Inflatables can provide advanced functionality such as structural support, controlled compliance, posability, self-actuation, etc when they are architecturally constrained. Unfortunately, some of the key benefits of inflatables, including their low architectural complexity and high deploy-and-stow capability, are typically adversely affected as functionalities become more sophisticated. A new architectural approach, a tendon constrained inflatable (TCI), is introduced to decouple functionality from deploy-and-stow capability. A TCI is a structure composed of a soft inflatable bladder with rigid end caps connected by inextensible internal constraint tendons. When a TCI is inflated, the tendons under pure tension impose constraints on the inflatable by pre-tensioning the TCI to be able to resist external loads, but when not pressurized, the soft bladder and flexible tendons collapse and provide a compact stow. This paper develops the fundamental case of TCI functionality, rigid axial load bearing, by providing predictive models, validation experiments, design space plots, and a design case study. A model predicting the rigid load-bearing capacity as a function of pressure for varying TCI architectural design parameters is derived through (a) an unconstrained inflation model, based on an axisymmetric nonlinear membrane deformation model and a hyperelastic material model, and (b) a constraint model that bounds specific inflation directions depending on the applied constraint. This axial rigid load-bearing TCI performance model is validated through experiments measuring TCI deformation due to pressure and external load. This model serves as the basis to develop versatile design space plots that provide various views of the design parameter dependencies. These plots are demonstrated in the context of a deployable modular support design case study. The insights gained enable the design of TCIs with rigid load-bearing functionality in a deployable and stowable package as well as setting the foundation to develop more sophisticated architecture with decoupled functionality and deploy-and-stow capability.