During nonlinear direct-integration time-history analysis, proper viscousspecial consideration may be needed for modeling stiffness-proportional damping is necessary to simulate the behavior of stiff elements which experience dynamic-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:
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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, CSI Software defaults to apply an elastic-perfectly-plastic force-deformation relationship. 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 deformation, and hence velocity. These damping forces, while properly in equilibrium with other forces at a joint connected to the stiff element, may cause an unexpected jump in stiffness forces between elements connected to Users may need to implement additional measures to capture the nonlinear response of stiff elements which soften under dynamic-inelastic behavior. Such a condition may occur, for example, when adjacent columns are expected to demonstrate comparable dynamic performance, but experience significant axial-force discrepancy. When initially-stiff columns are 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 skew those results generated through default settings.
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