Minimum principle of complementary energy of cable networks by using second-order cone programming

The minimum principle of complementary energy is established for cable networks involving only stress components as variables in geometrically nonlinear elasticity. It is rather amazing that the complementary energy always attains minimum value at the equilibrium state irrespective of the stability of cable networks, contrary to the fact that only the stationary principles have been presented for elastic trusses and continua even in the case of stable equilibrium state. In order to show the strong duality between the minimization problems of total potential energy and complementary energy, the convex formulations of these problems are investigated, which can be embedded into a primal–dual pair of second-order cone programming problems. The existence and uniqueness of solution are also investigated for the minimization problem of complementary energy.