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During nonlinear [direct-integration|Comparison between FNA and direct-integration time-history analyses] [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] elementsobjects may yield and then undergo significant [softening|kb:Material nonlinearity] due to yielding. If softening causes significant deformational velocity, significant damping forces may also result elementsin objects which are initially stiff. These damping forces, while properly While in equilibrium with other forces which occur at a [joint|kb:Joint] connected to the stiff elementobject, these damping forces may cause a jump in stiffness forces between the softening elementobject and connectedits interconnecting elementsobjects. 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 inwithin the samesubdivided column. While this jump in axial force does satisfiessatisfy dynamic equilibrium, such itbehavior may not be desirable, desiredand behavior.additional Usersmeasures may need to implementbe additionaltaken measures to achieve expected results proper response.


h2. Adjust Usersstiffness-proportional maydamping
solve
thisThis problem may be solved by transferring stiffness from the [load case|kb:Load case], general to the entire structure, to the material of individual elementsobjects which are affected by softening. This mayis be done through the following processas follows:

*# In the [time-history|kb:Time-history analysis] load case, leave the *{_}c{_}* *{_}{~}M{~}{_}* value, but change *{_}c{_}* *{_}{~}K{~}{_}* to zero.
*
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# For all materials, set *{_}c{_}* *{_}{~}K{~}{_}* to the value originally used in the load case. This is done throughusing [interactive database editing|kb:Interactive database editing] inunder VisStiff the> Material Properties 06 -- Material Damping table under the VisStiff column. UsersProperties may also managebe propertiesmanaged through the Define > Materials > Advanced Properties option.
* For.
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# Copy the softeningmaterial elements,of copysoftening their materialobjects, scale *{_}c{_}* *{_}{~}K{~}{_}* by a value between 10 {^\-2^}-2{^} and 10 {^\-3^}-3{^}, then apply this material locally to the affected elementsobjects.

Since material [damping|kb:Damping] sums with that specified in load cases, this procedure reduces stiffness-proportional damping only in affected elementsobjects, without affecting the rest of the model. [Nonlinear material behavior|kb:Material nonlinearity] will then account for energy dissipation.


h2. Convergence

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^1^/~3,~ while a value of \-1^1^/24~24~ or \-1^1^/12~12~ 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).


h1. See Also

* *Direct integration* -- [Direct-integration time-history analysis|kb:Direct-integration time-history analysis]