The type of stayed column considered was developed by the ASTRO Research Corporation for NASA to evaluate, and is designed to be deployed in space after forming a light, compact package in the Space Shuttle. It consists of a core with three identical stay frames equally spaced around it. Each stay frame has a central compressed spoke which pretensions stays which radiate from its end to the core. Three pretensioned battens connect the ends of the spokes. NASA performed vibration tests on a model of such a column and compared their results with approximate theoretical predictions. The present paper gives exact theoretical results, including natural frequencies and mode types not given in the NASA results. A substitute column method is used to achieve economy and a much deeper understanding of the modes. Approximations which give even greater economy and convenience are evaluated. A proposed design is analyzed alongside the NASA model. The proposed design is better than the NASA model because recent developments were implemented in the design method, and also because it has a graphite epoxy core whereas the NASA model had an aluminium one, due to temporary availability problems.