Tensegrities are prestressable trusses that have been proven to support various load distributions with minimum mass. This article presents a novel efficient method for designing lightweight tensegrities under local and global failure constraints. Local failure includes buckling and material yielding of individual members in the tensegrity. Global failure refers to global buckling of the tensegrity, where it loses stability without undergoing local failure at its individual members. The formulation and numerical approach to determine the critical global buckling forces and mode shapes of tensegrities with arbitrary shape and topology are first provided. Next, the design method considering local and global failure is presented, which starts with the local sizing of the member areas of the given tensegrity for the prevention of local failure. The method then determines the dominant failure mode by comparing the external forces and the critical global buckling force of the locally sized structure. If the critical global buckling force is larger than the external force, the dominant mode is a local failure and the locally sized design is returned as the minimum mass design. Conversely, if global failure is the dominant mode, different global reinforcement approaches are applied to raise the critical buckling force of the structure until it matches the external force, preventing global buckling. These reinforcement approaches include increasing the areas of the members and increasing the prestress in the tensegrity. Representative examples are provided to demonstrate the effectiveness of the design method considering box and T-bar tensegrities.