Lateral and vertical swelling pressures associated with expansive soils cause damages on structures. These pressures must be predicted before the structures are constructed in order to prevent the damages. The magnitude of the stresses can decrease rapidly when volume changes are partly allowed. Therefore, a material, which has a high compressibility, must be placed between expansive soils and the structures in both horizontal and vertical directions in order to decrease transmitted swelling pressure on structures. There are numerous techniques recommended for estimating the swelling pressures. However, these techniques are very complex and time-consuming. In this study, a new estimation model to predict the pressures is developed using experimental data. The data were collected in the laboratory using a newly developed device and experimental setup also. In the experimental setup, a rigid steel box was designed to measure transmitted swelling pressures in lateral and vertical directions. In the estimation model, approaches of artificial neural network (ANN) and adaptive neuro-fuzzy inference systems (ANFIS) are employed. In the first stage of the study, the lateral and vertical swelling pressures were measured with different thicknesses of expanded polystyrene geofoam placed between one of the vertical walls of the steel box and the expansive soil in the laboratory. Then, ANN and ANFIS approaches were trained using these results of the tests measured in the laboratory as input for the prediction of transmitted lateral and vertical swelling pressures. Results obtained showed that ANN-based prediction and ANFIS approaches could satisfactorily be used to estimate the transmitted lateral and vertical swelling pressures of expansive soils.