During direct-integration time-history analysis, proper viscous-proportional damping is necessary to simulate the behavior of stiff elements which soften under 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.
CSI Software applies an elastic-perfectly-plastic force-deformation relationship during analysis. Users may need to implement additional measures to capture the nonlinear response of stiff elements which experience inelastic behavior during time-history analysis. Such a condition may occur, for example, when adjacent columns are expected to demonstrate comparable performance, but experience axial-force discrepancy. When initially-stiff columns are subjected to dynamic loading, inelastic cyclic bending will soften the elements through cracking and the ratcheting of yielding tensile rebar. Axial velocity and excessive c K K j damping contribution may then skew the results when 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 effected 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 softening elements, copy their material, scale c K by a value between 10 -2 and 10 -3, and then locally apply this material to effected elements.
Since material damping sums with that specified in load cases, this procedure reduces stiffness-proportional damping only in effected elements, without effecting the rest of the model. Nonlinear material behavior will then serve as the energy-dissipation mechanism.
When reduced damping causes convergence problems, 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 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).