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Origami manufacturing has led to considerable advances in the field of foldable structures with innovative applications in robotics, aerospace, and metamaterials. However, existing origami are either load-bearing structures that are prone to tear and fail if overloaded or resilient soft structures with limited load capability. In this manuscript, we describe an origami structure that displays both high load bearing and high resilience characteristics. The structure, which is inspired by insect wings, consists of a prestretched elastomeric membrane, akin to the soft resilin joints of insect wings, sandwiched between rigid tiles, akin to the rigid cuticles of insect wings. The dual-stiffness properties of the proposed structure are validated by using the origami as an element of a quadcopter frame that can withstand aerodynamic forces within its flight envelope but softens during collisions to avoid permanent damage. In addition, we demonstrate an origami gripper that can be used for rigid grasping but softens to avoid overloading of the manipulated objects.