## 5 Case Study  
The dissipation performance of the excess pore pressures is investigated in the subsequent section, in which the common ramp and exponential loading in practical engineering are adopted. For the influencing factors, modeling sizes (i.e., \(S\) and \(N\)) and hydraulic-related parameters (i.e., \(\alpha_{\rm a}\), \(a_{\rm w}\), \(G_{\rm a}\) and \(G_{\rm w}\)) upon the consolidation of the unsaturated PVD foundation have been carried out comprehensively in the existing literature [15], and these parameters will not be discussed in this paper. The two time-dependent loadings are shown in Equations (26) and (27). Ramp loading:  
\[q(t)=\left\{ \begin{array}{ll}q_{0}+at,&0<t<\frac{q_{0}}{a}\\ 2q_{0},&\frac{q_{0}}{a}\leq t\end{array} \right. \tag{26}\]  
Exponential loading:  
\[q(t)=q_{0}+q_{0}\Big{(}1-e^{-bt}\Big{)} \tag{27}\]  
where \(a\) and \(b\) are the loading parameters of ramp and exponential loadings, respectively. Other parameter values are consistent with those noted above in the verification section if  
Figure 2: The comparison results of the current solutions versus existing solutions upon: (**a**) excess pore-air and (**b**) pore-water pressures. they are not involved in the special discussion. Substituting Equations (26) and (27) into Equations (23)-(25) can produce the final solutions under certain circumstances. The variations of the excess pore pressures in the unsaturated PVD foundation under different ramp loading parameters are illustrated in Figure 3, and the loading parameters \(a=5\times 10^{-2}\), \(5\times 10^{-3}\), \(5\times 10^{-4}\) and \(5\times 10^{-5}\) kPa/s are adopted. It can be concluded from Figure 3 that the loading parameters can lead to different dissipation rates of excess pore pressures during the dissipation process. Nevertheless, the final closure time seems to not be influenced, and this phenomenon is more obvious in Figure 3b. The first dissipation period of excess pore-water pressure (i.e., in the time interval \(10^{1}<t<10^{6}\)) is influenced by the change of excess pore-air pressure, and the variations of the dissipation rates are attributed to the changing disparities between the applied loading and the excess pore-air (or pore-water) pressure. In Figure 4, the dissipation process of excess pore pressures under the exponential loading is researched, with loading parameters \(b=5\times 10^{-2}\), \(5\times 10^{-3}\), \(5\times 10^{-4}\), \(5\times 10^{-5}\) and \(5\times 10^{-6}\) s\({}^{-1}\). Similarly, the loading parameter \(b\) of exponential loading poses significant influences upon the excess pore pressures, and the bigger values of \(b\) symbolize faster dissipation of excess pressures during a certain period. Specifically, the upward trends in the dissipation curves appear when the increase of the ramp or exponential loadings exceeds the reduction of excess pore pressures, and the excess pore pressures will dissipate (or reduce) gradually when the loading curves rise to the constant level. More importantly, the higher peak values occur when slower dissipation of the excess pore pressures is caused by the ramp or exponential loading (or the increase of the loading parameters). Above all, the increase of the loading parameters means the faster rise or decline of excess pore pressures at the loading influenced period, which is due to the same terminal dissipation time with different peak values. This phenomenon  
Figure 4: Dissipation of (**a**) excess pore-air and (**b**) pore-water pressures against various exponential loading parameters, \(b\). Figure 3: Dissipation of (**a**) excess pore-air and (**b**) pore-water pressures against various ramp loading parameters, \(a\). is harmful for real construction, because the sudden variations of excess pore pressures will lead to unpredicted accidents (i.e., the sudden rise of the excess pore pressures will lead to liquefaction of soil, and the sudden decline will cause a drop of the foundation at short notice).