Soft tensegrity is a recent concept, which can be described as a prestressed assembly of soft cables in tension and bars in compression. In this paper, we investigate the impact of the bar behavior on the mechanical properties of a tensegrity. This is achieved through the change of bar material, from soft (e.g., silicone) to stiff (e.g., steel) materials. First, we show that it is possible to make a self-supporting tensegrity where both cables and bars are made of soft material. Second, we highlight that the bar material impacts the isotropy of the stiffness matrix of the tensegrity and the range of achievable stiffness. Lastly, we demonstrate that the bar material has a significant impact on the precision of stiffness variation, when active elements are integrated into the tensegrity. From these effects, we extract design guidelines for the choice of bar material in a soft tensegrity.