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In the development of unit-cell theory for the analytical analysis of consolidation with vertical drains, the equal-strain assumption is often made with the intention of modelling consolidation under uniform settlement conditions. In contrast, the free-strain assumption for modelling consolidation under uniform load conditions is seldom employed, mainly because of the complexities involved in the analysis. This study derives a rigorous analytical solution to the generalised goving equations of free-strain consolidation with a vertical drain subjected to an instantaneous load. Calculated results from the newly proposed solution are compared with those from three available solutions derived based on the equal-strain assumption. Surprisingly good agreement is obtained in terms of excess pore-water pressure, degree of consolidation, and settlement. Horizontal profiles of settlement were not uniform before the end of consolidation. This indicates that the uniform settlement condition is not actually reproduced by the analytical solutions derived based on the equal-strain assumption. The equal-strain assumption is a sufficient but not necessary condition for deriving an analytical solution to unit-cell consolidation theory. The assumption plays no role in modelling consolidation under uniform settlement conditions but simplifies the analytical analysis of free-strain consolidation and results in an approximate solution of high accuracy for consolidation under uniform load conditions. Moreover, drain resistance and smear effects not only retard the consolidation rate, but also importantly shape the vertical and horizontal profiles of excess pore-water pressure, respectively.