## 3 Simulating HMAC Pavement Deformation in Curved Ramp  
### Road Alignment Feature  
Field investigation revealed that rutting in special road was more serious than in straight and flat highways [25]. However, the terminology'special road' includes profound implications. This section here presents four frequently encountered special roads and their geometric features as well as pavement stress characteristics. Moreover, the most unfavorable condition among them was selected for the following modeling.  
Typical special roads involve: First, a flat straight way with dramatic speed change, which usually appears near a tollgate or an intersection. In this case, asphalt pavement is subjected to a shear force resulting from acceleration. The second is consecutive steep route in which the shear force applied to pavement mainly originates from the gravity component along the gradient. Speed change will also contribute to the shear force to which the pavement is subjected. The third is a horizontally curved alignment. Due to the centrifugal force, a force departing from the curvature centre is imposed on the pavement. This is called the sideway force, which increases with the decrease of radius. The last special road is the horizontally curved and sloped highway, which is the combination of the second and third condition. In this case, the forces imposed onto the pavement are rather complicated due to dual action of centrifugal force and longitudinal friction. The fourth condition had been proven to be the most unfavorable by the authors of [25], so it was employed to represent special road in the following model.  
In curved ramp, the pavement unit bears two different shear forces, which are stemmed from the transverse and longitudinal friction between vehicle wheels and pavement surface as shown in Figure 6. One of them, the longitudinal friction force \(f_{l}\) should offset the component of wheel pressure \(p\) along the slope as well as the rolling friction force, according to Figure 6a. Based on the design speed and maximum gradient for roads at different levels in the Highway Alignment Design Specification [26], the longitudinal friction force \(f_{l}\) could be calculated as depicted in Table 2. The calculation shows that the longitudinal friction force was almost 0.6 of the standard 0.707 MPa vertical tire pressure at a profile of 5%.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Design Speed (km/h)** & **120** & **100** & **80** & **60** & **40** & **30** & **20** \\ \hline Max slope \(I\) (\%) & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ Slope angle \(A\) (\({}^{\circ}\)) & 1.718 & 2.291 & 2.862 & 3.434 & 4.004 & 4.574 & 5.143 \\ \(p\)sin\(A\) (MPa) & 0.021 & 0.028 & 0.035 & 0.042 & 0.049 & 0.056 & 0.063 \\ \(p\)cos\(A\) (MPa) & 0.707 & 0.706 & 0.706 & 0.706 & 0.705 & 0.705 & 0.704 \\ \(f_{l}=p\)sin\(A\) + \(\phi\)pcos\(A\) (MPa) & 0.375 & 0.381 & 0.388 & 0.395 & 0.402 & 0.409 & 0.415 \\ \hline \hline \end{tabular} \({}^{1}\)\(\phi\) is rolling friction coefficient. Wheel Contact Pressure \(p\) (MPa): Standard Ade Load BZZ100, \(p=0.707\) MPa, \(\phi=0.5\).  
\end{table}
Table 2: Pavement longitudinal friction force under different slopes (in pressure unit) \({}^{1}\).  
Figure 6: Pavement force analysis in curved and sloped road. (**a**) Longitudinal section; (**b**) transverse section.  
Similarly, the transverse friction force \(f_{t}\) could also be determined by multiplying tire pressure by a factor that reflects the effect of radius. The factor is called sideway force coefficient (SFC). According to the Highway Alignment Design Specification, this analysis took the largest SFC of 0.2 for considering the worst condition [26].