During direct-integration time-history analysis, proper viscous-proportional damping is necessary for realistic axial response in columns. As explained in the CSI Analysis Reference Manual (Material Properties > Material Damping > Viscous Proportional Damping), the 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.
Significant axial-force discrepancy between adjacent columns expected to demonstrate similar response indicates excessive c K K j damping contribution. This effect will magnify with shorter columns because their axial stiffness, and K j value, is larger. Given a dynamic loading condition, the cyclic bending of concrete sections will generate axial velocity. As cracking and the ratcheting of yielding tensile rebar increase axial extension, velocity can become significant. This will cause improper damping to further impact results.
Given such an instance, users should reduce stiffness-proportional damping in columns 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 the ‘5500psi-lowdamping’ material.
Since material damping sums with that specified in load cases, this procedure reduces stiffness-proportional damping only in columns, without effecting the rest of the model. Nonlinear material behavior will serve as the mechanism for 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 as much as 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).