This model is similar to the kinematic model, but accounts for the increasing strength with plastic deformation that is typical of bucking-restrained braces, causing the backbone curve, and hence the hysteresis loop, to progressively grow in size. It is intended primarily for use with axial behavior, but can be applied to any degree of freedom.
Two measures are used for plastic deformation:
Maximum plastic deformation in each of the positive and negative directions
Accumulated plastic deformation, which is the absolute sum of each increment of positive or negative plastic deformation. Plastic deformation is that which does not occur on the two elastic segments of the force-deformation curve.
Accumulated plastic deformation can occur under the cyclic loading of constant amplitude.
For this model, the following parameters are required:
Separately for tension (positive) and compression (negative) deformations
Hardening factor at maximum deformation, h, where h > 1.0
Maximum plastic deformation level at full hardening, x2, as a multiple of yield deformation, where x2 > 1.0
Maximum accumulated plastic deformation level at full hardening, x4, as a multiple of yield deformation, where x4 > 1.0
Accumulated deformation weighting factor, a, where 0.0 < a < 1.0.
The hardening factors scale the size of the backbone curve and hysteresis loop in the action (stress/force/moment) direction. Because the accumulated plastic deformation is constantly increasing, it is recommended that the weighting factor a generally be small or zero.
For each increment of deformation:
The absolute maximum positive and negative plastic deformations that have occurred up to this point in the analysis are determined, dmaxpos and dmaxneg, as well as the accumulated plastic deformation dacc.
Comparing dpos with the positive deformation level d2, obtained by multiplying x2 by the positive yield deformation, a hardening factor hmaxpos;can be determined by interpolation. If dmaxpos >d2, then hmaxpos = h.
Comparing dacc with the positive deformation level d4, obtained by multiplying x4 by the positive yield deformation, a hardening factor haccpos can be determined by interpolation. If dacc > d4, then haccpos = h
The net hardening factor due to positive deformation, hpos , is computed as
hpos = ahaccpos + (1-a) hmaxpos
Degradation does not occur during monotonic loading. However, upon load reversal, the curve for unloading and reverse loading in the opposite direction is modified according to the hardening factor computed for the last deformation increment. This is done by scaling the action values in that direction, including the backbone curve for further loading.
Important! Positive deformation and the corresponding hardening parameters only affect the negative strength, and vice versa.
Note that if the hardening factor is equal to 1.0, this model degenerates to the kinematic hysteresis model.
This behavior is illustrated the figure.
See Also: