An analytical solution for consolidation with vertical drains under multi-ramp loading

Various analytical solutions have been proposed for a unit-cell consolidation with a vertical drain under surcharge loading. These solutions involve different assumptions to address various aspects of consolidation. There is a lack of generalised solution for analysing consolidation of soil assisted by a vertical drain under various loading conditions. This paper presents a simplified solution for consolidation under multi-ramp loading. Generalised governing equations of equal-strain consolidation are solved. Simultaneous radial and vertical flow conditions, as well as the combined effects of drain resistance and smear, are taken into account fully. An increase in total stress due to multi-ramp loading is reasonably modelled as a function of both time and depth. An analytical solution to calculate excess pore-water pressure at any arbitrary point in soil is derived by using the method of separation of variables. The conventional definition of the degree of consolidation is given in terms of the dissipation of excess pore-water pressure as a result of the maximum increase in total stress in soil. This definition is interpreted in relation to the ultimate ground surface settlement due to surcharge preloading. Its validity and accuracy are verified by comparing the special cases of the proposed solution with two available analytical solutions. The proposed solution is also validated by a well-documented case history with settlement and pore-water pressure measurements. Reasonably good agreement is obtained. A new degree of dissipation is defined in terms of the dissipation of excess pore-water pressure as a result of currently induced increase in total stress in soil. By using this definition, an equation is proposed to estimate the gain in undrained strength of soil due to consolidation for assessing the stability of surcharge fills more effectively and correctly. The loading path over time and the compressibility of smeared soil are shown to have a potentially important influence on the degree of consolidation and the degree of dissipation.