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Vertical drains have been widely used in soil improvement projects to accelerate consolidation and improve the strength of soft soil. Previous consolidation models are mainly based on instantaneous loading, which cannot reflect the actual time-variable loading condition. This paper presents a set of analytical solutions for consolidation with vertical drains under depth and time-dependent loading, involving one-step loading, multi-step loading, and cyclic loading. A four-element model, which is a combination of the Merchant model and Maxwell model, is introduced to consider the rheological characteristics of the soil. By simulating the results of a consolidation test, it is found that the four-element model is more accurate than the Merchant model in predicting the changes in pore pressure and settlement during the clay consolidation process. Based on the solutions obtained, several factors affecting consolidation behavior are investigated. It has been shown that the rheological behavior becomes more and more obvious at the later stage of consolidation with the decrease both in the modulus of the spring in the Kelvin body and the viscosity coefficient of the independent dashpot. With the increase in the viscosity coefficient of the dashpot in the Kelvin body, the rate of consolidation becomes faster at an initial stage but slower at a later stage. For cyclic loading, the consolidation degree in each cycle reaches a maximum at the end of unloading and reaches a minimum at the beginning of the loading. more...