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During nonlinear [direct-integration|kb:Comparison of FNA and direct-integration time-history analysis] [time-history|kb:Time-history analysis] analysis, special consideration may be necessary for modeling the stiffness-proportional *damping of stiff elements* which experience *inelastic softening*. As explained in the {new-tab-link:http://www.csiberkeley.com/}CSI{new-tab-link} [_Analysis Reference Manual_|doc:CSI Analysis Reference Manual] (Viscous Proportional Damping, page 79), the [damping|kb:Damping] matrix for element *{_}j{_}* is computed as follows:


!Figure 1.PNG|align=center,border=0!


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|kb:Nonlinear] elements may undergo significant [softening|kb:Material nonlinearity] due to yielding. If softening causes significant deformational velocity, significant damping forces may also result elements which are initially stiff. These damping forces, while properly in equilibrium with other forces at a [joint|kb: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 in a concrete column modeled using multiple elements which contain [hinges|kb:Hinge]. When the initially-stiff column is subjected to [cyclic|kb:Material nonlinearity#Hysteretic cycle] 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 in axial force satisfies dynamic equilibrium, it may not be desired behavior. Users may need to implement additional measures to achieve expected results.

Users may solve this problem by transferring stiffness from the [load case|kb: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 [interactive database editing|kb:Interactive database editing] 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|kb:Material nonlinearity] will then account for energy dissipation.

If reduced damping creates convergence problems, users should apply [Hilber-Hughes-Taylor|Time-history output-acceleration accuracy] (HHT) 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 or \-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 {new-tab-link:http://www.csiberkeley.com/}CSI{new-tab-link} [_Analysis Reference Manual_|doc:CSI Analysis Reference Manual] (Nonlinear Direct-Integration Time-History Analysis > Damping, page 415).