Stiffness and stability of deployable pantographic columns

Abstract

Straight or curved planar pantographic columns are an integral part of the three-dimensional deployable structures made of scissor-like elements. It is shown that, under the assumption of linear behaviour, the stiffness characteristics of such straight columns can be derived analytically. Equivalent bending and axial stiffness for uniform pantographic columns are obtained by equating their displacements to those of uniform solid columns. The behaviour of pantographic structures in bending is found to be very similar to that of conventional structures. Under axial load, the geometric non-linearity caused by the finite rotations of bars in the scissor-like elements must be taken into account. The system of non-linear equations describing the structural response in this case is solved numerically. The buckling analyses for local snap-through and global column-like type of instability are performed. The analytical procedures developed for the structures made of identical scissor-like elements are modified for non-uniform columns.

The behaviour of three-dimensional triangular and square columns in bending is considered. The analytical procedure includes the force method of analysis for a single prismatic pantographic unit and the displacement method of analysis for the whole structure. The derived expressions indicate that deflections of a spatial column consist of primary and local parts. The primary part can be obtained by superposition of the equivalent bending stiffnesses of the planar columns forming the sides of the spatial structure. The contribution of each side depends on its orientation with respect to the plane of bending. The local part of the solution originates from the support conditions at the bottom of the column. The similarities and differences in the behaviours of pantographic and solid columns are discussed.