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During direct-integration time-history analysis, proper viscous-proportional damping is necessary for realistic axial response in columnsto 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.
Significant axial-force discrepancy between adjacent columns expected to demonstrate similar response indicates excessive c K K j damping contribution. This effect will magnify with shorter columns because their axial stiffness, and K j value, is larger. Given a dynamic loading condition, the cyclic bending of concrete sections will generate axial velocity. As CSI Software applies an elastic-perfectly-plastic stiffness relationship throughout analysis procedures. During time-history analysis, when stiff elements undergo inelastic behavior, additional measures may need to be taken to capture nonlinear response. This condition may be realized, 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 increase axial extension, velocity can become significant. This will cause improper damping to further impact results.Given such an instance, users should reduce stiffness-proportional damping in columns . Axial velocity and excessive c K K j damping contribution may then skew the 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 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 4.051E-03the value originally used in the load case. This is done through the interactive database editor Interactive Database Editor in the ‘Material Material Properties 06 – Material Damping’ Damping table under the ‘VisStiff’ VisStiff column. Users may also change manage properties through the ‘Define’ Define > ‘Materials’ Materials > ‘Advanced Properties’ Advanced Properties option.
- Add a copy of 5500psi material labeled ‘5500psi-lowdamping’ and set c K to a sufficiently small value such as 4.051E-05.
- Change the ‘column’ frame section property to use the ‘5500psi-lowdamping’ materialFor 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 columnseffected elements, without effecting the rest of the model. Nonlinear material behavior will then serve as the mechanism for energy-dissipation mechanism.
Users may also 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.
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