Pinching4
This command is used to construct a uniaxial material that represents a ‘pinched’ load-deformation response and exhibits degradation under cyclic loading. Cyclic degradation of strength and stiffness occurs in three ways: unloading stiffness degradation, reloading stiffness degradation, strength degradation.
uniaxialMaterial Pinching4 $matTag $ePf1 $ePd1 $ePf2
$ePd2 $ePf3 $ePd3 $ePf4 $ePd4
< $eNf1 $eNd1 $eNf2 $eNd2 $eNf3 $eNd3 $eNf4 $eNd4 >
$rDispP $rForceP $uForceP
< $rDispN $rForceN $uForceN >
$gK1 $gK2 $gK3 $gK4 $gKLim $gD1 $gD2 $gD3 $gD4 $gDLim $gF1 $gF2
$gF3 $gF4 $gFLim $gE $dmgType
matTag
|
integer tag identifying material |
|
floating point values defining force points on the positive response envelope |
|
floating point values defining deformation points on the positive response envelope |
|
floating point values defining force points on the negative response envelope |
|
floating point values defining deformation points on the negative response envelope |
rDispP
|
floating point value defining the ratio of the deformation at which reloading occurs to the maximum historic deformation demand |
fFoceP
|
floating point value defining the ratio of the force at which reloading begins to force corresponding to the maximum historic deformation demand |
uForceP
|
floating point value defining the ratio of strength developed upon unloading from negative load to the maximum strength developed under monotonic loading |
rDispN
|
floating point value defining the ratio of the deformation at which reloading occurs to the minimum historic deformation demand |
fFoceN
|
floating point value defining the ratio of the force at which reloading begins to force corresponding to the minimum historic deformation demand |
uForceN
|
floating point value defining the ratio of strength developed upon unloading from negative load to the minimum strength developed under monotonic loading |
|
floating point values controlling cyclic degradation model for unloading stiffness degradation |
|
floating point values controlling cyclic degradation model for reloading stiffness degradation |
|
floating point values controlling cyclic degradation model for strength degradation |
gE
|
floating point value used to define maximum energy dissipation under cyclic loading. Total energy dissipation capacity is defined as this factor multiplied by the energy dissipated under monotonic loading. |
dmgType
|
string to indicate type of damage (option: “cycle”, “energy”) |
NOTES:

Damage Models:
Stiffness and strength are assumed to deteriorate due to the imposed “load” history. The same basic equations are used to describe deterioration in strength, unloading stiffness and reloading stiffness:
-
\[k_i = k_0(1 -\delta k_i)\]
where \(k_i\) is the unloading stiffness at time \(t_i\), \(k_0\) is the initial unloading stiffness (for the case of no damage), and \(\delta k_i\) (defined below) is the value of the stiffness damage index at time \(t_i\).
-
\[d_{\text{max i}} = d_{\text{max 0}}(1 -\delta d_i)\]
where \(d_{\text{max i}}\) is the deformation demand that defines the end of the reload cycle for increasing deformation demand, $d_{} $ is the maximum historic deformation demand (which would be the deformation demand defining the end of the reload cycle if degradation of reloading stiffness is ignored), and \(\delta d_i\) (defined below) is the value of reloading stiffness damage index at time \(t_i\).
-
\[f_{\text{max i}} = f_{\text{max 0}}(1 -\delta f_i)\]
where \(f_{\text{max i}}\) is the current envelope maximum strength at time \(t_i\), $f_{} $ is the initial envelope maximum strength for the case of no damage, and \(\delta f_i\) (defined below) is the value of strength value index at time \(t_i\).
The damage indices \(\delta k_i\), \(\delta d_i\), and \(\delta f_i\), may be defined to be a function of displacement history only (dmgType = "cycle"
) or displacement history and energy accumulation (dmgType = "energy"
). For either case, all of the damage indices are computed using the same basic equation.
If the damage indices are assumed to be a function of displacement history and energy accumulation, the unloading stiffness damage index, \(\delta k_i\) is computed as follows:
-
\[\delta k_i = \left( \text{gK1} (d_{max})^\text{gK3} + \text{gK2} \left (\frac{E_i}{E_\text{monotonic}} \right )^\text{gK3} \right ) <= \text{gKLim}\]
where
-
\[(d_{max} = \text{max} \left[ \frac{d_\text{max i}}{\text{def}_\text{max}}, \frac{d_\text{min i}}{\text{def}_\text{min}} \right ]\]
Examples
DESCRIPTION:
Stiffness and strength are assumed to deteriorate due to the imposed “load” history. The same basic equations are used to describe deterioration in strength, unloading stiffness and reloading stiffness:
REFERENCES:
Code Developed by: Nilinjan Mitra, University of Washington