## 6 Concluding Remarks  
By taking the smear effect, drain resistance and external time-dependent loading into consideration, this paper presented analytical solutions for the PVD foundation under the unsaturated condition. The Fourier series expansion method, decoupling method and constant variation method were utilized in the derivative process, which can present the final solutions in neat and explicit forms, and verification work was conducted to prove the correctness of the solutions via existing solutions. To some extent, the proposed solutions filled the research gap of the unsaturated PVD foundation in analytical procedures by considering different external boundary conditions and time-dependent loading simultaneously.  
Smear effect and drain resistance can significantly influence the dissipation process of excess pore pressures, and the dissipation curves resemble certain external loading types. Compared with those semi-analytical solutions with complex numerical transforms and inversion processes (i.e., the Laplace transform and the inverse Laplace transform), the current solutions can be computed with high efficiency and accuracy. Additionally, the proposed analytical procedure may provide powerful guidelines to solve relevant coupled partial differential equations.  
Conceptualization, T.L.; writing--original draft preparation, Q.M.; writing--review and editing, Y.C., Q.X. and X.L. All authors have read and agreed to the published version of the manuscript.  
Figure 5: Consolidation patterns of the PVD foundation under (**a**) ramp and (**b**) exponential loadings against various loading parameters of \(a\) and \(b\).  
**Funding:** We sincerely acknowledge the National Natural Science Foundation of China (Grant No. 41807232) for financial support.  
**Institutional Review Board Statement:** This study did not require ethical approval.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** No new data were created or analyzed in this study.  
**Conflicts of Interest:** The authors declare that there are no conflict of interest.  
**Nomenclature**  
\(q_{0}\): the applied external loading at \(t=0\)  
\(a\): the ramp loading parameter  
\(b\): the exponential loading parameter  
\(r_{\rm w}\): the radius of the drain  
\(r_{\rm s}\): the largest radius of the smeared zone  
\(r_{\rm e}\): the largest radius of the influenced region  
\(H\): the thickness of the foundation  
\(u_{\rm a}\): the excess pore-air pressure  
\(u_{\rm w}\): the excess pore-water pressure  
\(u_{\rm a_{\rm a}}\): the excess pore-air pressure in the smeared zone  
\(u_{\rm w}\): the excess pore-water pressure in the smeared zone  
\(u_{0}^{0}\): the initial excess pore-air pressure  
\(u_{\rm w}^{0}\): the initial excess pore-water pressure  
\(u_{\rm a}\): the average excess pore-air pressure  
\(\overline{u}_{\rm w}\): the average excess pore-water pressure  
\(u_{\rm atm}\): the atmospheric pressure  
\(M\): the air mass molecular  
\(n\): the porosity  
\(S_{\rm r}\): the saturation  
\(R\): the universal air constant  
\(T\): the absolute temperature  
\(g\): the gravitational acceleration  
\(\gamma_{\rm w}\): the water unit weight