Modified Ibarra-Medina-Krawinkler Deterioration Model with Pinched  Hysteretic Response – MIMK_Pinched

The Modified Ibarra-Medina Krawinkler (MIMK) Deterioration curve with Pinched Hysteretic Response is based on the model that was primarily proposed by Ibarra et al. [2005] and was later modified by Lignos and Krawinkler [2009]. The model resembles to the Modified Ibarra-Medina-Krawinkler Deterioration Models with Bilinear and Peak-Oriented Hysteretic Response  since it is based on a backbone curve that represents the behavior for monotonic loading and establishes strength and deformation bounds, but uses a Pinched  hysteretic Model to model hysteresis of the backbone curve. The Pinched hysteretic is similar to the peak-oriented one, except that reloading consists of two parts. Initially the reloading path is directed towards a ‘break point’, which is a function of the maximum permanent deformation and the maximum load experienced in the direction of loading (Ibarra et al.[2005]. The model includes four modes of cyclic deterioration: a. basic strength deterioration, b. post-capping strength deterioration, c. unloading/reloading stiffness deterioration and d. accelerated reloading stiffness deterioration. Modified Ibarra-Medina Krawinkler Deterioration Model with Pinched Hysteretic Response is able to simulate the behaviour of reinforced concrete beams that primarily fail in a shear mode. This model is also able to simulate the hysteretic behaviour of shear connections, beam-to-column gusset plate connections and wooden components. In total 27 variables have to be defined for the Model.

Elastic Stiffness (Ke)
The default value is 200000

Effective yield strength for positive loading direction (fy(+))
The default value is 300

Effective yield strength for negative loading direction (fy(-))
The default value is 300

Plastic rotation capacity for positive loading direction (p(+))
Displacement difference between displacement at the yielding point and displacement at the maximum strength point (positive loading). The default value is 0.025

Plastic rotation capacity for negative loading direction (p(-))
Displacement difference between displacement at the yielding point and displacement at the maximum strength point (negative loading). The default value is 0.025

Post - capping rotation capacity for positive loading direction (pc(+))
Displacement difference between displacement at the maximum strength point and at the zero strength point (positive loading). The default value is 0.3

Post-capping rotation capacity for negative loading direction (pc(-))
Displacement difference between displacement at the maximum strength point and at the zero strength point (negative loading). The default value is 0.3

Ultimate rotation capacity for positive loading direction (u(+))
Displacement after which the strength equals the residual strength (positive loading). The default value is 0.4

Ultimate rotation capacity for negative loading direction (u(-))
Displacement after which the strength equals the residual strength (negative loading). The default value is 0.4

Residual Strength  Ratio for positive loading direction (k(+))
Ratio of the residual strength to the effective yield strength fy(+) for the positive loading direction. The default value is 0.3

Residual Strength Ratio for negative loading direction (k(-))
Ratio of the residual strength to the effective yield strength fy(+) for the negative loading direction. The default value is 0.3

Strain Hardening Ratio for positive loading direction (s(+))
Hardening ratio for the definition of the stiffness after the yielding point for positive loading. The default value is 0.03

Strain Hardening Ratio for negative loading direction (s(-))
Hardening ratio for the definition of the stiffness after the yielding point for negative loading. The default value is 0.03

Ratio between force at start of reloading to the force corresponding to the maximum experienced deformation for positive loading direction (Fpr(+))
Ratio used to define the maximum "pinched" strength corresponding to the breakpoint toward which the reloading branch is directed (positive loading). The default value is 0.8

Ratio between force at start of reloading to the force corresponding to the maximum experienced deformation for negative loading direction (Fpr(-))
Ratio used to define the maximum "pinched" strength corresponding to the breakpoint toward which the reloading branch is directed (negative loading). The default value is 0.8

Ratio of Reloading Stiffness (Kr)
Parameter used for defining the displacement corresponding with the breakpoint toward which the reloading branch is directed. The default value is 0.8

Cyclic deterioration parameter for strength deterioration (s)
Parameter affecting the deterioration of the effective yield strength fy (see Figure 6a in Ibarra et al. 2005). A smaller Λs leads to faster deterioration for example Λs equal to 0.5 causes higher deterioration than Λs equal to 1.5. Use of Λs equal to zero disables the deterioration mode. The default value is 0.6

Cyclic deterioration parameter for post-capping strength deterioration (c)
Parameter affecting the deterioration of the post-capping strength (see Figure 6b in Ibarra et al. 2005). A smaller Λc leads to faster deterioration for example Λs equal to 0.5 causes higher deterioration than Λc equal to 1.5. Use of Λc equal to zero disables the deterioration mode. The default value is 0.6

Cyclic deterioration parameter for accelerated reloading  deterioration()
Parameter affecting the deterioration of the reloading stiffness for each cycle (see Figure 6d in Ibarra et al. 2005). A smaller Λa leads to faster deterioration for example Λa equal to 0.5 causes higher deterioration than ?a equal to 1.5. Use of Λa equal to zero disables the deterioration mode. The default value is 0.6

Cyclic deterioration parameter for unloading stiffness deterioration (K)
Parameter affecting the deterioration of the unloading stiffness (see Figure 6c in Ibarra et al. 2005). A smaller ΛK leads to faster deterioration for example ΛK equal to 0.5 causes higher deterioration than ΛK equal to 1.5. Use of ΛK equal to zero disables the deterioration mode. The default value is 0.6

Strength deterioration rate(Cs)
Exponent used in the computations of the effective yield strength deterioration parameters (see equation (3) in Lignos and Krawinkler,2011) and is usually taken between 1 and 2.The default value is 1.0

Post-capping strength deterioration ratio (Cc)
Exponent used in the computations of the post-capping strength deterioration parameters (see equation (3) in Lignos and Krawinkler,2011) and is usually taken between 1 and 2. The default value is 1.0

Accelerated reloading deterioration ratio (C)
Exponent used in the computations of the reloading stiffness deterioration parameters (see equation (3) in Lignos and Krawinkler,2011) and is usually taken between 1 and 2. The default value is 1.0

Unloading stiffness deterioration rate (CK)
Exponent used in the computations of the effective unloading stiffness deterioration parameters (see equation (3) in Lignos and Krawinkler,2011) and is usually taken between 1 and 2. The default value is 1.0

Rate of cyclic deterioration in positive loading direction (D(+))
Parameter used for creating assymetric deterioration in the positive direction. If set larger than 1 the deterioration will be higher in the negative direction. The default value is 1.0

Rate of cyclic deterioration in negative loading direction (D(-))
Parameter used for creating assymetric deterioration in the negative direction. If set larger than 1 the deterioration will be higher in the negative direction. The default value is 1.0

Factor for elastic stiffness amplification (Ν)
Factor used for the amplification of the elastic stiffness Ke by (1+N). If N is set equal to zero, no amplification is carried out. The default value is 0.0

The capacity deformations are better described in the following graph: