k1=1.5706\times 1{0}^{4}{\frac {kN}{m}}}$ , ${\textstyle k2=1.5830\times 1{0}^{4}{\frac {kN}{m}}}$ as well as loading equipment for aircraft （ ${\textstyle {P}_{0}}$ =215000N）

When the velocity reaches 2.59 m/s and y=1.1m, Table 1 presents both the measured and theoretical deflection values, while Figure 4 illustrates the fitting curve.

When the velocity reaches 2.79 m/s and y=0.68 m, Table 2 presents both the measured and theoretical deflection values, while Figure 5 illustrates the corresponding fitting curve.

Table 1 Inversion Results of Foundation Modulus

coordinate	0.0	0.5	1.0	1.5	2.0	2.5	3.0
Measured deflection(mm)	0.395	0.365	0.335	0.270	0.195	0.115	0.050
Theoretical deflectio(mm)	0.402	0.388	0.349	0.285	0.201	0.104	0.073
K(N/cm3)	58.33

Table 2 Inversion Results of Foundation Modulus

坐标	0.0	0.5	1.0	1.5	2.0	2.5	3.0
Measured deflection(mm)	0.5	0.47	0.397	0.315	0.225	0.145	0.071
Theoretical deflection(mm)	0.485	0.469	0.420	0.343	0.242	0.127	0.060
K(N/cm3)	58.40

Figure 4 Comparison curve between theoretical deflection and measured deflection

Figure 5 Comparison curve between theoretical deflection and measured deflection

6 Conclusion

(1)The present study establishes a dynamic model for rigid thin plate pavement, considering boundary shear and bending under moving loads, based on the dual parameter foundation. By employing methods such as the variational method and reciprocal work theorem, an analytical solution is obtained for the deflection of finite-size rectangular plates;

(2)The least squares principle is employed to invert and identify the structural parameters of the rigid track panel based on the measured dynamic deflection. By comparing the theoretical deflection values with the measured deflection curves under the simultaneous transmission of bending moment and shear force at the boundary, it is verified that the dynamic model established in this study aligns more closely with real-world conditions while ensuring stability and convergence of the parameter inversion method;

(3)The dynamic calculation model proposed in this study provides a more precise representation of the operational condition of rigid pavement, thereby enabling the application of identified parameters in pavement overlay design.

Acknowledgments

The authors would like to thank The Youth Innovation Team of Shaanxi Universities(Key Technology Innovation Team for Urban Rail Transit Track Bed);Youth fund project at Xi’an Jiaotong Engineering Institute(2023KY-39) and The Scientific Research Program Funded by Education Department of Shaanxi Provincial Government (Program No.23JK0532).Hui-min Cao and Han-jun Zhao contributed equally to this work.

Reference：

[1] Zhang Wei-gang, Chen Xiang-ming, and Zuo Xing. Research on Airport Rigid Pavement Parameter Identification on Dynamic Loading [J]. Computer Simulation, 2012, 29 (09):73-76+202.(in chinese)

[2] Chen Xiang-ming, Liu Hong-bing. Dynamic Identification Simulation of Foundation Response Modulus of Airport [J]. Computer Simulation, 2009,26 (07):54-57(in chinese)

[3] Xing Yao-zhong, Liu Hong-bing. Foundation Parameter Identification Research of Airport Rigid Pavement [J].Central South Highway Engineering, 2006, (04):52-54.(in chinese)

[4] V. Patil, V. Sawant & Kousik Deb (2012) Finite element analysis of rigid pavement on a nonlinear two parameter foundation model, International Journal of Geotechnical Engineering, 6:3, 275-286, DOI: 10.3328/IJGE.2012.06.03.274-286

[5] Kumar, Y., Trivedi, A. & Shukla, S.K. Deflections Governed by the Cyclic Strength of Rigid Pavement Subjected to Structural Vibration Due to High-Velocity Moving Loads. J. Vib. Eng. Technol. (2023).

[6] Taheri M R, Zaman M M . Effects of a moving aircraft and temperature differential on response of rigid pavements[J].Computers & Structures, 1995, 57(3):503-511.DOI:10.1016/0045-7949(94)00625-D.

[7] Worku A, Lulseged A. Kinematic pile-soil interaction using a rigorous two-parameter foundation model[J]. Soil Dynamics and Earthquake Engineering, 2023, 165: 107701.

[8] Li C J, Zhang Y, Jia Y M, et al. The polygonal scaled boundary thin plate element based on the discrete Kirchhoff theory[J]. Computers & Mathematics with Applications, 2021, 97: 223-236.

[9] Alisjahbana S W, Wangsadinata W. Dynamic response of rigid roadway pavement under moving traffic loads[C]//The 8th International Structural Engineering and Construction Conference, Sydney, Australia. 2015.

[10] Stewart H , Aburn M J , Michael B .The Brain Dynamics Toolbox for Matlab[J].Neurocomputing, 2018, 315(NOV.13):82-88.DOI:10.1016/j.neucom.2018.06.026.

@@ Line 19: / Line 19: @@
 ===1 Introduction===
-With the rapid development of China's civil aviation industry, higher performance and safety requirements for airport pavement are inevitable. Currently, the commonly used foundation models include the E. Winkler foundation model and the elastic half-space foundation model. Although the Winkler foundation model theory is simple and intuitive, requiring only one elastic parameter to express ideal elastic foundation characteristics and relating displacement of any point solely to applied stress at that point, it fails to consider soil-ground continuity. In contrast, the elastic half-space foundation model overemphasizes stress diffusion between soil and ground, making calculations more cumbersome and impractical for engineering applications. Consequently, mechanics experts have developed several more reasonable foundation calculation models through extensive experimentation. Zhang Weigang and Chen Xiangming et al<span id='citeF-1'></span> <span id='citeF-2'></span>[[#cite-1|[1,2]]] utilized the Kelvin viscoelastic model under moving load conditions to obtain a relatively accurate foundation reaction modulus . The viscoelastic Winkler dynamic model verified by Xing Yaozhong<span id='citeF-3'></span>[[#cite-3|[3]]] better reflects actual airport operations. V. Patil<span id='citeF-4'></span>[[#cite-4|[4]]] employed Pasternak's two-parameter soil medium as a model to investigate material nonlinearity effects on pavement response in supporting soil medium. Yakshansh Kumar<span id='citeF-5'></span>[[#cite-5|[5]]] applied a novel finite element-based cyclic response model for rigid pavement design while establishing the period range of moving load within rigid pavement parameters. Airport pavements consist of multiple concrete panels working collectively. This paper selects the Pasternak two-parameter foundation model based on independent parameters: foundation reaction modulus K and shear modulus G; thereby establishing a dynamic rigid pavement model considering joint load transfer capacity.
+With the rapid development of China's civil aviation industry, higher performance and safety requirements for airport pavement are inevitable. Currently, the commonly used foundation models include the E. Winkler foundation model and the elastic half-space foundation model. Although the Winkler foundation model theory is simple and intuitive, requiring only one elastic parameter to express ideal elastic foundation characteristics and relating displacement of any point solely to applied stress at that point, it fails to consider soil-ground continuity. In contrast, the elastic half-space foundation model overemphasizes stress diffusion between soil and ground, making calculations more cumbersome and impractical for engineering applications. Consequently, mechanics experts have developed several more reasonable foundation calculation models through extensive experimentation.Zhang Weigang and Chen Xiangming et al<span id='citeF-1'></span> <span id='citeF-2'></span>[[#cite-1|[1,2]]] utilized the Kelvin viscoelastic model under moving load conditions to obtain a relatively accurate foundation reaction modulus.The viscoelastic Winkler dynamic model verified by Xing Yaozhong<span id='citeF-3'></span>[[#cite-3|[3]]] better reflects actual airport operations.V.Patil<span id='citeF-4'></span>[[#cite-4|[4]]] employed Pasternak's two-parameter soil medium as a model to investigate material nonlinearity effects on pavement response in supporting soil medium.Yakshansh Kumar<span id='citeF-5'></span>[[#cite-5|[5]]] applied a novel finite element-based cyclic response model for rigid pavement design while establishing the period range of moving load within rigid pavement parameters. Airport pavements consist of multiple concrete panels working collectively. This paper selects the Pasternak two-parameter foundation model based on independent parameters: foundation reaction modulus K and shear modulus G; thereby establishing a dynamic rigid pavement model considering joint load transfer capacity.
 ===2 The formulation of dual-panel motion equations===