A design methodology is proposed for deployable tube flexures made of ultra-thin carbon fiber composite. The methodology is developed to achieve desired stowage and deployment performance in the absence of material failure. The cut-out shape in the tube flexure is determined by means of a Bayesian optimization technique. The objective function is defined based on both desired moment characteristics and Hashin failure criteria. Multiple cut-out parameterization schemes are presented. Bayesian optimization is shown to be effective and significant performance improvements are achieved thought use of spline-based optimization in comparison to more conventional forms. The efficiency of the approach is validated via experiment.