During direct-integration time-history analysis, proper viscous-proportional damping is necessary to simulate the behavior of stiff elements which experience softening under dynamic-inelastic response. As explained in the CSI Analysis Reference Manual (Material Properties > Material Damping > Viscous Proportional Damping), the damping matrix for element j 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.
During analysis, CSI Software defaults to apply an elastic-perfectly-plastic force-deformation relationship. Users may need to implement additional measures to capture the nonlinear response of stiff elements which soften under dynamic-inelastic behavior. Such a condition may occur, for example, when adjacent columns are expected to demonstrate comparable dynamic performance, but experience significant axial-force discrepancy. When initially-stiff columns are subjected to cyclic bending, cracking and the ratcheting of yielding tensile rebar will soften element response. Axial velocity and excessive c K K j damping contribution may then skew those results generated through default settings.
Users may solve this problem by transferring stiffness from the load case, general to the entire structure, to the material of individual elements affected by softening. This may be done 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 the value originally used in the load case. This is done through the Interactive Database Editor in the Material Properties 06 – Material Damping table under the VisStiff column. Users may also manage properties through the Define > Materials > Advanced Properties option.
- For the softening elements, copy their material, scale c K by a value between 10 -2 and 10 -3, then apply this material locally to the affected elements.
Since material damping sums with that specified in load cases, this procedure reduces stiffness-proportional damping only in affected elements, without affecting the rest of the model. Nonlinear material behavior will then serve energy dissipation.
When damping reduction disrupts convergence, users should 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 three.
Additional details and descriptions may be found in the CSI Analysis Reference Manual (Nonlinear Time-History Analysis > Nonlinear Direct-Integration Time-History Analysis > Damping).
