Abstract: For ultra-soft soils such as sludge, traditional single-method reinforcement often fails to achieve satisfactory stabilization. Assuming free strain and accounting for the smear effect, multi-stage loading, and time-dependent vacuum pressure, we derive an analytical solution for radial consolidation of soil under the combined effects of electro-osmosis, vacuum, and staged loading, using Bessel functions and the method of characteristic functions. The solution was verified by comparing the model degradation with existing solutions and laboratory test data. Subsequently, we examined the effects of multi-stage loading, the permeability coefficient, and vacuum pressure on soil consolidation. Results indicate that, compared with single-stage loading, multi-stage loading more effectively reduces excess pore water pressure during loading and prevents sharp increases in pore water pressure. Reductions in the electro-osmotic coefficient within the smear zone significantly delay pore pressure dissipation and slow soil consolidation. The hydraulic conductivity has little effect on pore pressure in the early consolidation stage but markedly affects the later stage; thus the smear effect under combined preloading cannot be ignored. The consolidation in undisturbed soil zones is more significantly influenced by electroosmosis, while consolidation in smear zones is more affected by vacuum pressure.

Key words: radial consolidation, electroosmotic-vacuum preloading, free strain, multi-stage surcharge preloading, analytical solution