Dynamic Response Analysis of JPCP with Different Roughness Levels under Moving Axle Load Using a Numerical Methodology

Abstract

In-service Portland cement concrete (PCC) pavements are subject to repeated dynamic loads from moving vehicles; thus, the actual stress generated in a PCC pavement may significantly differ from the static stress, which is normally used in the design and evaluation of pavement performance. Calculating the stress in PCC pavements under moving vehicle loads is of importance to assess their actual service condition, particularly for pavements with different surface roughness levels as the deteriorated roughness might cause large stress in PCC pavement subject to dynamic loads. In this paper, a method is proposed to compute the dynamic response in terms of loads and stresses generated in jointed plain concrete pavements (JPCPs) under a moving axle load, considering the effects of the pavement surface roughness, the vehicle parameters (including vehicle speeds and axle weights), and the pavement structure parameters (including thickness and elastic modulus of different layers and the existence of dowel bars). The dynamic axle load is firstly generated based on the quarter-car model, running through three successive slabs of which the surface roughness is determined by the power spectral density method, and the critical locations in slabs where the largest tensile stresses occur are identified. The combined effects of various pavement surface roughness levels, vehicle speeds, axle weights, and pavement structure parameters are evaluated in terms of the stress and the dynamic factor defined as the ratio of the tensile stress under dynamic load to the tensile stress under static load. For the roughness level D, the tensile stress can reach a maximum value of 3.13 MPa, and the dynamic factor can reach a maximum value of 2.46, which is much larger than the dynamic factor of 1.15 or 1.2 currently used in design guidebooks. Increasing the thicknesses of pavement slab or the subbase layer is an effective way to reduce the tensile stress in JPCP, while increasing the thickness of base layer is not effective. The results of this study can benefit future pavement design and pavement performance evaluation by providing the actual stress and the useful dynamic factor values for various conditions of field pavements. Moreover, preventive maintenance, particularly the improvement of pavement surface roughness, can be planned by referring to the results of this study, to avoid large tensile stress generated in JPCPs.

1. Introduction

Portland cement concrete (PCC) pavements have the advantages of high stiffness, enhanced durability, and low life-cycle cost, and have been widely used in the construction of airport runways, highways, wharf pavements, etc. These rigid pavements are subject to simultaneous moving vehicle loads and weathering loads due to their service conditions with most of their surfaces exposed to the environments. Such loading conditions make rigid pavement susceptible to the fatigue damage due to the tensile stress generated from dynamic loads. Currently, many studies [1,2] on PCC pavement focus on static axle loading and environmental loading, and many design guidebooks for PCC pavements [3,4] utilize the static vehicle load to calculate the stress and strain in PCC pavements. To consider the impact of dynamic vehicle loads, these guidebooks often magnify the results of the static method by a dynamic factor (DF), e.g., DF for highways pavement design in Chinese guidebooks is taken as 1.15, and as 1.2 in the PCA design guide for heavy truck roads [3,4].

However, the actual DF in PCC pavements is not a constant, as greater dynamic vehicle load might generate in pavement with deteriorated roughness [5]. The pavement surface roughness level deteriorates significantly with the increasing repeated vehicle loads and weathering from environment [6], which adversely affects driving comfort and most importantly may generate greater tensile stress in pavement under moving vehicle loads. This implies that the amplitude of the dynamic vehicle loads applied to the pavement under severe roughness conditions might be much greater than that of the static loads, resulting in greater stress and vibration response of the pavement. This can significantly shorten the service life of PCC pavement. However, pavement damage induced by surface roughness and vehicle speed have attracted little attention.

There are limited studies investigating the structural responses of rigid pavements to dynamic vehicle loads, with different opinions on which type of loading results in greater tensile stress or deflection in pavement [7,8,9,10,11]. Chatti et al. [7] first conducted finite element analysis to compute the stress generated in pavement under moving vehicle load, which was simulated by using the local displacement shape functions in the finite-element formulation, considering the surface profiles produced by faulting, day- and night-time warping, and breaks of different levels of severity. Their analysis indicated that dynamic analysis is generally not needed for the design of rigid pavements and that it usually leads to decreased pavement response. On the other hand, Darestani et al. [10] found that dynamic vehicle load doubled the stress, which reached 0.65 MPa compared with the static stress of 0.31 MPa, when a semitrailer truck moved on a new PCC pavement at a speed of 47 km/h. The deflection of jointed plain concrete pavement (JPCP) was compared with the case of static load through field measurements as well as finite element analysis, which adopted a similar dynamic load application method to that of Chatti et al. However, it should be noted that in Chatti et al.’s method, the moving axle load was not generated by the 1/4 vehicle model and neither the effect of pavement surface roughness level indexed by the power spectrum density of the vertical road profile displacement as A, B, C, and D nor the variation of vehicle speed were considered. In other words, the dynamic effects were not investigated by including the road roughness as extra excitation between wheels and pavements. Kuo et al. [11] input the measured road surface profile to their designed quarter-car–pavement coupled model and found that higher vehicle speed caused higher stress in the pavement. However, Kuo et al. treated the pavement as an equivalent lump model which did not take the slab size and dowel bars into account.

In review of the above discussion, there are different opinions on how rigid pavements respond to dynamic vehicle loads, due to the facts that many factors are involved during analysis and computing, including the definition of surface profiles, the vehicle model and parameters, the pavement structures, materials properties, the dynamic load generation method, etc., which deserve further comprehensive investigation.

Altogether, it is very rare to find research analyzing the dynamic response of JPCPs considering the combined effects of pavement roughness, dynamic vehicle loads, and pavement structure. JPCP does not contain any steel reinforcement. However, there may be load transfer devices (e.g., dowel bars) at transverse joints and deformed steel bars (e.g., tie bars) at longitudinal joints [12]. In this paper, a numerical model is established to investigate the dynamic response of JPCPs under moving axle load, considering the effects of the pavement roughness level, the dynamic load parameters, and the pavement structure parameters. The result of this study can benefit future pavement design by providing useful dynamic factor values for various conditions. Moreover, preventive maintenance to avoid excessive fatigue stress, particularly the improvement of pavement surface roughness, can be planned by referring to the results of this study.

2. Assumption

To simplify the procedure of computing dynamic response of JPCP, some reasonable assumptions are made by referring to previous studies, which are as follows:

(1)

The pavement roughness is a uniform random field obeying Gaussian distribution with zero mean value [5,13].

(2)

The vehicle can be simplified as a quarter-car model when considering the pavement response under the vehicle load [13,14].

(3)

The contact area between the tire ribs and the pavement can be simplified as a rectangle, and the amplitude of the contact force is sinusoidally distributed [15,16].

3. Dynamic Load Generation

The pavement roughness data are a set of two-dimensional information representing the deviation of the pavement surface elevation relative to the reference plane, which is one of the main excitations causing vehicle vibration. Critical roughness can cause extreme responses which are related to the vehicle speed and the suspension system of the vehicle. One way of considering roughness effect on the pavement stress from the moving vehicle load is to reflect the roughness directly by the surface profile of the pavement model in the finite element analysis (FEA); however, this imposes extremely high requirements on the model meshing and significantly increases the convergence difficulty of the FEA. Liu et al. [17] used a sinusoidal form to simulate the asphalt pavement surface unevenness by finite element method, based on which the dynamic loads were generated through a quarter-car model. This greatly simplified the calculation of pavement surface unevenness; however, the dynamic effects were not investigated by including the standard road roughness level as the excitation source, and there might be difficulties in comparing the results with other dynamic response cases. The other dynamic load generation method is based on the roughness index, as discussed below, which has been used in a few studies [11,13,18,19].

3.1. Pavement Roughness Generation

There are two major indexes for pavement roughness, including the international roughness index (IRI) and the power spectral density (PSD). The IRI was proposed by the World Bank and is widely used in transportation departments as a direct reflection of ride quality; the index measures the overall relative velocity between the axle and the sprung mass of the quarter car. PSD is widely used in the vehicle industry and the research community to analyze the dynamic interaction between the vehicle and the pavement, suspension optimization, and energy consumption of the vehicle. The PSD method is used in this study to generate the pavement roughness, as it is more mechanically related to the pavement behavior.

A large number of measurements show that the pavement roughness is a uniform random field obeying Gaussian distribution with zero mean value, the characteristics of which can be described by a PSD function G_q(n) as shown in Equation (1). PSD is defined as the limiting mean-square value of a signal per unit frequency bandwidth, according to the definition of the PSD of vertical road profile displacement in ISO 8608 [20].

$$G_q\left(n\right)=\left\{\begin{array}{l}{G}_q\left(n_0\right){\left(\frac{n}{n_0}\right)}^{-w}\hspace{1em}{n}_l<n<n_u\hfill \\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}other\hfill \end{array}\right.$$

(1)

where n represents the spatial frequency in m⁻¹; n₀ is the reference spatial frequency, taken as 0.1 m⁻¹; w is the frequency index, taken as 2 [20]; G_q(n₀) is the PSD of pavement roughness at the reference spatial frequency n₀, also known as the pavement roughness coefficient, in m³.

In this paper, the dynamic load is calculated by using a quarter-car model which is in point contact with the pavement. By referring to [21], if the width of the transverse joint in concrete pavement is small and no faulting exists, the influence of the transverse joint on the roughness is minor. Therefore, the influence of transverse joint or the slab length on the generation of the pavement roughness is ignored in this study. Due to the vehicle vibration suspension system, the displacement or acceleration response of the vehicle to a certain spatial frequency of pavement roughness excitation is extremely small; it is not necessary to consider all spatial frequency when generating the pavement roughness. Therefore, the spatial frequency is intercepted, and the lower and the upper limits of the effective frequencies are set to be n_l and n_u, respectively, as shown in Equation (1). The lower and upper limits of n_l and n_u are related to the lower and upper limits of the natural frequency range (f_l, f_u) of the vehicle vibration, respectively:

$$n_l=\frac{f_l}{v}, n_u=\frac{f_u}{v}$$

(2)

where v is the speed of the vehicle in m/s.

In general, the natural frequency of the body part of a vehicle is 1~2 Hz, and the natural frequency of the wheel part is 10~15 Hz [20]. The frequency range of [0.033 m⁻¹, 1.5 m⁻¹] is calculated according to Equation (2). For engineering application, [0.011 m⁻¹, 2.83 m⁻¹] was proposed by ISO 8608 as the effective spatial frequency range which covers the natural frequency of most vehicle vibrations when the vehicle speed ranges 36~108 km/h.

The discrete Fourier transform method is used to construct the discrete pavement roughness data from Equation (1), which is used as the input for the dynamic load calculation for the quarter-car model, as illustrated in Section 2. The sampling space and the sampling length for the discrete Fourier transform should be controlled by satisfying Equation (3) to avoid PSD confounding and to ensure a meaningful lower limit of the effective spatial frequency n_l:

$$\Delta l\le \frac{1}{2n_u}, L\ge \frac{1}{n_l}$$

(3)

where ∆l is the sampling spacing distance of the pavement roughness in m; L is the total sampling length of the pavement roughness in m.

In this paper, n_l is taken as 0.01 m⁻¹, n_u is taken as 3 m⁻¹, ∆l is taken as 0.1 m, L is taken as 1000 m. Referring to the classification method by ISO 8608 [20] and the literature [6], four pavement roughness levels ranging from level A to D were selected, as shown in

Table 1. The classification criteria for pavement surface roughness levels A, B, C, and D [20,22].

Level	Gq(n0) (×10−6 m3)			IRI (m/km) Corresponding to the Average Value of Gq(n0)
	Lower Limit	Geometric Value	Upper Limit
A	0	16	32	2.4
B	32	64	128	4.8
C	128	256	512	9.6
D	512	1024	2048	19.2

. It can be found that G_q(n₀) increases as the pavement roughness deteriorates, the average value of which becomes four times the original when the roughness level develops to the next level, and the average G_q(n₀) values of levels A, B, C, and D are taken as 16, 64, 256, and 1024, respectively. The discrete roughness is then generated following the steps below:

(1) Let x_m (m = 0, 1, 2, …, N − 1, N is an even number) be the pavement discrete roughness data. X_k is the result of the discrete Fourier transform of x_m. The relationship between X_k and the pavement roughness PSD function G_q(n) is shown in Equations (4)~(6):

$$X_k=\sqrt{\frac{N}{2\mathsf{\Delta }l}{G}_q\left(n_k\right)}{e}^{j{\phi }_k}\left(k=0,1,\dots ,\frac{N}{2}\right)$$

(4)

$$n_k=k\mathsf{\Delta }n$$

(5)

$$\mathsf{\Delta }n=\frac{1}{L}$$

(6)

where X_k is the result of Fourier transform at spatial frequency n_k; ${\phi }_k$ is the phase angle, which obeys normal distribution in the interval [0, 2π] with mean π; ∆n is the sampling resolution in m⁻¹; j represents an imaginary number, j² = −1.

(2) Since X_N/2-i and X_N/2+i are conjugate to each other, the value of X_k (k = N/2 + 1, …, N − 1) can be obtained from the value of X_k (k = 1, …, N/2 − 1), which is obtained based on Equation (4). Then, the discrete Fourier inverse transform of X_k gives the discrete data of the pavement roughness (x_m) in the spatial domain, as shown in Equation (7):

$$x_m=\frac{1}{N}\sum _{k=0}^{N-1}{X}_k{e}^{\frac{2\pi kmj}{N}}(m=0,1\dots ,N-1)$$

(7)

where x_m is the roughness information at sampling location m∆l in m.

The generated pavement roughness curves in the spatial domain for different roughness levels are shown in Figure 1a. The statistical distribution information is shown in Figure 1b and

Table 2. The statistical distribution information for different roughness conditions.

Pavement Level	A	B	C	D
Standard deviation (mm)	3.21	6.30	12.61	29.60
Mean value (mm)	0.04	0.07	0.21	0.02
Minimum (mm)	−10.91	−19.78	−34.94	−84.22
25% value (mm)	−2.05	−4.22	−8.30	−20.70
Median value (mm)	0.28	0.14	0.04	−1.17
75% value (mm)	2.29	4.37	8.57	21.86
Maximum (mm)	9.61	15.86	49.76	100.61

. It can be seen that the pavement roughness fluctuation increases from level A to level D, and the mean and median values remain near 0. Therefore, the generated roughness data are consistent with the assumption of a Gaussian distribution with a mean value of 0. In particular, when G_q(n₀) expands by a factor of 4, the standard deviation of pavement roughness expands by about a factor of 2, which is consistent with Equation (4) in the generation method.

To validate the accuracy of the generated discrete roughness data in the spatial domain, their PSD functions are calculated by Equations (8)~(10). The PSD functions of the generated discrete roughness data for different pavement roughness levels are the black curves shown in Figure 2. The straight lines are the theoretical PSD functions of level A, B, C, and D pavement roughness, obtained based on Equation (1). It can be observed that the PSD functions based on the generated discrete roughness data fluctuate around the theoretical PSD functions, which indicates that the generated discrete roughness data represent very well the different roughness levels as shown in Table 1.

$$X(k)=\sum _{m=0}^{N-1}{x}_m{e}^{\frac{-2\pi jmk}{N}}(k=0,1,2,\dots ,N-1)$$

(8)

$$G\left(n_k\right)=\left\{\begin{array}{c}\frac{\mathsf{\Delta }l}{N}|X\left(k\right){|}^2\hspace{1em} k=0\hfill \\ \frac{2\mathsf{\Delta }l}{N}|X\left(k\right){|}^2 \hspace{1em} k=1, 2,\dots ,\frac{N}{2}\hfill \end{array}\right.$$

(9)

$$n_k=k\mathsf{\Delta }n\left(k=0, 1, 2,\dots ,\frac{N}{2}\right)$$

(10)

Taking a PCC pavement in New York with the SHRP-ID of 4018 according to the Long Term Pavement Performance (LTPP) database [23] as verification, the elevation data with good and poor conditions are selected as shown in Figure 3a. According to the LTPP database, the IRI values corresponding to the good and poor conditions are 1.6 m/km and 2.7 m/km, respectively, and their PSD results are shown in Figure 3b,c. From Figure 3b,c, it can be seen that the roughness level of good conditions is better than A, and the roughness level of poor conditions is between A and B, which corresponds to Table 1 referring to their IRI values. Based on the above verification of real pavement roughness, it is believed that the generated roughness based on PSD in this study conforms to the actual pavement roughness characteristics, which are used in the following sections for calculating the dynamic load to better analyze the influence of roughness.

3.2. Dynamic Load Generation

Although the quarter-car model cannot simulate some conditions including the contact between tires and pavement surface, the acceleration, the steering, and so on, existing research shows that it can be well used for pavement dynamic response analysis, with the advantages of simplicity and efficiency [13,14]. The quarter-car model has two degrees of freedom with the vertical displacements of the suspension mass and the nonsuspension mass. The schematic diagram of the quarter-car model is shown in Figure 4, and its motion differential equations are shown in Equations (11)~(13):

$$m_1{\ddot{x}}_1=-k_1\left(x_1-x_2\right)-c_1\left({\dot{x}}_1-{\dot{x}}_2\right)$$

(11)

$$m_2{\ddot{x}}_2=k_1\left(x_1-x_2\right)+c_1\left({\dot{x}}_1-{\dot{x}}_2\right)-k_2\left(x_2-q\right)-c_2\left({\dot{x}}_2-\dot{q}\right)$$

(12)

$$F_d=k_2\left(x_2-q\right)+c_2\left({\dot{x}}_2-\dot{q}\right)$$

(13)

where m₁ is suspension mass; m₂ is nonsuspension mass; k₁ is stiffness of suspension system; k₂ is stiffness of tire; c₁ is damping coefficient of suspension system; c₂ is damping coefficient of tire; x₁ is vertical displacements of suspension mass; x₂ is vertical displacements of nonsuspension mass; q is vertical displacement caused by pavement roughness; ${\dot{x}}_1$ and ${\dot{x}}_2$ are the speed of the suspended mass and the non-suspended mass, respectively; ${\ddot{x}}_1$ and ${\ddot{x}}_2$ are the acceleration of the suspended mass and the non-suspended mass, respectively; F_d is the dynamic load of the tire; and q is the pavement roughness, which can be generated by the method discussed in Section 2.

The wheel total random dynamic load applied to the pavement can be obtained based on the summation of F_d and the system static load (G), as shown in Equations (14) and (15):

$$F_t=F_d+G$$

(14)

$$G=\left(m_1+m_2\right)g$$

(15)

where F_t is the total random dynamic load; G is the system static load; g is the acceleration of gravity, taken as 9.8 m/s².

The value of the standard axle weight specified in Chinese PCC pavement design guidebook is 10 tons [3], and by referring to the study by Lu et al. [24], the values of the vehicle mass, the bogie mass, and the wheel mass are set as 4450 kg, 500 kg, and 50 kg, respectively. Normally, the vehicle body mass is treated as the suspension mass, and the nonsuspension mass includes the bogie mass and the wheel mass [15], this means that the suspension mass (m₁) and the nonsuspension mass (m₂) are 4450 kg, and 550 kg, respectively, in this study. To analyze the effect of overloading on the stress generated in the pavement, the value of the nonsuspension mass (m₂) is fixed as 550 kg, and the value of the suspension mass (m₁) increases from 4450 kg to 9450 kg and 14,450 kg, which corresponds to axle loads of 200 kN and 300 kN, respectively. The parameters of the quarter-car vehicle model used in this study are listed in

Table 3. The parameters of quarter-car model of medium trucks.

Parameters	Value
m1	4450 kg (axle weight = 10 tons, and first natural frequency = 11.86 Hz) 9450 kg (axle weight = 20 tons, and first natural frequency = 8.17 Hz) 14,450 kg (axle weight = 30 tons, and first natural frequency = 6.62 Hz)
m2	550 kg
k1	1 × 106 N/m
k2	1.75 × 106 N/m
c1	15 × 103 N·s/m
c2	2 × 103 N·s/m

. The first natural frequency in Table 3 is calculated according to reference [25]. To simplify the description of the loading conditions, “X km/h-Y tons-Z” is used to represent the conditions with vehicle speed of X km/h, axle weight of Y tons, and roughness level Z; i.e., “30 km/h-10 tons-A” represents that the vehicle speed is 30 km/h, the axle weight is 10 tons, and the roughness level is A.

4. Numerically Simulated Dynamic Response of JPCP

4.1. Structure Model

The geometric and the material information of each structural layer and the dowel bar parameters of JPCP are summarized in

Table 4. The geometric and materials parameters for each structural layer and dowel bars in JPCP modeling.

	Geometric Parameters x (m) × y (m) × z (m)	Elastic Modulus	Poisson’s Ratio	Density (kg/m3)
Slab	5 × 4 × 0.26	31 GPa	0.15	2400
Base layer	16 × 5 × 0.2	1.3 GPa	0.20	2300
Subbase layer	16 × 5 × 0.2	1.0 GPa	0.20	2300
Foundation	16 × 5 × 4	60 MPa	0.35	2000
Dowel bar	30 mm in diameter; 500 mm in length; 300 mm spacing	206 GPa	0.30	7850

, which refer to the Chinese concrete pavement design guide [3]. Slab thickness of 220~280 mm is recommended for heavy traffic roads, and correspondingly the elastic modulus and the flexural strength of concrete are recommended to be 31.0 GPa and 5.0 MPa, respectively. The thicknesses of the base layer and the subbase layer are recommended to be between 150~200 mm if their material category is inorganic binder-stabilized aggregate. The elastic modulus of the cement-stabilized aggregate after shrinkage cracking ranges from 1.0 GPa to 2.5 GPa. The elastic modulus of the foundation is recommended to be greater than 60 MPa. The schematic diagram of the numerical model of JPCP is shown in Figure 5.

Three successive slabs with size of 4 m × 5 m are arranged along the vehicle traveling direction to simulate the dynamic response of JPCP under the moving vehicle load. Slab1 and Slab3 are used for initial loading and final unloading, and Slab2 is selected for monitoring the dynamic response at the specific locations of P1 (bottom of mid-slab edge), P2 (bottom of slab center), P3 (top of slab edge 0.25 m from the joint), and P4 (top of joint 1 m from the longitudinal joint). The coordinate system of the numerical model is shown in Figure 5, the x direction is parallel to the direction of vehicle travel, the y direction is parallel to the direction of the transverse joints, and the z direction is perpendicular to the pavement. The damping ratio of the model is taken as 0.05. The effects of both gravity and the dynamic axle load are considered, and the time increment is fixed as 0.002 s. Thus, the signal sampling rate is fixed as 500 Hz.

A tie connection is used between the soil base and the subbase as well as between the subbase and the base. The friction coefficient between the surface slab and the base is set as 1.5; it was proved that the magnitude of friction coefficient has no significant effect on the calculated stress under dynamic loads [26]. The width of the transverse joint is 4 mm, and a total of 13 dowel bars are arranged at the transverse joint, with spacing of 300 mm. One end of the dowel bar is tied to one concrete slab and the other end is in frictional contact with the other concrete slab, with a friction coefficient of 0.05. According to previous study [21], the width of the transverse joint is of little significance to dynamic response of the pavement under conditions of no faulting. Therefore, the effect of transverse joints is not considered in this study.

The finite element method is used to calculate the stress generated in JPCP, and dynamic analysis is conducted to calculate the dynamic response of JPCP, following the motion equation as shown in Equation (16). It has been shown that second-order elements in the finite element analysis can significantly reduce the number of elements along the thickness of the slab [27]. Therefore, three layers of C3D20R elements are used in Slab2 with a horizontal dimension of 7 cm, two layers of C3D20R elements are used for Slab1 and Slab3 with a greater horizontal dimension, and three layers of C3D8R elements are used for the base and the subbase with a horizontal dimension of 10 cm. The soil foundation is meshed with C3D8R elements with a horizontal dimension of 10 cm, and the height dimension of the elements increases with the increasing depth.

$$M\ddot{u}+C\dot{u}+Ku=F\left(t\right)$$

(16)

where M is the mass matrix of the structure; C is the damping matrix of the structure; K is the stiffness matrix of the structure; u is the node displacement array of the structure; F(t) is the external load.

4.2. Computing Procedure

The dynamic response of JPCP is influenced by numerous factors which can be roughly divided into the pavement structure parameters and the dynamic load parameters. The pavement structure parameters mainly include the dimensions of each structural layer and the elastic modulus of different materials. The dynamic load parameters mainly include the vehicle speed, the axle weight, and the pavement roughness level.

A comprehensive but simplified procedure is critical for accurately assessing the dynamic response of JPCP in relation to various factors, which includes the following steps. First, determining the input values of both the structure and the dynamic load parameters: the pavement structure and materials parameters are determined as the values shown in Table 4, which are the recommended values in the Chinese PCC pavement design guidebook [3]. The dynamic load parameters are considered for a total of 36 cases (36 = 3 vehicle speed × 3 axle weight × 4 roughness level), as shown in

Table 5. Dynamic loading parameters for calculation of stress in JPCP.

Vehicle Speed (km/h)	Axle Weight (t)	Roughness Level
30 60 100	10 20 30	A B C D

. It is worth mentioning that the roughness level of many real pavements is A or B [28], and it is difficult for a vehicle to travel at 100 km/h on a pavement with roughness level D. The purpose of calculating such conditions is to demonstrate the methodology proposed in this study and to comprehensively analyze the influence of roughness level and vehicle speed on PCC pavement. The second step is to selecting the critical dynamic load parameters (100 km/h-30 tons-D, as illustrated in Section 5.3) based on which the large tensile stresses in JPCP are calculated with the structure parameters shown in Table 4. Finally, the effect of the pavement structure parameters on the dynamic response of JPCP is investigated using the critical dynamic load parameters (100 km/h-30 tons-D). For this purpose, the values of the typical pavement structure parameters are varied as shown in

Table 6. Sensitivity analysis of pavement structure parameters for dynamic stress calculation.

Slab		Base Layer		Subbase Layer		Foundation	Dowel Bar
Thickness (m)	Elastic Modulus (GPa)	Thickness (m)	Elastic Modulus (GPa)	Thickness (m)	Elastic Modulus (GPa)	Elastic Modulus (MPa)	Existence
0.22	20	0.10	0.3	0.10	0.4	20	No Yes
0.26	31	0.20	1.3	0.20	1.0	60
0.30	40	0.30	2.3	0.30	1.6	100

Note: The values in bold for each of the pavement structure parameters are the same as the values in Table 4.

, including three values of the thicknesses and the elastic moduli for each structure layer and the existence of dowel bars (23 cases). The materials parameters of different layers are shown in Table 4. It should be noted that the case corresponding to the values in bold in Table 6 was computed in the first step. By design, one pavement structure parameter in Table 6 increases or decreases relative to the bolded value, and other pavement structure parameters (bold values) remain unchanged. The computing procedure is illustrated in Figure 6. It is seen that the total number of computing cases is 51.

4.3. Axle Load

To obtain a more realistic dynamic response of JPCP, it is important to consider the tire–pavement contact. Single-axle double-wheel loads have been selected in this study as the vehicle load, as shown in Figure 7, which is often used as the standard form of axle load in PCC pavement design guidebooks [3,4]. The tire is in contact with the pavement through the multiple ribs between which the groove is set for drainage, to improve the friction between the tire and the pavement. This study divides a single tire into five ribs to better reflect the contact area between the tire and the pavement, as shown in Figure 7. In general, the pressure of the tire applied on the pavement is sinusoidally distributed along the direction of vehicle travel (as shown in Figure 5b, expressed as Equations (17) and (18)) [15].

$$P\left(t\right)=a\times \frac{F\left(t\right)}{2\times S}\times \cos \left[\frac{\pi }{b}\times \left(x-X_0\left(t\right)\right)\right]$$

(17)

$$X_0\left(t\right)=X_{ini}+v\times t$$

(18)

where P(t) is the pressure applied to the pavement surface by the tire along the traveling direction at the moment t; F(t) is the random dynamic load of the quarter-car model at the moment t, which corresponds to the random dynamic load of two tires on one side of the single-axle double-wheel model used in this study; S is the area between an individual tire and the pavement, equal to the sum of the areas of the five ribs; b is the length of the rib in the direction of travel, taken as 14 cm for R1, R2, R4, and R5, and taken as 16 cm for R3; X₀(t) is the x coordinate of the rib center point along the traveling direction at the moment t; X_ini is the initial x coordinate of the rib center point along the traveling direction, which takes −7.5 m corresponding to the left edge of Slab1 in Figure 5; v is the speed of the axle load; x is the x coordinates of the load integration points in the loading area; a is the loading coefficient, taken as 0.5 for R1 and R5, taken as 0.9 for R2 and R4, and taken as 1.0 for R3.

The moving path of the axle load starts from Slab1 to Slab3, and Slab2 is selected as the slab to be monitored where the dynamic response is analyzed in detail. Since the critical loading path of JPCP is along the longitudinal edge of the slab [29], the same traveling mode is used in this study, as shown in Figure 5b.

5. Result and Analysis

5.1. Generated Dynamic Load along the Traveling Direction

The generated dynamic load from the quarter-car model along the driving direction is assessed for different pavement roughness levels, with vehicle speeds of 30 km/h, 60 km/h, and 100 km/h, and axle weights of 10 tons, 20 tons, and 30 tons, respectively, as shown in Figure 8. It can be seen that the dynamic load generated from the moving quarter-car fluctuates around its static load, and the fluctuation becomes larger with the deterioration of the pavement roughness level, reaching 2.81 times the static load in the case of “100 km/h-10 tons-D”. Heavy axle load causes greater dynamic load and affects a greater slab area. When the axle weight is equal to 10 tons or 20 tons, the dynamic load may sometimes be less than 0 kN in Level D pavement, which is considered a “jumping” phenomenon [30]. In this case, the wheel loses contact with the pavement surface, causing an axle load of below 0, and then drops after a short period of time. As shown in Figure 8a,d,e,g,h, the jumping might increase as the vehicle speed increases. For stress calculation, it should be noted that because of the randomness of the generated roughness and consequently the dynamic load, for conservation, the dynamic load is applied in such a way that it reaches its maximum when the moving axle travels near the critical loading position of the mid-slab edge of Slab2.

5.2. Recognizing Critical Locations with Large Tensile Stress in JPCP under Dynamic Load

It is important to determine the critical locations where the tensile stress might reach its maximum and to study the effect of the pavement structure parameters and the dynamic load parameters on the dynamic responses of JPCP. According to engineering practice, bottom-up transverse cracks normally occur at the middle of the slab, while top-down cracks normally occur at the corners of the slab. The following locations are normally recognized as the key locations where the maximum tensile stress might be generated, denoted as P1, P2, P3, and P4, respectively, for the bottom midpoint of the longitudinal slab edge, the bottom of the slab center, the top of the longitudinal slab corner edge, and the top of the transverse slab corner edge, as shown in Figure 9. For simplification, σ_x-d and σ_x-s are denoted as the dynamic and the static stress in the traveling direction, respectively, and σ_y-d and σ_y-s are denoted as the dynamic and the static stress in the transverse direction, respectively. In this study, the magnitude of stress is defined as positive in tension and negative in compression.

To determine the critical location where the tensile stress might reach its maximum, the case of “100 km/h-30 tons-D” is selected for computation. The values listed in Table 4 are used as the input for pavement structure parameters. The calculated stresses at the four locations of P1~P4 are plotted in Figure 9, with the position of the dynamic moving quarter-car model.

It is seen that the maximum tensile stresses are generated at locations P1 and P2. The tensile stress reaches maximums of 3.13 MPa and 2.67 MPa, respectively, when the axle load moves to the middle of Slab2. Under well-supported slab conditions, the tensile stresses σ_x-d and σ_y-d at P3 and P4 are small and negligible, and are not in the critical location to develop top-down corner cracks. Considering the actual engineering scenario, it is believed that the top-down cracks developed at the slab corner are mainly caused by the loss of slab support either from voids beneath the slab or slab curling/warping due to temperature/moisture gradient. Comparing the tensile stresses at the four potential critical locations, σ_x-d at P1 is the largest, and thus the bottom midpoint of the longitudinal Slab2 edge (P1) is selected as the location to be monitored for dynamic response analysis.

5.3. Effect of Dynamic Loading Parameters

As illustrated above, the bottom midpoint of the longitudinal Slab2 edge (P1) is selected as the critical location to be monitored, where the tensile stress will be calculated under various conditions. The dynamic axle loads generated in Figure 8 are used as the external excitation in the JPCP numerical model, and the maximum tensile stresses at P1 under different loading conditions shown in Table 5 are computed to analyze the effect of the dynamic loading parameters including the vehicle speed, the axle weight, and the pavement roughness level; the results are shown in Figure 10. The JPCP structure parameters are fixed as shown in Table 4.

It is seen that the tensile stress at P1 (σ_x-d) increases with the deterioration of pavement roughness and the increase of axle weight and vehicle speed, reaching a maximum value of 3.13 MPa under the loading conditions of “100 km/h-30 tons-D”. Meanwhile, σ_x-d is more sensitive to the deterioration of the roughness under a small axle weight. At a vehicle speed of 100 km/h, the maximum tensile stress under axle weights of 10 t, 20 t, and 30 t is increased by 117.6%, 183.8%, and 56.5%, respectively, when the roughness deteriorates from level A to level D. On the other hand, σ_x-d is sensitive to vehicle speed under the more severe roughness; i.e., at an axle weight of 30 tons, σ_x-d under roughness conditions of levels A, B, C, and D is increased by 1.5%, 3.1%, 6.0%, and 23.7%, respectively, when the vehicle speed increases from 30 km/h to 100 km/h.

It is difficult to measure directly the stresses or the dynamic loads generated in pavements subject to moving vehicle loads; indirect measurement of the strain is normally conducted. Some researchers have found that the strain in pavements subject to dynamic load may increase by up to 100% for asphalt concrete pavement [31] and 75% for PCC pavement [32,33] compared with that when loading is static, and the increase of strain is related to vehicle velocity, vehicle weight, pavement structure, etc. Therefore, it is reasonable to conclude that the stresses correspondingly increase in pavement subject to moving vehicle load, which verifies the numerical results obtained in this study.

The dynamic factor is calculated as the ratio of the tensile stress under dynamic load to that under static load under the same axle weight conditions, and is normally used for assessing the effect of dynamic load on pavement response, as shown in Figure 11. It is seen from Figure 11 that the dynamic factor is always greater than 1 under various conditions. It increases with the roughness deterioration or the decreasing axle weight, with ranges of 0.98~1.17, 1.02~1.39, 1.11~1.77, and 1.26~2.46, respectively, for roughness levels A, B, C, and D. The dynamic factor increases more significantly with the vehicle speed under roughness level D, reaching a maximum of 2.46 which is more than double that of 1.15 used in the Chinese concrete pavement design guide [3] and 1.2 in the PCA design guide [4]. This suggests that a greater dynamic factor should be selected in the PCC pavement design guide to reflect the actual dynamic behavior under moving vehicle loads.

5.4. Effect of Pavement Structure Parameters

The maximum tensile stresses at P1 (σ_x-d) are computed for different structure parameters listed in Table 6. The vehicle speed, the axle weight, and the roughness level are taken as 100 km/h, 30 tons, and D, respectively, as the tensile stress in JPCP is the largest under such loading conditions (Figure 10). The calculated σ_x-d at P1 is shown in Figure 12.

It is seen from Figure 12 that the maximum tensile stress σ_x-d at P1 can be reduced by either increasing the slab thickness or reducing the elastic modulus of the slab. On the other hand, increasing the thickness or elastic modulus of the base layer improves the support conditions of the slab and thus reduces the tensile stress at P1 as well; however, the tensile stresses are all greater than 2.25 MPa, beyond which fatigue failure is expected under repeated vehicle loads, while the lower fatigue limit is 45% of the flexural strength of 5 MPa [12]. It is therefore concluded that increasing the thickness or elastic modulus of the base layer is not an effective way to reduce the dynamic stress or to improve the fatigue life of JPCP. Increasing the thickness or the elastic modulus of the subbase layer and the foundation reduces the maximum dynamic tensile stress slightly. With the presence of the dowel bars, the tensile stress at P1 and the dynamic factor are reduced slightly by 7.4% and 2.6%, respectively. It is considered that the dowel bars are not effective in reducing the tensile stress under dynamic loads.

The pavement structure parameters have minor influence on the dynamic factor, ranging from 1.48 to 1.67 under the “100 km/h-30 tons-D” loading conditions (Figure 12c), compared to which the dynamic load parameters have significant influence on the dynamic factor, ranging from 0.98 to 2.46 (Figure 11). In addition, the tensile stress of JPCP does not vary linearly with the change of the dynamic load parameters or the pavement structure parameters. This implies that the Duhamel integral, which is normally used for theoretically analyzing the dynamic response of a linear system, is not applicable for the dynamic analysis of JPCP.

5.5. Optimization of Pavement Structure Parameters

It is possible to reduce the tensile stresses in JPCP under dynamic loads by optimizing the pavement structure parameters to ensure its long life. As shown in Figure 10, the tensile stresses are all greater than 2.25 MPa in the cases of “60 km/h-30 tons-C”, “100 km/h-30 tons-C”, “30 km/h-30 tons-D”, “60 km/h-20 tons-D”, “60 km/h-30 tons-D”, “100 km/h-20 tons-D”, and “100 km/h-30 tons-D” dynamic loads, beyond which fatigue failure is expected under repeated vehicle loads. It can be seen from Figure 12a that the maximum tensile stress in JPCP can be reduced by varying the thickness and the elastic modulus of different layers. Considering that it is difficult to keep a vehicle traveling at 100 km/h under conditions of level D roughness in practice, “60 km/h-20 tons-D” and “60 km/h-30 tons-D” are selected in this part to optimize the pavement structure parameters. The optimization methods and their results are shown in

Table 7. Optimization of pavement structure parameters to reduce the tensile stresses in JPCP under “60 km/h-20 tons-D” and “60 km/h-30 tons-D” dynamic loading conditions.

		Before Optimization: Structure Parameters Are Shown in Table 4	Optimization 1: Increasing Thickness of Slab from 0.26 m to 0.3 m	Optimization 2: Increasing Thickness of Base Layer from 0.2 m to 0.3 m	Optimization 3: Increasing Elastic Modulus of Base Layer from 1.3 GPa to 2.3 GPa	Optimization 4: Increasing Thickness of Subbase Layer from 0.2 m to 0.3 m
Dynamic tensile stress (MPa)	20 t	2.63	1.83 (−30.5%)	2.44 (−7.3%)	2.46 (−6.6%)	1.92 (−27.1%)
	30 t	3.05	2.10 (−31.1%)	2.80 (−8.1%)	2.84 (−6.8%)	2.21 (−27.4%)
Static Tensile stress (MPa)	20 t	1.37	0.98 (−28.5%)	1.27 (−7.3%)	1.32 (−3.6%)	1.05 (−23.5%)
	30 t	2.01	1.43 (−28.9%)	1.85 (−8.0%)	1.94 (−3.5%)	1.52 (−24.4%)

Note: The values in brackets show the reduction of the stress after the optimization.

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It is found that increasing the thickness of the pavement slab from 0.26 m to 0.30 m or the thickness of subbase layer from 0.2 m to 0.3 m is an effective method to reduce the tensile stress by at least 27% and maintain less than 2.25 MPa under the “60 km/h-20 tons-D” and “60 km/h-30 tons-D” loading conditions and consequently improve the fatigue life of JPCP. However, it is not effective to reduce the tensile stress by increasing the thickness of the base layer or by improving the elastic modulus of base layer, as the tensile stress is reduced only by less than 8% and still maintains at high stress level of 2.25 MPa.

6. Conclusions

In this paper, a method is proposed to compute the dynamic response of JPCPs under the moving axle load, considering the combined effects of the pavement surface roughness, the dynamic load parameters, and the pavement structure parameters. The dynamic axle load is generated based on the quarter-car model, running through three successive slabs of which the surface roughness level is determined by the power spectral density function, and the critical locations in slabs where the largest tensile stresses occur are identified.

As the pavement roughness deteriorates from level A to level D, both the tensile stress and the dynamic factor of JPCP are significantly increased. Under conditions of deteriorated pavement surface roughness, increasing vehicle speed also increases the tensile stress and the dynamic factor. For the roughness level D, the tensile stress reaches a maximum value of 3.13 MPa, and the dynamic factor reaches a maximum value of 2.46, which is much larger than the dynamic factor of 1.15 or 1.2 currently used in design guidebooks [3,4]. Although the generated maximum tensile stress is less than the tensile strength of concrete, it is greater than the fatigue tensile strength of 2.25 MPa [12], which means that fatigue failure is to be expected under repeated vehicle loads.

Increasing the thicknesses of pavement slab or the subbase layer is an effective way to reduce tensile stress in JPCP, while increasing the thickness of base layer is not effective. Considering that the construction cost will be tremendously increased by thicker slab or subbase layer, it is suggested to improve the surface roughness as the primary preventive maintenance method to reduce tensile stress and consequently to improve the service life of JPCP. Overloading should be avoided as well, due to the greater dynamic factor under the coupled effect from heavy wheel load and the deteriorated surface roughness.

The results of this study can benefit future pavement design and pavement performance evaluation by providing the tensile stress and useful dynamic factor values for various conditions of field pavements. A greater dynamic factor should be included in the PCC pavement design guide to reflect the actual dynamic behavior under moving vehicle loads. Moreover, preventive maintenance, particularly the improvement of pavement surface roughness, can be planned by referring to the results of this study to avoid generation of high tensile stress in JPCPs. Typical methods include void repairment by polymer grouting, pothole repairment by epoxy resin, crack repair by cement concrete cover, and so on. For flexible pavement, it is also necessary to analyze its dynamic response under different roughness conditions and vehicle loads. It should be noted that the effects of temperature curling and moisture warping of JPCPs are not considered in this study, and are certainly worthy of further investigation. Also, axle load spectra from the LTPP database or other sources can be considered in the future to further verify the methodology proposed in this paper.