On the rigidity of triangulated hyperbolic paraboloids

The paper investigates the rigidity of two classes of three-dimensional pin-jointed assemblies which describe triangulated surfaces of hyperbolic paraboloidal shape: type 1 with straight boundaries, and type 2 with parabolic boundaries. The two arrangements have identical horizontal plans. The investigation begins with a brief historic review of techniques for detecting lack of rigidity. The behaviour of both type-1 and type-2 assemblies depends on the number n of bars on each edge of the structure and, although both geometric arrangements satisfy Maxwell rule (3j ═ b, where j is the number of joints and b is the number of bars) for any value of n, all type-1 assemblies with n even and greater than 2 and type-2 assemblies with n > 1 are in fact not rigid. This result is first proved for some specific cases by the zero-load test; and then for arbitrary values of n. The equilibrium equations of each joint are written down in terms of a stress function, and are then assembled in an equilibrium matrix whose rank is found by general matrix manipulations. The number of inextensional mechanisms and static redundancies is also obtained. The rigidity of assemblies consisting of four interconnected type-1 hyperbolic paraboloidal sheets is also discussed, and it is concluded that they are equally prone to the same kind of misbehaviour.