A Semianalytical Solution for Consolidation of Soft Soil with Partially Penetrating Vertical Drains Using a Radial Segmented Drainage Boundary

Foundations with partially penetrated vertical drains are common in areas with thick, soft soils. This study presents a semianalytical solution for the consolidation of foundations with partially penetrating vertical drains. The model assumes the concept of a virtual vertical drain and introduces a piecewise function to define a radial segmented drainage boundary, effectively capturing differences in consolidation behavior between the penetrated and unpenetrated layers. A novel solution for excess pore-water pressure and the average degree of consolidation across the entire foundation is derived. The governing equation for consolidation is formulated using an axisymmetric consolidation model, which is solved through Fourier series expansion and Laplace transforms. The solution theoretically degenerates to Terzaghi’s one-dimensional (1D) consolidation theory under a permeable surface. The validity of the proposed solution is verified by comparing it with other solutions. The distribution of excess pore-water pressure and consolidation behavior are analyzed, revealing that the dissipation of excess pore-water pressure is strongly influenced by drainage boundaries, with faster dissipation observed near the permeable surface and vertical drains. Consolidation is observed to occur in two distinct phases: an early stage dominated by the penetrated layer and a later stage governed by the unpenetrated layer. Additionally, the effects of key parameters, such as the depth ratio of the vertical drain, the thickness–width ratio of the model, and the permeability anisotropy coefficient, on consolidation are examined. The parametric analysis shows that the depth ratio primarily affects the time ratio between the two consolidation stages, while the thickness–width ratio and permeability anisotropy coefficient primarily influence radial seepage. These results offer valuable theoretical insights for optimizing drainage system designs.