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

ePf1 ePf2 ePf3 ePf4

floating point values defining force points on the positive response envelope

ePd1 ePd2 ePd3 ePd4

floating point values defining deformation points on the positive response envelope

eNf1 eNf2 eNf3 eNf4

floating point values defining force points on the negative response envelope

eNd1 eNd2 eNd3 eNd4

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

gK1 gK2 gK3 gK4 gKLim

floating point values controlling cyclic degradation model for unloading stiffness degradation

gD1 gD2 gD3 gD4 gDLim

floating point values controlling cyclic degradation model for reloading stiffness degradation

gF1 gF2 gF3 gF4 gFLim

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:

Piinching4.jpg

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:

\[\delta k_i = \left( \text{gK1} (d_{max})^\text{gK3} + \text{gK2} \left (\frac{E_i}{E_\text{monotonic}} \right )^\text{gK3} \right ) &lt;= \text{gKLim}\]

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

Pinching4MaterialExample


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:

PEER 2003/10


Code Developed by: Nilinjan Mitra, University of Washington

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