Basic (Discontinuous) and Smoothing-Up (Conjugated) Solutions in Transfer-Matrix Method for Static Geometrically Nonlinear Beam and Cable in Plane

Abstract

A new approach to a geometrically nonlinear (GN) static two-dimensional (2D) beam-cable problem is given. It has four new features. First, the search for the ultimate solution (US) is presented as sum of basic (BS) and smoothing (SS) solutions, where BS is discontinuous and SS is to ensure geometrical continuity and forces equilibrium in each point of the structure. Second, BS establishes the constant curvature reference geometry, in which basic bending moment and basic axial force are already embedded together with the resulting basic elongation and curvature. BS geometry gives the local system of coordinates according to which SS has to be determined. Third, a number of analytically exact SS for distributed loading are derived in a form convenient for applying the transfer matrix method. These exact SS are given for four different elements: a cable, a linear curved beam, and geometrically nonlinear tensed or compressed curved beams. Fourth, the efficient procedure of correction of BS based on SS derived at each iteration is proposed, taking into account whether or not SS is oscillating to slow down or speed up this correction. Four elaborated examples are investigated. They explore the optimal number and length of elements, as well as ranges of application of particular SS. It is shown that the GN beam element is the best choice able to reduce the required number of elements and iterations in tens of times. For tensed geometries, combined application of GN beam and cable elements is found to be very effective. The method is very stable, and its convergence is insensitive not only to the initial geometry configuration, but also to the order and position of each particular element. The advantage of special meshing is discussed, where the size/number of elements is controlled by sections where the bending moment is nearly constant.