This article belongs to a series where the influence of anisotropic pressure on the gravitational properties of rigidly rotating fluids is studied using new exactsolutions of GR constructed for the purpose. For mathematical simplification, stationarity and cylindrical symmetry implying three Killing vectors are considered. Moreover, two pressure components are set to vanish in turn. In Papers 1 and 2 the pressure is axially directed, while it is azimuthal in Paper 3. In present Paper 4, a radially directed pressure is considered. Since a generic differential equation, split into three parts, emerges from the field equations, three different classes of solutions can be considered. Two could only be partially integrated. The other one, that is fully integrated, yields a set of solutions with negative pressure. Physical processes where a negative pressure is encountered are depicted and give a rather solid foundation to this class of solutions. Moreover, these fully integrated solutions satisfy the axisymmetry condition while they do not verify the so-called "regularity condition". But, since their Kretschmann scalar does not diverge on the axis, this feature must be considered as reporting a mere coordinate singularity. Finally, the matching of these solutions to an exterior appropriate vacuum enforce other constraints on the two constant parameters defining each solution in the class. The results displayed here deserve to be interpreted in the light of those depicted in the other four papers in the series.