During nonlinear direct-integration time-history analysis, special consideration may be needed necessary for modeling the stiffness-proportional damping of stiff elements which experience inelastic softening. As explained in the CSI CSI Analysis Reference Manual (Viscous Proportional Damping, page 79), 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 objects may yield and then undergo significant softening due to yielding. This may generate . If softening causes significant deformational velocity, significant damping forces in elements that may also result in objects which are initially stiff if the softening results in significant deformational velocity. These damping forces, while properly . While in equilibrium with other forces which occur at a joint connected to the stiff elementobject, these damping forces may cause a jump in stiffness forces between the softening element and connected elementsobject and its interconnecting objects. Such a condition may occur , for example, in a concrete column modeled with using multiple elements containing which contain 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 within the same subdivided column. While this jump in axial force satisfies does satisfy dynamic equilibrium, it such behavior may not be the desired behavior. Users desirable, and additional measures may need to implement additional measures be taken to achieve the expected results.Users may solve this problem proper response.
Adjust stiffness-proportional damping
Problems associated with inelastic softening may be solved by transferring stiffness from the load case, general to the entire structure, to the material of individual elements objects which are affected by softening. This may be done through the following processis done as follows:
- 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
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- using interactive database editing under VisStiff > Material Properties 06
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- - Material Damping
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- .
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- Properties may also
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- be managed through
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- Define > Materials > Advanced Properties
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- .
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- Copy the
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- material of softening objects, scale c K by a value between 10-2 and 10-3, then apply this material locally to the affected
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- objects.
Since material 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 will then provide account for energy dissipation.
Convergence
If reduced damping reduction creates convergence problems, users should apply Hilber-Hughes-Taylor (HHT) integration to the load case using a small negative HHT-alpha value. The prescriptive range is 0 to -1/3. A , while a value of -1/24 to 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 CSI Analysis Reference Manual (Nonlinear Time-History Analysis > Nonlinear Direct-Integration Time-History Analysis > Damping, page 391415).
See Also
- Direct integration – Direct-integration time-history analysis