## 3 Seismic Numerical Analysis
### Modeling Procedure in OpenSees
#### 3.2.2 Modified Ibarra-Krawinkler Relationships  
In recent years, the Ibarra-Krawinkler model has been used for better modeling of structural behavior. Among the newer models is the Lignus model, which is based on laboratory observations. Below are some of the modifications made in the model [26]. The conventional approach to relating stiffness after yielding to initial stiffness, stiffness coefficient to, is high and far from expected, and it is better to use the ratio to define the stiffening behavior after the elastic state, which is the value for the beam. Additionally, the columns are obtained between 1.05 and 1.1. In the main model, the ratio is used as the input parameter defining the plastic deformation capacity. Based on the new data, providing better and more direct parameters in determining the behavior of steel components after elastic state and behavior after, respectively, gives a definition of basic energy dissipation capacity in the main model. This definition was considered a multiple of the yield force in the yield deformation. However, in this model, a more stable parameter (plastic deformation) is used [27]. \[\mathrm{E_{t}}=\gamma\times\mathrm{F_{Y}}\times\mathrm{\delta_{P}} \tag{3}\]  
It is better to use the parameter \(\Delta=\gamma\times\mathrm{\theta_{P}}\) to rotate the plastic joints. Since the parameter \(\mathrm{\theta_{P}}\) is dimensionless, the \(\Delta\) parameter will be compatible with the relation \(\mathrm{\beta_{i}}=\left[\frac{\mathrm{E_{i}}}{\mathrm{E_{t}}-\mathrm{\Sigma }_{i}-\mathrm{E_{i}}}\right]^{c}\), giving a more tangible understanding of plastic rotation. For example, if \(\mathrm{\theta_{P}}=0.5\) and \(\gamma=20\), then \(\Delta=1\). Figure 9 shows the modified Ibarra-Krawinkler Model used in this study for modeling the connections. In figure the parameters are:  
\(\delta_{\rm c}\): Deformation at the apex point  
\(\rm F_{\rm y}\): Effective yield strength (taking into account strain hardening)  
\(\rm\delta_{\rm y}\): Effective yield deformation  
\(\rm K_{\rm c}\): Effective elastic stiffness  
\(\rm f_{\rm r}\): Residual capacity  
\(\rm\delta_{\rm r}\): Residual deformation  
\(\rm\delta_{\rm u}\): Final deformation capacity  
\(\rm\delta_{\rm p}\): Plastic deformation  
\(\rm\delta_{\rm pc}\): Deformation capacity after Yielding  
\(\rm F_{\rm pp}\): Predicted yield resistance  
\(\rm F_{\rm pn}\): Nominal yield resistance  
K: Residual ratio  
The above model is based on the database collected from cyclic and uniform loading tests on steel members around the world, and after nonlinear regression analysis on the data, the following values are proposed for different sections [27]. (a) Beam with W-shape section  
\[\theta_{\rm p}=0.07\biggl{(}\frac{\rm h}{\rm t_{\rm w}}\biggr{)}^{-0.35}\cdot \biggl{(}\frac{\rm b_{\rm f}}{2\cdot\rm t_{\rm f}}\biggr{)}^{-0.09}\cdot\biggl{(} \frac{\rm L}{\rm d}\biggr{)}^{0.310}\cdot\biggl{(}\frac{\rm d}{\rm c_{\rm unit} ^{2.21}}\biggr{)}^{-0.281}\cdot\biggl{(}\frac{\rm c_{\rm unit}^{2}}{50}\biggr{)} ^{-0.383} \tag{4}\]  
\[\theta_{\rm pc}=4.645(\frac{\rm h}{\rm t_{\rm w}}\biggr{)}^{-0.449}\cdot(\frac {\rm b_{\rm f}}{2\cdot\rm t_{\rm f}}\biggr{)}^{-0.837}\cdot(\frac{\rm d}{\rm c _{\rm unit}^{2}\cdot 21})^{-0.265}\cdot(\frac{\rm c_{\rm unit}^{2}\cdot\rm F_{\rm y}}{50}) ^{-1.136} \tag{5}\]  
\[\Lambda=26.36(\frac{\rm h}{\rm t_{\rm w}})^{-0.589}\cdot(\frac{\rm b_{\rm f}}{ 2\cdot\rm t_{\rm f}})^{-0.574}\cdot(\frac{\rm c_{\rm unit}^{2}\cdot\rm F_{\rm y }}{50})^{-1.454} \tag{6}\]  
(b) Beam with Box-Shape section  
\[\theta_{\rm p}=0.572\cdot(\frac{\rm D}{\rm t})^{-1.00}\cdot(1-\frac{\rm N}{ \rm N_{\rm y}})^{1.210}\cdot(\frac{\rm c_{\rm unit}^{2}\cdot\rm F_{\rm y}}{50 })^{-0.838} \tag{7}\]  
\[\theta_{\rm pc}=14.51\cdot(\frac{\rm D}{\rm t})^{-1.217}\cdot(1-\frac{\rm N}{ \rm N_{\rm y}})^{3.035}\cdot(\frac{\rm c_{\rm unit}^{2}\cdot\rm F_{\rm y}}{50 })^{-0.498} \tag{8}\]  
Figure 9: Modified Ibarra–Krawinkler Model.