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

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Assumption

- (1)
- (2)
- (3)

## 3. Dynamic Load Generation

#### 3.1. Pavement Roughness Generation

_{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].

^{−1}; n

_{0}is the reference spatial frequency, taken as 0.1 m

^{−1}; w is the frequency index, taken as 2 [20]; G

_{q}(n

_{0}) is the PSD of pavement roughness at the reference spatial frequency n

_{0}, also known as the pavement roughness coefficient, in m

^{3}.

_{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:

^{−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.

_{l}:

_{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. It can be found that G

_{q}(n

_{0}) 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}(n

_{0}) 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:

_{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):

_{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, j

^{2}= −1.

_{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):

_{m}is the roughness information at sampling location m∆l in m.

_{q}(n

_{0}) 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.

#### 3.2. Dynamic Load Generation

_{1}is suspension mass; m

_{2}is nonsuspension mass; k

_{1}is stiffness of suspension system; k

_{2}is stiffness of tire; c

_{1}is damping coefficient of suspension system; c

_{2}is damping coefficient of tire; x

_{1}is vertical displacements of suspension mass; x

_{2}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.

_{d}and the system static load (G), as shown in Equations (14) and (15):

_{t}is the total random dynamic load; G is the system static load; g is the acceleration of gravity, taken as 9.8 m/s

^{2}.

_{1}) and the nonsuspension mass (m

_{2}) 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 (m

_{2}) is fixed as 550 kg, and the value of the suspension mass (m

_{1}) 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 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

#### 4.2. Computing Procedure

#### 4.3. Axle Load

_{0}(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.

## 5. Result and Analysis

#### 5.1. Generated Dynamic Load along the Traveling Direction

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

_{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.

_{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

_{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.

#### 5.4. Effect of Pavement Structure Parameters

_{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.

_{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.

#### 5.5. Optimization of Pavement Structure Parameters

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) The pavement roughness of levels A, B, C, and D in the spatial domain generated by the PSD method, and (

**b**) the box plot corresponding to the pavement roughness of levels A, B, C, and D.

**Figure 2.**PSD of (

**a**) simulated level A roughness data, (

**b**) simulated level B roughness data, (

**c**) simulated level C roughness data, and (

**d**) simulated level D roughness data, where the straight lines represent theoretical PSD functions of level A, B, C, and D pavements, obtained based on Equation (1).

**Figure 3.**(

**a**) The elevation of an actual PCC pavement referring to LTPP database, and PSD of pavements in (

**b**) good condition and (

**c**) poor condition.

**Figure 8.**The influence lines of dynamic axle load for different loading cases: (

**a**) 30 km/h-10 tons, (

**b**) 30 km/h-20 tons; (

**c**) 30 km/h-30 tons, (

**d**) 60 km/h-10 tons, (

**e**) 60 km/h-20 tons, (

**f**) 60 km/h-30 tons, (

**g**) 100 km/h-10 tons, (

**h**) 100 km/h-20 tons, and (

**i**) 100 km/h-30 tons with the other parameters of quarter-car model fixed as shown in Table 3.

**Figure 9.**The stress influence lines at locations P1~P4 of Slab2 under the moving axle loading conditions of “100 km/h-30 tons-D” with the pavement structure parameters fixed as shown in Table 4.

**Figure 10.**Tensile stress at the bottom midpoint of the longitudinal Slab2 edge (P1) under different vehicle speeds (30 km/h, 60 km/h, and 100 km/h), axle weights (10 tons, 20 tons, and 30 tons), and roughness conditions (level A, level B, level C, and level D) with the pavement structure parameters fixed as shown in Table 4.

**Figure 11.**The development of the dynamic factor at the bottom midpoint of the longitudinal Slab2 edge (P1) under different vehicle speeds (30 km/h, 60 km/h, and 100 km/h), axle weights (10 tons, 20 tons, and 30 tons), and roughness conditions (level A, level B, level C, and level D) with the pavement structure parameters fixed as shown in Table 4.

**Figure 12.**The development of (

**a**) the tensile stress under moving load, (

**b**) the static tensile stress, and (

**c**) the dynamic factor at the bottom midpoint of the longitudinal Slab2 edge (P1) for the “100 km/h-30 tons-D” example with varied pavement structure parameters as shown in Table 6.

Level | G_{q}(n_{0}) (×10^{−6} m^{3}) | IRI (m/km) Corresponding to the Average Value of G_{q}(n_{0}) | ||
---|---|---|---|---|

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 |

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 |

Parameters | Value |
---|---|

m_{1} | 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) |

m_{2} | 550 kg |

k_{1} | 1 × 10^{6} N/m |

k_{2} | 1.75 × 10^{6} N/m |

c_{1} | 15 × 10^{3} N·s/m |

c_{2} | 2 × 10^{3} N·s/m |

**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/m^{3}) | |
---|---|---|---|---|

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 |

Vehicle Speed (km/h) | Axle Weight (t) | Roughness Level |
---|---|---|

30 60 100 | 10 20 30 | A B C D |

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 | NoYes |

0.26 | 31 | 0.20 | 1.3 | 0.20 | 1.0 | 60 | |

0.30 | 40 | 0.30 | 2.3 | 0.30 | 1.6 | 100 |

**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%) |

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## Share and Cite

**MDPI and ACS Style**

Yan, C.; Wei, Y.
Dynamic Response Analysis of JPCP with Different Roughness Levels under Moving Axle Load Using a Numerical Methodology. *Appl. Sci.* **2023**, *13*, 11046.
https://doi.org/10.3390/app131911046

**AMA Style**

Yan C, Wei Y.
Dynamic Response Analysis of JPCP with Different Roughness Levels under Moving Axle Load Using a Numerical Methodology. *Applied Sciences*. 2023; 13(19):11046.
https://doi.org/10.3390/app131911046

**Chicago/Turabian Style**

Yan, Chuang, and Ya Wei.
2023. "Dynamic Response Analysis of JPCP with Different Roughness Levels under Moving Axle Load Using a Numerical Methodology" *Applied Sciences* 13, no. 19: 11046.
https://doi.org/10.3390/app131911046