Analytical solution of the bending of a bi-convex boom

Abstract

This paper derives closed-form solutions for the local deformation of a bi-convex boom under circular bending, and the resulting strain energy and self-extending force. Convex tapes and bi-convex booms that consists of a pair of convex tapes can be stored into a small volume and have high specific rigidity. They extert a self-extending force when stored cylindrically. Therefore, they have been proposed as members of deployable space structures. In this paper, two types of bi-convex booms are considered. In the first, the tapes of the bi-convex boom are bonded to each other at their edges; in the second, the tapes are wrapped in a cylindrical braid mesh. The latter is called a BCON (braid-coated bi-convex) boom. The tape of a BCON boom can slip on each other, and do not separate from each other because of the tension of the mesh net. Consequently, the BCON boom can be used in an ultralight self-deployable structure with quite high stowage volume efficiency and specific rigidity. However, structures using convex tapes or BCON booms have been designed and developed through a trial-and-error process because there is no appropriate formula for the self-extending force of convex tapes. This paper proposes a formula for the deformation of a convex tape that is initially bent into a circular shape. The deviation from the circular shape is obtained by solving the equilibrium equations. The deformation of a bi-convex boom is also derived by using the solution for a convex tape. Thus the theory described in this paper contributes to the design of space structures using convex tapes in bi-convex booms, as well as to the structural mechanics of flexible beams.

Keywords

• flexible structure
• self-deployable structure
• convex boom
• nonlinear deformation
• analytical solution