GEOMETRICAL STIFFNESS OF A BEAM IN SPACE - CONSISTENT VW APPROACH

The application of the standard virtual work expressions to the large displacement-small strain domain merely requires the replacement of the standard linear strain-displacement relations by the quadratic ones. In fact, the geometrical stiffness matrix of an arbitrary finite element can be derived immediately from the virtual work of the second order terms in the strains. A correct geometrical stiffness is, however, only obtained if the prerequisites of the energy theorems are strictly observed. This proves to be a rather difficult task as soon as the elements contain rotational freedoms. In this case, the solution demands a comprehensive understanding of the true nature of the strains, stresses and nodal displacements, rotations as well as forces, moments. After an extensive study of the relevant entities the above principle is successfully applied to the derivation of the geometrical stiffness of a beam element in space. The consistency of the present approach is demonstrated by the full agreement with the prior results of the authors based on the natural mode technique [2,3]. Some numerical examples demonstrate the practical importance of the present development for the geometrically nonlinear analysis of three-dimensional frame structures. © 1979.