Vertical-drain consolidation using stone columns: An analytical solution with an impeded drainage boundary under multi-ramp loading

An analytical solution is derived to predict consolidation with vertical drains under impeded drainage boundary conditions and multi-ramp surcharge loading. The impeded drainage is modelled by adopting the third type boundary condition with a dimensionless characteristic factor of drainage efficiency developed by Gray (1945) for one-dimensional consolidation. Fully drained and undrained boundary conditions can also be modelled by applying an infinite and a zero characteristic factor, respectively. The combined effects of drain resistance and smear are taken into account fully. An explicit, rigorous analytical solution is derived using the method of separation of variables to calculate excess pore-water pressure at any arbitrary point in soil and to derive the overall average degree of consolidation. The proposed solution can also be used to analyse one-dimensional consolidation without vertical drains but with an impeded drainage boundary. Its validity and accuracy are verified by comparing the proposed solution with the solutions developed by Gray (1945) and Terzaghi (1943). Its practical applicability is also evaluated by analysing a case history involving a fill embankment, which was constructed over a crust layer of hard soil overlying soft clay improved with stone columns. The crust layer is modelled as an impeded drainage. Reasonably good agreement is obtained between the consolidation results obtained from the proposed analytical solution and available three-dimensional finite-element predictions. With the further consideration of smear effects, good agreement is achieved between the consolidation results obtained from the proposed analytical solution and field measurements.