For direct-integration time-history analysis, improper viscous-proportional damping may lead to unrealistic axial response. As explained in the CSI Analysis Reference Manual (Material Properties > Material Damping > Viscous Proportional Damping), a damping matrix is computed as follows:
Here, c M and c K are the mass- and stiffness-proportional damping coefficients, M j is the mass matrix, and K j is the initial stiffness matrix. Dynamic equilibrium is then computed as the sum of stiffness forces, damping forces, inertial forces, and applied loading.
When significant differences are reported for axial forces in adjacent columns expected to demonstrate similar response, the c K K j contribution is generating large damping forces. This effect will magnify with shorter columns because their axial stiffness is larger, increasing K j. Given a dynamic loading condition, the cyclic bending of a concrete section will generate axial velocity. As axial extension increases with cracking and the ratcheting of yielding tensile rebar, velocity can become significant. This will cause improper damping to further effect results.
In such a case, users should reduce stiffness-proportional damping in column elements through the following process:
- In the time-history load case, leave the c M value, but change c K to zero.
- For all materials, set c K to 4.051E-03. This is done through the interactive database editor in the ‘Material Properties 06 – Material Damping’ table under the ‘VisStiff’ column. Users may also change properties through the ‘Define’ > ‘Materials’ > ‘Advanced Properties’ option.
- Add a copy of 5500psi material labeled ‘5500psi-lowdamping’ and set c K to a sufficiently small value such as 4.051E-05.
- Change the ‘column’ frame section property to use ‘5500psi-lowdamping’ material.
Since material damping sums with that specified in the load case, this procedure reduces stiffness-proportional damping only on the column, without effecting the rest of the model. Nonlinear material behavior will serve energy dissipation.
Users may also apply Hilber-Hughes-Taylor integration to the load case using a small negative HHT-alpha value. The prescriptive range is 0 to -1/3. A value of -1/24 should improve the rate of convergence, cutting analysis duration by up to a factor of 3.
Additional details and descriptions may be found in the CSI Analysis Reference Manual (Nonlinear Time-History Analysis > Nonlinear Direct-Integration Time-History Analysis > Damping).
