Modified Ibarra-Medina-Krawinkler Deterioration curve with Bilinear Hysteretic Response – MIMK_bilin

The Modified Ibarra-Medina Krawinkler (MIMK) Deterioration curve is based on the model initially proposed by Ibarra et al. [2005] and later modified by Lignos and Krawinkler [2011]. The MIMK is based on a backbone curve that represents the behaviour for monotonic loading and establishes strength and deformation bounds, and a set of rules that define the basic characteristics of the hysteretic behavior between the bounds. Hysteresis is modelled by standard bilinear hysteretic rules with kinematic strain hardening. The model also includes three modes of cyclic deterioration namely: a. basic strength deterioration, b. post-capping strength deterioration and c. unloading/reloading stiffness deterioration. The model can be utilized to model the cyclic moment-rotation relationship in plastic hinge areas of beams and the model variables for the case of steel beams can be defined by empirical relationships deduced from experimental data, from more than 300 experiments, by Lignos and Krawinkler [2011]. The model can be applied to any force-deformation relationship, though it has initially been described in terms of moment and rotation quantities. In total 22 variables have to be defined for the model (an online tool for defining the model parameters in the case of steel beams can be found in http://dimitrios-lignos.research.mcgill.ca/databases/component/).

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 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

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

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

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

post-capping strength deterioration rate (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

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 negatve 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 (N)
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: