### Nuzhdin L.V. Mikhaylov V.S. Konovalova S.V.

## Modeling of material heterogeneous damping properties of foundation structures and subsoil in SCAD FEA software

### Reporter: Mikhaylov V.S.

One of the most difficult aspects of performing dynamic calculations for building structures is the consideration of the damping properties of materials. In contrast to engineering tasks, building materials and soils have not only much greater variability of properties and tolerance of initial imperfections, but often enough within the same task they require simultaneously taking into account heterogeneous damping in structures made of different materials and in soils.

When performing linear dynamic calculations of bearing structures, three main methods are widely used: calculations based on modal analysis, taking into account the coincidence of external influences with their own oscillation periods; calculation of amplitude-frequency characteristics and response spectra of structures in the frequency range; direct integration of the equations of motion over time. In the absence of damping, with a small step of integration over time and with full account of the natural vibration forms, all three methods give almost identical results for simple systems. However, each method has its own strengths and weaknesses.

As a rule, for homogeneous material design schemes, as a reference comparison, the calculation results are given on the basis of a modal analysis with the same modal damping coefficients, which are assigned in tabular form according to regulatory documents and generally accepted engineering and scientific reference publications. However, a comparative analysis of the exact solutions obtained in the frequency range reveals a certain error in the modal approach, which can be ignored with a slight damping below 5%. This error in complex continual systems and at a higher level of damping is related to the fact that a dynamic reaction can be associated not only with critical modal reactions in the main forms of oscillations, but also in lower forms of oscillations that are supercritical or subcritical, which are numerical methods calculated with errors.

The disadvantage of most computational complexes is that the numerical computation using the direct integration of the equations of motion over time uses the "Rayleigh" damping model, which makes it possible to obtain a dissipation matrix based on an approximate analysis of the frequencies of free oscillations only for systems of many degrees of freedom that are homogeneous in material. This method, moving away from the physicality of the damping model by taking into account the mass matrix, which should not physically affect the dissipative forces, proposes to reproduce the specified modal damping coefficient at two a priori defined frequencies. The "sagging" dependence of attenuation on frequency is guaranteed to underestimate the modal damping in the interval between a pair of selected frequencies, which by the middle of the interval can be significant and lead to an overestimated conservatism of the calculation results. In this case, an artificial underestimation of damping sometimes leads to an "anomalous" underestimation of seismic reactions, which is explained by the connection of the reactions with the frequency composition of the impact relative to the natural frequencies of the system. The “anomaly” of underestimation of seismic reactions can be detected by a variational comparison of the response spectra at various given damping values, instead of choosing one particular spectrum.

Thus, modal method has an advantage for homogeneous structures. The advantage of direct dynamic calculation with the Rayleigh dissipation matrix is the ability to take into account the inhomogeneity of the damping of oscillations in the base, by modeling it using special finite elements of viscous dampers.

In SCAD version 21.1.9.3, a fundamentally new approach has been introduced in which the dissipation matrix C is represented as damping by material type, regardless of the oscillation frequency of the structure, which extremely simplifies the use of direct dynamics methods for solving standard engineering problems. As a result, in the direct integration mode of the equations of motion, the damping can be set unique for each material assigned to the finite elements in the dialog box for describing the stiffness properties. This opens up the possibility of creating direct physical models of foundations and structures with different properties of soils and structures using standard and generally accepted values of the damping coefficient in fractions of the critical.

The authors, in their study, demonstrate the coincidence of the exact numerical-analytical calculation of the pile in the soil array using the mathematical package Mathcad with the results of the numerical calculation in the SCAD computer complex by direct integration of the equations of motion over time with material damping in the properties of the pile materials and the soil foundation, as well as when replacing the soil viscous dampers. Recommendations are given on methods for constructing direct dynamic models of the base from bulk finite elements and continually-rod models of structures with inhomogeneous damping, distant and absorbing borders of the soil mass.

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