Gaussian closure technique for chain like structures with elasto-plastic elements described by the Bouc hysteresis

Abstract

Hysteretic materials can be described using the Bouc model. Based on this model, analytic solutions for the Gaussian closure technique for one mechanical degree of freedom were presented previously. The solution is based on classical moments of a multivariate Gaussian distribution and, for the non-linear portion of the model, analytic solutions can be derived when the necessary multi-dimensional integrals are solved in the correct order. These results can be extended to multiple degrees of freedom as long as a description in relative coordinates is possible. This is true for chain-like structures of which the model of a multi storey building under wind or earthquake excitation or the vertical displacements of a vibration insulation are examples. A procedure is presented that allows the solution of all integrals by a sequence of substitutions similar to the single degree-of-freedom model. The method is evaluated using a two and a five storey frame and the results are in good agreement with Monte-Carlo results although a slight underestimation occurs for larger loads. As no numerical integration is necessary, the method is very fast. Furthermore, contrary to statistical linearization the method takes also the covariances between the nodes into account and allows the calculation of the transient case using a fast explicit time step procedure. Finally, the high number of moment equations which increases almost with the square of degrees of freedom does not seem to lead to instabilities.