During nonlinear direct-integration time-history analysis, special consideration may be needed for modeling stiffness-proportional damping of stiff elements which experience inelastic softening. 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, nonlinear elements may undergo significant softening due to yielding. This may generate significant damping forces in elements that are initially stiff if the softening results in significant deformational velocity. These damping forces, while properly in equilibrium with other forces at a joint connected to the stiff element, may cause a jump in stiffness forces between the softening element and connected elements. Such a condition may occur, for example, in a concrete column modeled with multiple elements containing hinges. When the initially-stiff column is 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 generate large differences in the axial force between adjacent elements in the same column. While this jump is in axial force satisfies dynamic equilibrium, it may not be the desired behavior. Users may need to implement additional measures achieve the expected results.
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 provide energy dissipation.
If damping reduction creates 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 to -1/12 should improve the rate of convergence without significantly affecting the accuracy of the results.
Additional details and descriptions may be found in the CSI Analysis Reference Manual (Nonlinear Time-History Analysis > Nonlinear Direct-Integration Time-History Analysis > Damping).