Mechanical Analysis of Semi-Rigid Base Asphalt Pavement under the Influence of Groundwater with the Spectral Element Method

Abstract

Over prolonged exposure to groundwater conditions, semi-rigid base asphalt pavements can undergo significant changes in their internal moisture field, resulting in substantial variations in the pavement’s stiffness and, consequently, affecting the overall load-bearing capacity and stability of the road structure. This paper employs FWD non-destructive testing equipment to assess its mechanical performance and conduct data analysis and conducts a mechanical response study of asphalt road surfaces, considering the influence of roadbed moisture levels. Using the dynamic analytical theory, the fundamental equations and stiffness matrices for a linear elastic half-space model were established, leading to the development of a computational model for the mechanical response of semi-rigid base asphalt pavements under FWD dynamic loading, with an examination of the surface deflection in relation to changes in groundwater levels. Numerical examples and engineering applications were employed to validate the proposed model. The research findings indicate: With the passage of time, surface deflection values initially increase and then decrease, exhibiting a sinusoidal variation pattern similar to that of the applied load. As the distance from the loading center increases, the moment of peak deflection continually lags behind. The average absolute relative error between the results obtained using the method proposed in this study and the traditional ABAQUS finite element method was only 0.70%. The correlation coefficient between the theoretically computed deflection curve and the measured deflection curve using the spectral element method was greater than 0.9, with an average absolute relative error of 4.92% between the theoretical peak deflection and the measured peak deflection. As the groundwater level rises, surface deflection noticeably increases, with an approximately 40% increase in deflection values at the loading center. These research findings can be utilized to analyze the dynamic deflection of semi-rigid base asphalt pavements under various groundwater conditions, providing significant practical value for assessing road structural performance and serviceability.

1. Introduction

In recent years, China’s highway construction has experienced rapid development, with the scale and quantity of construction growing at a remarkable pace. As of the end of 2022, the total length of highways in China has reached 5.3548 million kilometers, including 177,300 km of expressways. Among these pavement structures, semi-rigid base asphalt pavements constitute the most widely used type [1]. The combined effects of dynamic loading and external water environments on semi-rigid base asphalt pavements can significantly impact their load-bearing capacity and material stability. This can lead to various types of water-related damage, such as potholes, delamination, and subsidence, severely affecting the operation and normal use of highways. As a result, relevant road maintenance departments must allocate substantial resources in terms of funding and manpower for maintenance and repairs [2]. The emergence of these situations has imposed higher demands on asphalt pavement testing equipment, road maintenance techniques, and performance evaluation systems. Under the influence of groundwater, the internal humidity of roadbeds may undergo significant changes over time, and the rebound modulus of roadbed materials may continuously deteriorate. This directly affects the overall mechanical characteristics and service life of asphalt road surfaces. Therefore, the rapid and accurate assessment of the performance of semi-rigid base asphalt road surfaces, the exploration of mechanical response patterns within road structures under water environments, the identification of hidden defects, and their subsequent repair have become crucial issues in current engineering research. Currently, there is a lack of rigorous mechanical models for asphalt road surfaces under dynamic loading, which overlook the impact of changes in roadbed moisture levels on pavement structural performance. Consequently, it is challenging to accurately evaluate the dynamic mechanical response of asphalt road structures affected by groundwater.

The rapid development of highway construction in China has significantly driven the research, development, and application of road detection technologies and equipment. Both domestic and international efforts have led to the development of a series of relevant technologies and equipment. Road surface inspection equipment has undergone a transition from static to dynamic testing devices. The FWD (Falling Weight Deflectometer) is one of the most typical devices in this regard and has been widely adopted for the comprehensive assessment of road bearing capacity and quality when conducting tests and evaluations. This method offers not only high testing speed but also minimal susceptibility to human factors, resulting in high precision in measurements. It has garnered increasing attention from inspection authorities. FWD-measured deflection data can reflect the stress and deformation states of road structures and, at the same time, can be used to back-calculate crucial mechanical parameters of road materials, providing significant references for road design [3]. Currently, research in the field of mechanical response of asphalt road surfaces under load focuses on two main analysis methods: analytical methods and numerical methods.

Analytical methods are primarily based on fundamental dynamic equations. These methods involve the derivation of mechanical models for asphalt road surfaces using various complex variable functions and integral transformations, leading to theoretical solutions. Zafir et al. [4], based on a continuous finite layer model and considering the pavement structure as a half-space problem, conducted a research analysis of the dynamic response patterns of pavement structures under the influence of moving loads. Grundmann et al. [5,6] studied the dynamic response changes of a half-space structural model under dynamic loads, and then obtained the integral inverse transform solution using wavelet decomposition. Zhou et al. [7] studied the steady-state response solutions of the elastic half-space system under moving loads and obtained the general integral form solution using Fourier integral transformation. Shan et al. [8,9,10,11] studied surface bending under moving loads in elastic layered systems, combining integral transformation with the generalized Duhamel integral to obtain a dynamic analytical solution, and developed a computing program to address more complex Gaussian integral inverse transformation problems. Compared to numerical methods, analytical methods offer simplicity in calculations and have the potential to significantly reduce computational workload and time requirements [12].

In addition to analytical methods, numerical methods represent another commonly used approach in the analysis of pavement mechanics response. Al-Qadi et al. [13] analyzed the dynamic response of flexible asphalt pavements under FWD loading using the finite element method. Ahmed et al. [14] used ABAQUS to analyze the response of transversely isotropic viscoelastic asphalt surface layers under non-uniform loading and further studied the influence of temperature on their mechanical response. Chen et al. [15,16] constructed a multilayer asphalt pavement structure model using ANSYS (v12.0) finite element numerical simulation software and studied the dynamic response of multilayer asphalt pavement structures under dynamic loading. Lu et al. [17,18,19] conducted a study and analysis of the nonlinear dynamic response of asphalt pavements under external loads using 3D nonlinear finite element numerical simulation software (Abaqus 6.14). Tao et al. [20] considered the effects of rainfall and groundwater on the roadbed, established an ABAQUS three-dimensional model, and analyzed the influence of humidity on the dynamic response of the pavement structure. Wu [21] established a three-dimensional finite element calculation model under the coupling of driving load and water and analyzed the influence of pore dynamic water pressure on the existing cracks in the asphalt pavement structural surface layer.

Both of the aforementioned methods for analyzing the dynamic response of pavement structures have certain limitations. Analytical methods tend to encounter singular points during the integration process, limiting their applicability to relatively simple pavement models. Therefore, analytical methods may not be universally applicable when dealing with complex model calculations. Numerical methods generally require detailed pavement modeling during the solution process. The accuracy of mesh division in the preprocessing stage is vital, as it directly impacts the calculation results, often leading to a significant time investment. The spectral element method is a mechanical response analytical approach that combines spectral analysis with finite element computation. This method integrates the dynamic control equations of structures with the Helmholtz displacement potential function using integral transformations to obtain solutions in the transformed domain, simplifying the complex solving process of conventional equations. Doyle [22] was the first to systematically propose a mechanical response analytical solution analysis method combining spectral analysis and finite element calculation, naming it the spectral element method. He combined the dynamic control equations of the structure with the Helmholtz displacement potential function, using integral transformation to obtain solutions in the transformation domain. Wu et al. [23,24,25] applied the spectral element method to the vicinity of roads, solving for the dynamic response solution of road layered structures under impact loads, using unconstrained nonlinear optimization methods and BP networks for modulus inversion. The spectral element method allows grouping layers of the same type within a structure as a single unit during dynamic analysis, reducing the number of elements significantly compared to finite element methods and thereby enhancing computational efficiency in dynamic analysis [26,27,28]. It also offers several advantages over conventional mechanical analysis methods, including faster computation, higher accuracy in results, greater performance efficiency, and overall precision [29,30].

This paper establishes a dynamic response analysis model for semi-rigid base asphalt pavement based on the theory of the spectral element method. A computational analysis software for dynamic deflection has been developed through extensive research. This method can accurately assess and monitor the mechanical performance of pavement structures under the influence of groundwater, providing a scientific basis for pavement design, maintenance, and management decisions. It aims to improve the quality, performance, and economic efficiency of road pavements. The research results have been successfully applied to the analysis of critical engineering indicators affected by humidity.

2. Fundamental Equation of the Spectral Element Method

The spectral element method calculates the dynamic deflection of pavements by performing Fourier transforms on the dynamic response control equations of structures, and then uses a combination of spectral analysis and finite element methods to solve them in the frequency domain. This yields displacement results for computational points in the frequency domain. The displacement time history curves in the time domain can then be obtained through the inverse Fourier transform. This article adopts a model of a layered road structure under axisymmetric conditions with isotropic properties. It assumes that each layer is a continuous, homogeneous, and isotropic elastic material that extends infinitely in the horizontal and vertical downward directions.

2.1. Governing Equation

The governing equations describe the mechanical behavior of elastic body structures under external forces through balance equations, geometric equations, and physical equations, reflecting the interrelationship among internal stresses, strains, and displacements [31,32].

The dynamic balance equation under the cylindrical coordinate system is as follows:

$$\left\{\begin{array}{c}\frac{\partial {\sigma }_r}{\partial r}+\frac{\partial {\tau }_{rz}}{\partial z}+\frac{{\sigma }_r-{\sigma }_{\theta }}{r}=\rho \frac{{\partial }^{2}u}{\partial t^2}\\ \frac{\partial {\sigma }_z}{\partial z}+\frac{\partial {\tau }_{rz}}{\partial r}+\frac{{\tau }_{rz}}{r}=\rho \frac{{\partial }^{2}w}{\partial t^2}\end{array}\right.$$

(1)

where $u$ is the radial displacement, and $w$ is the vertical displacement; ${\sigma }_r$ represents the radial stress, ${\sigma }_{\theta }$ is the tangential stress, and ${\sigma }_z$ is the vertical stress; ${\tau }_{rz}$ is the shear stress; $\rho$ is the density of the object.

The geometric equation is as follows:

$$\left\{\begin{array}{c}{\epsilon }_r=\frac{\partial u}{\partial r}\\ {\epsilon }_{\theta }=\frac{u}{r}\\ {\epsilon }_z=\frac{\partial w}{\partial z}\\ {\gamma }_{rz}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial r}\end{array}\right.$$

(2)

in which ${\epsilon }_r$, ${\epsilon }_{\theta }$, and ${\epsilon }_z$ denote the radial strain, tangential strain, and vertical strain; ${\gamma }_{rz}$ represents shear strain.

The physical equation is as follows:

$$\left\{\begin{array}{c}{\sigma }_r=2G{\epsilon }_r+\lambda e\\ {\sigma }_{\theta }=2G{\epsilon }_{\theta }+\lambda e\\ {\sigma }_z=2G{\epsilon }_z+\lambda e\\ {\tau }_{rz}=G{\gamma }_{rz}\end{array}\right.$$

(3)

where $e={\epsilon }_r+{\epsilon }_{\theta }+{\epsilon }_z$ represents volumetric strain; $\lambda =2vG/\left(1-2v\right)$, $G=0.5E/\left(1+v\right)$; $G$, $E$, and $v$ represent shear modulus, elastic modulus, and Poisson’s ratio, respectively.

2.2. Potential Energy Equation of Displacement Field

According to Helmholtz’s theorem, the displacement field can be represented by the sum of scalar potential $\phi$ and vector potential $\delta$ [33,34], as shown below:

$$\overrightarrow{u}=\nabla \phi +\nabla \overrightarrow{\delta }$$

(4)

The axially symmetric vector potential $\delta$ only has a single component ${\delta }_{\theta }$. In this study, $u$ represents the displacement in the $r$ direction, $w$ represents the displacement in the $z$ direction, and the relationship between displacement and potential is as follows:

$$u=\frac{\partial \phi }{\partial r}-\frac{\partial \delta }{\partial z}$$

(5)

$$w=\frac{\partial \phi }{\partial z}+\frac{1}{r}\frac{\partial r\delta }{\partial r}$$

(6)

By substituting Equations (5) and (6) into Equation (2), and combining Equations (1) and (3), we obtain:

$$\frac{{\partial }^2\phi }{\partial r^2}+\frac{1}{r}\frac{\partial \phi }{\partial r}+\frac{{\partial }^2\phi }{\partial z^2}=\frac{1}{c_p^2}\frac{{\partial }^2\phi }{\partial t^2}$$

(7)

$$\frac{{\partial }^2\delta }{\partial r^2}+\frac{1}{r}\frac{\partial \delta }{\partial r}+\frac{{\partial }^2\delta }{\partial z^2}-\frac{\delta }{r^2}=\frac{1}{c_s^2}\frac{{\partial }^2\delta }{\partial t^2}$$

(8)

where $c_p={\left(\left(\lambda +2G\right)/\rho \right)}^{1/2}$ and $c_s={\left(G/\rho \right)}^{1/2}$ represent the compression and shear wave velocities, respectively.

Applying a Fourier transform to Equations (7) and (8) can convert the expressions from the time domain to the frequency domain for further solution:

$$\frac{{\partial }^2\tilde{\phi }}{\partial r^2}+\frac{1}{r}\frac{\partial \tilde{\phi }}{\partial r}+\frac{{\partial }^2\tilde{\phi }}{\partial z^2}=-\frac{1}{c_p^2}{\omega }^2\tilde{\phi }$$

(9)

$$\frac{{\partial }^2\tilde{\delta }}{\partial r^2}+\frac{1}{r}\frac{\partial \tilde{\delta }}{\partial r}+\frac{{\partial }^2\tilde{\delta }}{\partial z^2}-\frac{\tilde{\delta }}{r^2}=-\frac{1}{c_s^2}{\omega }^2\tilde{\delta }$$

(10)

Since the function $\tilde{\phi }$ is related to $r$ and $z$, the function can be constructed by multiplying two separate functions $r$ and $z$.

$$\tilde{\phi }=\tilde{R}•\tilde{Z}$$

(11)

In this equation, the function $\tilde{R}$ and the function $\tilde{Z}$ represent those related only to $r$ and $z$, respectively.

By substituting Equation (11) into Equation (9) and simplifying both sides, two independent ordinary differential equations in terms of $\tilde{R}$ and $\tilde{Z}$ are derived, as follows:

$$\frac{{\mathrm{d}}^2\tilde{R}}{\mathrm{d}{r}^2}+\frac{1}{r}\frac{\mathrm{d}\tilde{R}}{\mathrm{d}r}+k^2\tilde{R}=0$$

(12)

$$\frac{{\mathrm{d}}^2\tilde{Z}}{\mathrm{d}{z}^2}+\left(\frac{{\omega }^2}{c_p^2}-k^2\right)\tilde{Z}=0$$

(13)

Assuming, and using the chain rule, Equation (12) can be transformed into a Bessel equation, as shown below:

$$s^2\frac{{\mathrm{d}}^2{\tilde{R}}_s}{\mathrm{d}{s}^2}+s\frac{\mathrm{d}{\tilde{R}}_s}{\mathrm{d}s}+s^2{\tilde{R}}_s=0$$

(14)

The solution form of Equation (14) consists of a sum of the first kind Bessel function $J_0$ and the second kind Bessel function $Y_0$. When $r$ equals zero, $Y_0$ is singular, which means it will go to infinity. However, because the oscillation at the origin is not infinite, the solution of $Y_0$ should be discarded. The function for ${\tilde{R}}_r$ can be simplified as:

$${\tilde{R}}_r=A_1{J}_0\left(s\right)=A_1{J}_0\left(kr\right)$$

(15)

in which $A_1$ is a constant that can be determined by boundary conditions; $k$ indicates the radial wavenumber.

For the purpose of obtaining a discretized solution, it is necessary to introduce a boundary condition. This paper assumes that at the boundary where $r=R$, the oscillation amplitude completely vanishes. Therefore, at $r=R$, Equation (15) becomes:

$${\tilde{R}}_R=J_0\left(kR\right)=0$$

(16)

Assuming that the m-th root of the Bessel function J₀ is ${\beta }_m$, and as stated in equation $kR={\beta }_m$, thus $k=k_m={\beta }_m/R$. Therefore, for the m-th root ${\beta }_m$, the equation is:

$${\tilde{R}}_{mr}=A_{1m}{J}_0\left(k_{m}r\right)=A_{1m}{J}_0\left(\frac{{\beta }_m}{R}r\right)$$

(17)

From Equation (17), it can be seen that the function has infinitely many solutions, but because the $J_0$ function has an oscillating decay characteristic, a finite number of solutions can be used to represent the approximate solution of the function. Similarly, the solution to Equation (13) is:

$${\tilde{Z}}_{mn}=A_{2m}{e}^{-iz{k}_{pzmn}}$$

(18)

Consider $A_{2m}$ to represent a constant, which can be determined according to boundary conditions; $i$ to represent $\sqrt{-1}$; and $k_{pzmn}$ to represent the wavenumber in the direction of $z$. The relationship among these variables can be encapsulated in the following equation:

$$k_{pzmn}={\left(\frac{{\omega }_n^2}{c_p^2}-k_m^2\right)}^{1/2}$$

(19)

In this context, $c_p$ is the compression wave speed, and ${\omega }_n$ is the angular frequency.

The solution to Equation (11) at any angular frequency ${\omega }_n$ is:

$${\tilde{\phi }}_{mn}=A_{mn}{e}^{-iz{k}_{pzmn}}{J}_0\left(k_{m}r\right)$$

(20)

where $A_{mn}$ is a constant that can be determined according to boundary conditions.

Similarly, the solution to Equation (10) can be calculated as shown below:

$${\tilde{\delta }}_{mn}=B_{mn}{e}^{-iz{k}_{szmn}}{J}_1\left(k_{m}r\right)$$

(21)

3. Stiffness Matrix of Spectral Element Method

The forward analysis algorithm forms the basis for pavement modulus inversion, and it employs the spectral element method for dynamic elastic mechanical response analysis of asphalt pavement. In their seminal work, Doyle [22] proposed the fundamental structural form of spectral elements. During the calculation and analysis process, two types of spectral elements are used to partition the pavement structure. Dual-node spectral elements simulate the finite thickness of the pavement structure layer, while single-node spectral elements simulate the infinite depth of the subgrade layer. The schematic diagrams for these two types of spectral elements are shown in Figure 1. In this method, a single spectral element can accurately describe the distribution of inertial forces within a pavement structure layer [35]. Therefore, the number of elements required is equal to the number of layers in the pavement structure. Compared to the finite element method, this not only significantly reduces the number of element divisions, improving computational efficiency, but also yields precise results.

The single-node spectral element is a specific instance of the two-node spectral element and exhibits semi-infinite characteristics. When an incident wave propagates within the single-node spectral element, the wave’s energy gradually dissipates along its path until it diminishes completely. In the case of a single-node spectral element, the incident wave encounters no boundaries during its propagation, resulting in the absence of reflected waves. By utilizing the solution process that relies on the stiffness matrix of a two-node spectral element, it can derive the stiffness matrix of a one-node spectral element. It can be shown as Equation (22).

$$\left[\begin{array}{c}{\tilde{T}}_{rmn1}\\ {\tilde{T}}_{zmn1}\end{array}\right]=\frac{\mu }{p_{mn}}\left[P\right]\left[\begin{array}{c}{\tilde{u}}_{mn1}\\ {\tilde{w}}_{mn1}\end{array}\right]$$

(22)

4. Time Domain Response Solution of Mechanical Model

Similar to the finite element method, combining the above theory, the element stiffness matrix formed in the given frequency-wavenumber domain in the spectral element method analysis can be determined by the boundary conditions and interlayer continuity conditions of the pavement structure. Subsequently, the stiffness matrix and boundary moment matrix of each spectral element are assembled in the same manner as the finite element stiffness matrix. Following this, the global stiffness matrix, the global boundary force matrix, and the global node displacement matrix are combined into a system of linear equations. Finally, by solving the system of linear equations, the response solution in the frequency domain is obtained, and the response solution in the time domain is obtained through the Fourier inverse transform. Equation (23) provides the equations for the overall stiffness matrix, overall displacement matrix, and overall nodal force matrix.

$$\tilde{F}=\left[\begin{array}{c}{\tilde{T}}_{rmn1}\\ {\tilde{T}}_{zmn1}\\ \cdots \\ {\tilde{T}}_{rmni}\\ {\tilde{T}}_{zmni}\end{array}\right]=\left[K\right]\left[\begin{array}{c}{\tilde{u}}_{mn1}\\ {\tilde{w}}_{mn1}\\ \cdots \\ {\tilde{u}}_{mni}\\ {\tilde{w}}_{mni}\end{array}\right]={\tilde{K}}_{mn}{\tilde{X}}_{mn}$$

(23)

where $i$ represents the nodes of the spectral element, ${\tilde{K}}_{mn}$ is the overall stiffness matrix, $\tilde{F}$ is the overall nodal force matrix, and ${\tilde{X}}_{mn}$ represents the overall displacement matrix.

The overall displacement in the time domain can be obtained by summing and applying the Fourier inverse transformation, as shown in Equations (24) and (25):

$$u={\displaystyle \sum }_{n=1}^N{\displaystyle \sum }_{m=1}^M{\tilde{u}}_{mn}{\tilde{F}}_n{F}_m{J}_0(k_{m}r)e^{i\omega t}$$

(24)

$$w={\displaystyle \sum }_{n=1}^N{\displaystyle \sum }_{m=1}^M{\tilde{w}}_{mn}{\tilde{F}}_n{F}_m{J}_0(k_{m}r)e^{i\omega t}$$

(25)

5. Dynamic Mechanical Response Analysis of Asphalt Pavement under the Influence of Groundwater

Therefore, based on multilayer elasticity theory, this paper employs the spectral element method to consider factors such as the elastic modulus, thickness, and humidity of asphalt pavement materials. It performs a dual summation over a limited range of wave numbers and frequencies, obtaining a frequency-domain solution represented by coefficients. This study fully considers the influence of groundwater on the moisture content of the subgrade, and then analyzes the law of road surface deflection value changing with the groundwater level. This can be used to estimate the deflection and stress levels of the pavement, achieving the goal of rapidly and accurately analyzing the mechanical response of asphalt pavement under the influence of groundwater. This study compares the dynamic deflection calculation results obtained with the results from the Abaqus (6.14) finite element software. We analyze and evaluate the feasibility and accuracy of the proposed dynamic response calculation method. This article evaluates the applicability of using SEM to analyze the mechanical response of semi-rigid base asphalt pavement under the influence of humidity.

5.1. Essential Parameter

In typical semi-rigid asphalt pavement structures in China, there are three layers, and the parameters for each layer are shown in

Table 1. Parameters’ configuration for semi-rigid base pavement.

Pavement Structure Layers	Modulus (MPa)	Thickness (cm)	Poisson’s Ratio	$\mathit{\rho }$ (kg/m3)
Surface	1200	18	0.35	2400
Base	1400	32	0.30	2300
Subgrade	60	$\infty$	0.40	1800

. The thicknesses of the layers in the road structure are 18 cm, 38 cm, and infinite, respectively. The layout of the related structural layers is shown in Figure 2. For the analysis of pavement structure dynamic response, this paper utilized the impulse load from the Dynatest 8000 model FWD. To facilitate computation, the FWD load was simplified to a half-cycle sine function with a peak value of 714 KPa, a duration of 30 ms, and an applied load radius of 0.15 m.

The simplified FWD pulse load-time history curve is shown in Figure 3.

The arrangement of FWD sensors is shown in

Table 2. Arrangement of Dynatest 8000 FWD sensors.

Sensor Number	D1	D2	D3	D4	D5	D6	D7	D8	D9
Distance from Load Center (mm)	0	203	305	457	610	914	1219	1574	1829

.

5.2. Finite Element Modeling of Asphalt Pavement

In the construction of a road model utilizing ABAQUS finite element software, complete continuity is assumed across all layers. The model is defined with geometric dimensions of 7.5 m in width and 15 m in length, and a Falling Weight Deflectometer (FWD) dynamic load is applied at the central top surface. Boundary conditions are configured to restrict displacements in the X and Y directions, with the base of the model being fully fixed. The semi-rigid base asphalt pavement’s finite element model is illustrated in Figure 4. An eight-node linear hexahedral element with reduced integration for stress analysis (C3D8R, Dassault System, Inc., Providence, RI, USA) was chosen, featuring a grid division ratio coefficient of 0.2; the detailed grid division is presented in Figure 5.

5.3. Analysis of Asphalt Pavement Deflection Calculation

Asphalt is a road-binding material with extremely high temperature sensitivity. Therefore, this study fully considers the impact of temperature on the deflection of asphalt pavements, conducting research on the mechanical response variations of asphalt pavements under both ambient (20 °C) and high-temperature (45 °C) conditions. The related deflection time history curve calculations and analysis results are shown in Figure 6 and Figure 7.

It can be observed that as the time increased, the deflection values at various sensor locations initially increased and then decreased, exhibiting a sinusoidal pattern similar to the applied load. Additionally, the arrival time of the peak deflection values at different sensor locations lagged as the sensor locations moved farther away. This observed lag phenomenon aligns with the real-world scenario, indicating a certain degree of reasonableness in the computed results. At the same time, it was found that temperature has a significant impact on the dynamic deflection changes of asphalt pavements. As the temperature increased, the deflection values of the asphalt pavement continuously grew, with the most noticeable increase in the maximum deflection value at the load center position. However, as the distance from the loading center point increased, the influence of temperature on dynamic deflection gradually diminished.

To further validate the accuracy of the computed results, a comparison was made between the maximum deflection values obtained using finite element analysis and spectral element analysis, as shown in

Table 3. Comparison of maximum deflection values at various sensor locations between the two methods.

Measurement Point Locations		D1	D2	D3	D4	D5	D6	D7	D8	D9
20	ABAQUS (μm)	169.27	138.97	124.11	96.41	90.55	75.19	61.62	48.50	40.17
	Proposed Algorithm (μm)	168.22	138.47	122.57	97.12	90.14	73.83	61.22	49.24	40.65
	Error (%)	0.62	0.36	−1.24	0.74	0.45	1.81	−0.65	1.52	1.19
45	ABAQUS (μm)	322.58	240.42	218.14	193.54	176.83	143.03	115.42	90.66	74.44
	Proposed Algorithm (μm)	320.82	239.70	216.92	195.13	175.90	141.68	113.32	91.19	74.58
	Error (%)	−0.55	−0.30	−0.56	0.81	−0.53	−0.95	−1.85	0.58	0.19

and Figure 8.

From Table 3 and Figure 8, it can be observed that the maximum absolute value of the relative error between the results obtained by the two methods was only 1.85%, and the average absolute value of the relative error was only 0.70%. This demonstrates that the accuracy of the calculation method in this paper meets the engineering requirements. Both methods were computed on the same computer, with the finite element method taking approximately 7 min for the pavement deflection calculation, while the spectral element method took approximately 12 s. It is necessary to note that numerical experiments were conducted on an assembled desktop computer equipped with an Intel Core i7-12700KF (5.0 GHz,12 cores) (Intel, Santa Clara, CA, USA), 64 GB DDR4 RAM, 1 TB SSD (Lenovo, Beijing, China), and a 2 TB hard disk (DELL, Austin, TX, USA), and the system ran on Windows 10. It can be seen that the spectral element method has significantly improved computational efficiency compared to the finite element method.

5.4. Impact of Groundwater on the Variation Patterns of Road Surface Deflection

During the service life of road structures, the roadbed is significantly influenced by groundwater factors, resulting in fluctuations in internal moisture within a certain range. The mechanical properties of the roadbed, such as the resilient modulus, also undergo substantial changes with variations in the moisture field, thereby affecting the overall stiffness of the road. Roadbed materials, as a typical porous medium, are typically studied based on the relationship between porous media and moisture characteristic curves to understand their hydraulic characteristics. Currently, the Van Genuchten model provides a reasonable description of the relationship between soil moisture content and potential and has been widely applied. It is represented by Equation (26).

$$\theta ={\theta }_{\mathrm{r}}+\frac{{\theta }_s-{\theta }_r}{{\left(1+{\left(\alpha h\right)}^n\right)}^m}$$

(26)

where $\theta$ is volumetric water content; ${\theta }_s$ is saturation water content; ${\theta }_r$ is residual water content; $\alpha ,m,n$ are the fitting parameters; and $h$ is soil water potential.

By altering the groundwater table’s elevation, the matric suction at a specific point within the roadbed was calculated. The moisture content at different distances from the groundwater table in the roadbed was determined using the soil moisture characteristic curve. The results are depicted in Figure 9.

The migration of moisture within the roadbed follows Darcy’s law, where the velocity of groundwater through the roadbed medium is directly proportional to the magnitude of the hydraulic gradient and the permeability of the fill material. This relationship can be expressed as Equation (27):

$$Q=K•A•\frac{\Delta h}{L}$$

(27)

in which $Q$ is the seepage flow rate, $K$ is the permeability coefficient, $A$ is the cross-sectional area for water flow, $\Delta h$ is the head loss, and $L$ is the path length.

This paper comprehensively analyzes the variations in the mechanical response of road structures under different humidity conditions, considering the water–soil characteristics of roadbed fill material and the migration patterns of groundwater within the roadbed based on control equations. In this research, medium plasticity silt was selected as the roadbed fill material, and the impact of varying groundwater levels on road surface deflection was thoroughly examined. The roadbed was divided into layers, with each layer representing a 1% change in moisture content. Assuming the groundwater level was at distances of 1 m, 2 m, 4 m, 6 m, and 7 m from the top of the roadbed, we calculated the corresponding pavement deflection values. The relevant results are shown in Figure 10.

From the data analysis results, it can be observed that the groundwater table had the following impact patterns on road surface deflection: (1) As the groundwater table rose, the overall road surface deflection gradually increased. When the distance between the groundwater level and the subgrade surface changed from 7 m to 1 m, the maximum deflection value at D1 increased by 42.36%. (2) When the groundwater table was more than 6 m below the roadbed surface, the road surface deflection tended to stabilize. (3) Changes in roadbed moisture had a less noticeable impact on the deflection growth rate at points D1 and D9.

6. Engineering Application

To validate the practical applicability of the road mechanical characteristic analysis method and computational program proposed in engineering, we conducted field measurements and data analysis of FWD deflection on the Shanghai-Shanxi Expressway. The road structure type of the engineering application section is as follows: 4 cm asphalt concrete surface layer + 5 cm asphalt concrete intermediate layer + 6 cm asphalt concrete subbase layer + 50 cm cement stabilized crushed stone base layer + 150 cm subgrade. In the engineering application section, calculations and analyses are solely conducted on the dynamic deflection of asphalt pavements under actual temperature and humidity conditions. The assumed conditions are as follows: a pavement temperature of 20 degrees, a subgrade volume moisture content of 11.5%, and a groundwater level 8 m below the subgrade surface. The measured load-time curve is shown in Figure 11.

In order to demonstrate the accuracy of the proposed calculation program under the influence of groundwater, the calculated deflection values will be compared with the measured deflection values based on the measured load-time curve and sensor positions. The related analysis results are shown in

Table 4. Comparison of peak deflection values for each sensor.

Measurement Point Locations	D1	D2	D3	D4	D5	D6	D7
Observed data (μm)	100.58	79.87	63.94	51.24	43.11	37.53	33.72
Proposed Algorithm (μm)	103.34	82.53	67.50	54.35	45.37	39.53	35.83
Error (%)	2.74	3.22	5.57	6.07	5.24	5.33	6.27

.

From the test results under the influence of groundwater environment, it can be observed that the trend of the calculated theoretical deflection curve using the finite element method was generally consistent with the measured deflection curve. The correlation coefficient between the two was greater than 0.9. The maximum absolute value of the relative error between the theoretical deflection peak and the measured deflection peak was 6.27%, while the average absolute value of the relative error was 4.92%. Considering the influence of humidity factors, these errors are within acceptable limits for practical engineering calculations and analysis.

7. Conclusions

Grounded in dynamic response theory, this paper establishes a spectral element computational model considering humidity for isotropic semi-rigid base asphalt pavements under dynamic loading. Focusing on typical road structure types in China, the paper compares the proposed method with classical finite element results and applies it in actual engineering projects under the influence of humidity. It analyzes the variation patterns of dynamic deflection of road surfaces and confirms the reliability and applicability of the spectral element method under the influence of groundwater. The main conclusions are as follows:

(1) Based on wave propagation theory, the chain rule, and the method of separation of variables, this paper calculates the general solution of the potential function in the frequency domain. For finite layers and semi-infinite layers, potential functions were used to describe the principles of wave propagation, which resulted in stiffness matrices for two types of spectral elements.

(2) The paper derives related formulas for the overall stiffness matrix, displacement matrix, and nodal force matrix of the road structure. The overall displacement in the time domain was obtained through the Fourier inverse transform. A spectral element mechanical model of the semi-rigid base pavement structure under FWD loading was established, along with a computational program for asphalt pavement dynamic response.

(3) The maximum absolute error in relative terms between the spectral element method and the ABAQUS finite element method was only 1.85%, and the average absolute error was only 0.70%; the finite element method took about 7 min to compute, while the spectral element method took approximately 12 s.

(4) When considering humidity changes in engineering applications, the correlation coefficient between the computed deflection curve and the measured one was greater than 0.9, with a peak deflection error of less than 6.5%. Compared to traditional finite element methods, the spectral element method model for calculating dynamic deflection of road surfaces offers higher computational speed and accuracy under the influence of groundwater.