A simple theory of geometrical stiffness with applications to beam and shell problems

Abstract

Geometrical stiffness is the basis for any attempt to study the behaviour of slender beams and thin shells under conditions in which large deflections may occur with small strains. Not all problems require high accuracy in the representation of the geometrical stiffness. These are generally certain self-equilibrating stress systems (natural modes) which are the principal contributors to the geometrical stiffness. In particular, stress systems which produce rigid body moments due to rigid body rotations of the element are generally most important. Also, very great differences in bending stiffness about different axes may make it necessary to consider otherwise unimportant natural forces.

Although beams are considered among the simplest of structural elements their analysis when bent and twisted in three dimensions is by no means simple and the same is true of the consideration of their geometrical stiffness in space. Thus the beam in space may be considered as a test case for the general methods developed here.

Large deflection theory of plate and shells is generally concerned with deflections of the order of the thickness which are sufficient to induce considerable membrane stresses. Thus the non-linear effect arises from the induced membrane stresses rather than from gross changes in geometry. The problem of snap through and the perhaps rather academic problem of the three dimensional elastica pose some very difficult finite element applications in which the geometry changes are of the order of the structural dimensions. To tackle such problems using a highly sophisticated shell element such as SHEBA is not an easy undertaking. For other more immediately practical reasons it has been necessary to develop a simple flat facet shell element with transverse shear deformation also. This element, which is a displacement but not a Rayleigh-Ritz element, has only 18 nodal freedoms and is adaptable to thin, thick and sandwich type applications, is especially suitable for large deflection problems.

The paper presents some large deflection examples for beams and it is hoped also to have ready some non-trivial applications to shells.