Tubular hydroforming provides a number of advantages over conventional stamping processes, including reduction in part counts and weight reduction, improved strength and stiffness of structural components, lower tooling costs, and higher dimensional accuracy. In order to provide hydroforming guidelines for product designers and process engineers, and to obtain fundamental understandings of such forming process, we begin a series of study on failure analysis of tubular hydroforming under internal pressure and end feeding. The focus of this paper is placed on the condition of the onset of bursting. Bursting is an instability phenomenon where the tube can't sustain any more internal pressure. Splitting usually follows due to extreme deformations in the bursting area. Onset of bursting depends on process parameters such as pressure and end feeding, as well as on material properties. A mathematical analysis is conducted in this paper for the conditions of onset of bursting. Numerical experiments with Finite Element Analysis (FEA) calibrates the analysis remarkably well. The onset conditions are determined for a variety of materials and process parameters. Other factors affecting the validity of the analysis is also investigated, including the end constraint effect and the effect from loading path deviating from proportional loading.