Modified Ibarra-Medina-Krawinkler Deterioration Model with Peak-Oriented Hysteretic Response – MIMK_peak

The Modified Ibarra-Medina Krawinkler (MIMK) Deterioration curve with Peak-Oriented Hysteretic Response is based on the model initially proposed by Ibarra et al. [2005] and later modified by Lignos and Krawinkler [2009]. The model resembles to the Modified Ibarra-Medina-Krawinkler Deterioration Model with Bilinear 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 Peak-Oriented hysteretic Model to model hysteresis of the backbone curve. The Peak-Oriented hysteretic Model follows the rules proposed by Clough and Johnston [1966] and later modified by Mahin and Bertero [1976]. 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 Peak-Oriented Hysteretic Response is able to simulate the behaviour of reinforced concrete beams that primarily fail in a flexural mode and has been calibrated with more than 200 Reinforced Concrete beams by Lignos and Krawinkler [2012]. In total 24 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

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 ?c 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 Λα leads to faster deterioration for example Λα equal to 0.5 causes higher deterioration than Λα equal to 1.5. Use of Λα 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: