Supported ceramic membranes based on mixed ionic and electronic conductors are a promising technology for oxygen separation applications. In addition to chemically induced stress under oxygen activity gradients in the materials, strain mismatch between membrane and support gives rise to considerable stress that may compromise mechanical reliability. This paper presents an analysis of stress generated in tubular supported membranes during operation. Closed-form analytical solutions for stresses due to external pressures, strain gradients, and mismatch in materials properties are derived. Stress distributions in two membrane systems have been analyzed and routes to minimize stress are proposed. For a Ba0.5Sr0.5Co0.8Fe0.2O3−δBa0.5Sr0.5Co0.8Fe0.2O3−δ membrane supported on a porous substrate of the same material under pressure-vacuum operation, the optimal configuration in terms of minimizing the risk of fracture by ensuring only compressive stresses in the component is achieved by placing the support on the feed side of the membrane. For a Ce0.9Gd0.1O1.95−δCe0.9Gd0.1O1.95−δ membrane on a MgO support, stress due to thermal strain mismatch is as large as that due to oxygen activity gradient. Tailoring the thermal expansion coefficient of the support is an effective method to alleviate the total stress. Failure criteria for membrane fracture under compression are thereafter presented. It is found that the tolerable flaw size for fracture in compression is in the millimeter range for both membrane systems at operating conditions in the range of practical interest.