Application of method of discontinuous basic and enhanced smoothing solutions for 3D multibranched cable. (15th January 2022) - Orynyak, Igor; Mazuryk, Roman
Graphical abstract: Highlights: Looking for solution for 3D cable is presented as a sum of discontinuous basic solution and smoothing one. Smoothing solution is an alternative to catenary solution and is able to account for any loading. New retardation-acceleration procedure of form finding is proposed, which reacts on whether or not the solution is converging. Cable system always reaches its correct position inrespective to any random initial position of its elements. Abstract: The paper is devoted to the development of new algorithm of form finding for 3D multibranched cable under static loading. Each cable is presented as a number of elements. For each element the looking for Ultimate solution, US, is presented as a sum of Basic solution, BS, and Smoothing solution, SS. BS present itself as the part of ideal circle with embedded axial force, the length of which corresponds to this force and initial length. The radius, axial force and initial position of BS is gradually refined at each iteration to maximally equilibrate the outer forces. BS is discontinuous one, it does not exactly equilibrate outer forces, there are no continuity at the border between two neighboring elements, and the boundary conditions are not fulfilled for it. All this is solved by SS, which is derived in local coordinate system related to the basic geometry. Here new analytical SS for 3D cable is proposed, which takes into account the uniformly and linearly distributed both normal and tangent forces. Algorithm of the accelerated or decelerated refinement of BS based on the current convergent or divergent behavior of SS is further elaborated. A lot of practical examples are considered, which show a very good accuracy of the proposed approach for the form finding with no relation to the initial random placement of basic geometry.