Geosynthetic reinforcement over a grid of piles in soft soil can help transfer loads to the piles, and reduce settlements. This first part of a three-part study utilises a three-dimensional thin-plate model of the geosynthetic. The von Ka´rma´n theory, which includes the effects of bending and stretching, is used. The geosynthetic layer is represented using a linear stress–strain relationship. Owing to symmetry, a one-eighth triangular section of a square unit cell is considered, and the pile caps have either a square, diamond or circular shape. The soil between the pile caps and the geosynthetic, and the soft soil between the piles, are modelled as linear foundations with specified stiffnesses. The embankment stresses over the piles and over the soft soil are also specified. The numerical procedure involves finite differences and minimisation of the total energy. For two sets of parameter values, the effects of the shape of the pile caps on the displacements, strains and net stress reduction ratio are investigated. Sharp spikes in strain and stress occur at the edges of the pile caps. The second paper in this study adopts a three-dimensional cable-net model of the geosynthetic, and an axisymmetric model is analysed in the final paper.