Method for calculating strongly damped systems with non-proportional damping

The calculation of systems under seismic excitations is performed both dynamically by time integration and quasistatically under inertial seismic loads using linear response-spectra method (RSM). Dynamic timing calculation can be performed either using direct integration of the initial system of motion equations, or by using the spectral decomposition of motion equations by shape modes. RSM is completely based on spectral decomposition. However, the spectral decomposition was worked out only for systems with proportional damping, when the eigenvectors of the undamped and damped systems coincide. With regard to RSM, even for proportional damping, the existing Guide Lines do not allow to take into account the actual damping in the system. There are proposals for the explicit calculation of damping within the framework of the RSM for proportional damping in literature. Their results can be used both for constructing RSM and for integrating motion equations with arbitrary damping using the spectral decomposition of the motion equations. But so far the mentioned mathematical results have not been connected with calculating structures. The authors propose a variant of the RSM for calculating highly damped systems under earthquake impact. To this aim, complex eigenvectors and eigenvalues of the motion equation system were obtained, and this system was reduced to a tridiagonal form. As a result, the assumed equation system of the order equal to N was decomposed into N pairs of independent real equations. The base oscillation accelerogram and its derivative present the input in the right part of the motion equations. In this way two matrices of seismic forces are generally obtained. To sum up these forces, the shape mode correlation coefficients were analyzed. An example of mass damper calculation is given in the paper. For 4-6-story buildings constructed in ordinary conditions the proposed variant of the RSM leads to the same data that existing Guide Lines. But the proposed variant of RSM makes it possible to calculate systems with heterogeneous damping including seismic isolated systems, mass dampers and systems with soil-structure interaction which is impossible to do on the base of the existing Guide Lines.