A direct method to predict cyclic steady states of elastoplastic structures
The asymptotic steady state behavior of an elasticperfectly plastic structure under cyclic loading may be determined by time consuming incremental time-stepping calculations. Direct methods, alternatively, have a big computational advantage as they attempt to find the characteristics of the cyclic state right from the start of the calculations. Most of these methods address an elastic shakedown state through the shakedown theorems and on the basis of mathematical programming algorithms. In the present paper, a novel direct method that has a more physical basis and may predict any cyclic stress state of a structure under a given loading is presented. The method exploits the cyclic nature of the expected residual stress distribution at the steady cycle. Thus, after equilibrating the elastic part of the total stress with the external load, the unknown residual stress part is decomposed into Fourier series whose coefficients are evaluated iteratively by satisfying compatibility and equilibrium with zero loads at time points inside the cycle and then integrating over the cycle. A computationally simple way to account for plasticity is proposed. The procedure converges uniformly to the true cyclic residual stress for a loading below the elastic shakedown limit or to an unsafe cyclic total stress, which may be used to mark the regions with plastic straining inside the cycle. The method then continues to determine whether the applied loading would lead the structure to ratcheting or to regions that alternate plastically. The procedure is formulated within the finite element method. A von Mises yield surface is typically used. Examples of application of one and two dimensional structures are included.
direct methods cyclic loading elastic shakedown fourier series ratcheting alternating plasticity