COMPLEMENTARY ENERGY PRINCIPLE FOR CABLE NETWORKS IN TERMS OF FORCE VECTOR VARIABLE

An alternative approach to the variation formulation of static equilibrium  problem for cable networks is proposed. The dual minimization principle is derived in terms of force variables. The full vector force in the cable member is chosen as the dual variable. Unlike the previously proposed treatments of the problem using the scalar axial force variable the elastic energy retains the most general form until the closing steps. It is only assumed to be a convex function of the member end-to-end vector. As a result the nonsmooth asymmetric response of the cable does not appear in the form of inequality in the very beginning. Instead it shows up in the final form of the complementary energy functional, which turn out to be non-differentiable for zero member forces. The kinematic relations between member elongations and nodal displacements convert into the force equilibrium conditions in terms of nodal reactions and member force vectors. The Dirichlet boundary conditions appear in the dual form as a linear term in the complementary energy functional. For the linear elastic cables this approach lead to a second-order cone optimization problem with two sets conical constraints. Such formulation provides flexibility and thus can be potentially extended to account for phenomena such as inter-fiber friction and adhesion.
Keywords: complementary energy, variational principles, cable networks