Effects of pavement roughness on shakedown limits under moving traffic loadings

This study proposes a lower-bound shakedown analysis method for pavement systems under moving traffic loadings considering pavement roughness. The effects of key variables, including moving speed, wavelength irregularity, and the modulus and thickness of pavements, on the shakedown limit are discussed. The calculation results indicate that pavement roughness has a considerable effect on shakedown limits, which can be overestimated in some cases when the pavement roughness is not considered. The effect of pavement roughness on the shakedown limit of a homogeneous subgrade layer gradually decreases with increasing speed. Additionally, the shakedown limit decreases with wavelength irregularity, particularly under high-speed traffic loadings. Furthermore, the effect of pavement roughness on the shakedown limit of a two-layered pavement system is more pronounced for a larger pavement modulus and thickness and a lower moving speed.


Introduction
Since its introduction into the field of transportation geotechnics, the shakedown theorem has significantly been developed in the field of pavement engineering [1]- [3].The shakedown analysis considers the elastic-plastic behavior of pavement structures and ensures their stability under variable or cyclic loads.During the design and application of pavement engineering, a pavement system is considered to be in a shakedown state as plastic deformation ceases to increase after repeated load cycles [4].
The lower-and upper-bound shakedown theorems proposed by Melan [5] and Koiter [6], respectively, constitute the classical shakedown theorems, based on which subsequent studies have been conducted.Yu and Wang [7] proposed a lower-bound shakedown analysis method for solving design problems of flexible pavements.Accordingly, Liu et al. [8] investigated dilation angle effects on the lower-bound shakedown limits of pavements.Considering previous studies have been conducted on static analyses, Qian et al. [9]- [10] extended the dynamic shakedown theorem by considering the dynamic stress effect of moving loads to provide a theoretical basis for the shakedown analysis of pavement systems under high-speed moving loads.Recently, Wang et al. [11] established a simplified and unified shakedown equation to evaluate the shakedown limits of pavements and railways under repeated moving loads.
Traffic loads can be categorized into two components, static and dynamic, with the former dominated by the weight of vehicles and the latter by the additional effects of pavement roughness [12].In actual pavement engineering, the periodicity and amplitude of vehicle dynamic loads are significantly affected by the roughness of the pavement surface.Pavement roughness induces additional dynamic stresses, thus affecting the shakedown behavior of pavement systems.However, previous studies on the shakedown analysis of pavement systems have almost ignored the effects of pavement roughness on the shakedown limit.Therefore, this study develops a dynamic shakedown analysis method that considers pavement roughness based on a dynamic finite-infinite element model and the lower-bound shakedown theorem.Moreover, the effects of substantial factors, including moving speed, wavelength irregularity, and the modulus and thickness of pavements, on the shakedown limit of pavement systems are investigated.

Shakedown analysis method considering pavement roughness
The lower-bound shakedown theorem proposed by Melan [5] and Ceradini [13] was used to develop a dynamic shakedown theorem that considered inertial and damping forces.According to the lower-bound shakedown theorem, a structure would be in a shakedown state; in this case, searching for a selfequilibrium residual stress field r ij  combined with the elastic stress field   e , ij x t  obtained from finite element calculations always satisfied the yield condition f , as shown in equation (1).
where λ denotes the load multiplier and is a dimensionless scale factor.

Dynamic stress responses
For this study, a numerical method combining infinite and finite element analyses was adopted to investigate the elastic dynamic stress fields of pavement structures under moving traffic loads.The effectiveness of the method has been verified in previous studies [9].The simulation of infinite domains using hemispherical infinite-element artificial boundaries could better absorb reflected stress waves, as shown in figure 1.The load distribution adopted a hemispherical moving Hertz load with radius r = 0.125 m.When considering the effect of pavement roughness, the traffic dynamic load was simplified as a superposition of the vehicle self-weight, which did not change with time, and the harmonic dynamic load induced by pavement roughness [14], as expressed in equation ( 2).The moving harmonic load was reduced to moving a constant load when the effect of pavement roughness was neglected, such that α0 = 0.

Dynamic shakedown identification
The pavement material considered was elastic and perfectly plastic, according to the Mohr-Coulomb criterion.The total stress field at any k point in the structure required to satisfy the conditions is given by equations ( 3)-( 5) to ensure its shakedown state [15].
2 tan tan e 2 e 4 1 tan ( ) ( ) tan( ) where c denotes cohesion and φ denotes the internal friction angle of the material.
Considering the moving harmonic load, the load within a cycle period T was divided into n equal parts, and the stress field at each moment was extracted for shakedown analysis, as expressed in equation (6).The shakedown limit was the minimum of the shakedown multipliers at all moments in time, as expressed in equation (7).

Effect of moving speed
The shakedown limit of a homogeneous-subgrade layer comprising uniform materials with E = 20 MPa, υ = 0.48, and ρ = 1800 kg/m 3 was calculated.The Rayleigh wave speed ( R v ) was ~60 m/s.The coefficient of the additional dynamic load as α0 = 0.3 and the wavelength irregularity was L = 2 m. Figure 2 presents the variations in the shakedown limit of a single subgrade layer with different speeds when considering pavement roughness.The shakedown limit of the homogeneous subgrade layer decreases with speed and increases with internal friction angle.The shakedown limit is overestimated when the effect of pavement roughness is neglected.Furthermore, as the speed of the moving load increases, the effect of pavement roughness on the shakedown limit decreases gradually until it becomes equal to the shakedown limit, as speed reaches R v .When the speed exceeds R v , the effect of pavement roughness on the shakedown limit begins to reemerge.

Effect of wavelength irregularity
Figure 3 shows the variation in the shakedown limit of a single subgrade layer with different wavelength irregularities.The shakedown limit decreases with wavelength irregularity.Particularly, the effect of wavelength irregularity on the shakedown limit is substantially considerable at shorter wavelengths (such as L ≤ 2 m).Furthermore, the shakedown limit decreases more significantly when the speed is close to R v .However, when the irregularity wavelength (L) exceeds a critical value (such as L = 4 m), the effect of wavelength irregularity on the shakedown limit tends to be neglected.

Shakedown limit of two-layered pavement
A multi-layered pavement system can be simplified as a two-layered pavement system comprising top (surface and base layers) and bottom (subgrade) layers.The material parameters are listed in table 1.

Effect of modulus of pavement
Figure 4 shows the variations in the shakedown limit of a two-layered pavement with different pavement moduli.The shakedown limit slightly decreases and gradually stabilizes with the pavement modulus, in the case of lower c1/c2.The shakedown limit increases to its peak and then decreases with a further increase in the pavement modulus.The modulus ratio corresponding to the maximum shakedown limit is referred to as the optimal modulus ratio.For a larger c1/c2, the shakedown limit increases significantly.When considering pavement roughness, the shakedown limit is larger than that when the pavement roughness is neglected.The effect of pavement roughness on the shakedown limit increases with the pavement modulus.When the modulus ratio (E1/E2) exceeds a critical value, the influence of pavement roughness gradually stabilizes.

Effect of pavement thickness
Figure 5 shows the variations in the shakedown limit of a two-layered pavement with different pavement thicknesses for E1/E2 = 5 and c1/c2 = 5.The shakedown limit increases with pavement thickness.The effect of pavement roughness on the shakedown limit is more pronounced for a higher pavement thickness, particularly at lower speeds.Furthermore, a diminishing effect on the shakedown limit with an increase in velocity is observed.

Conclusion
This study proposed a dynamic shakedown analysis method considering pavement roughness and examined the effect of key factors on the shakedown limit of pavement systems.The following conclusions were drawn: (1) The shakedown limit of a pavement system can be overestimated when the effect of pavement roughness is neglected.The effect of pavement roughness on the shakedown limit decreases gradually with the moving speed, and it can be neglected when the moving speed exceeds a critical value.Additionally, the shakedown limit decreases with wavelength irregularity, particularly at higher speeds.
(2) For a two-layer pavement system with rigid upper and soft lower layers, the shakedown limit is affected by the moving speed along with the pavement modulus and thicknesses.Furthermore, the effect of pavement roughness on the shakedown limit is more pronounced for a larger pavement modulus and thickness and a lower moving speed.
where P0 represents the moving constant load and α0 denotes the coefficient of additional dynamic load induced by pavement roughness.0 = 2v/L denotes the frequency of the additional dynamic load, wherein v is the speed of the moving load and L is the wavelength irregularity.

Figure 1 .
Figure 1.3D finite element model with infinite element boundary.

Figure 2 .
Figure 2. Variations in the shakedown limit of a single subgrade layer with different speeds.

Figure 3 .
Figure 3. Variations in the shakedown limit of a single subgrade layer with different wavelength irregularities.

Figure 4 .
Figure 4. Variations in the shakedown limit of a two-layered pavement with different pavement moduli: (a) v = 10 m/s; (b) v = 60 m/s.

6 Figure 5 .
Figure 5. Variations in the shakedown limit of a two-layered pavement with different pavement thicknesses.