# Improvement of Performance Level of Steel Moment-Resisting Frames Using Tuned Mass Damper System
## 6 Modeling and Verification  
A six-story steel frame is selected in this study (see Figure 2). The frame has already been studied by several researchers (see [39; 46]). A finite element software named OpenSees [47] is used to model the frame. Additionally, the modified Ibarra-Krawinkler (IMK) deterioration model (see [54; 55; 56]) is utilized to simulate the nonlinear behavior of the plastic joints. According to this constitutive model, the zero-length elements are considered as concentrated plasticity at the beam-column connections. The IMK model takes into account the cyclic response of the springs, which shows the nonlinear behavior of the frame. Due to the presence of the gravitational loads, as well as the TMD mass, the P-delta effect must be taken into account to consider the enhanced structural responses. For this purpose, a virtual column with truss elements attached to the base of the structure is analyzed. Additionally, all columns are considered to be fixed at the base of the structure. Since the rotational springs and frame elements are connected in series, the stiffness of the rotational elements must be modified in such a way that the stiffness value of elements and the actual stiffness of the frame are the same. For this purpose, the stiffness of the rotational springs is considered to be \(n=10\) times higher than the rotational stiffness of the elastic elements. The stiffness and also the moment of inertia of the elastic element are multiplied into the \((n+1)/n\). Finally, to match the nonlinear behavior of the elements and the actual behavior of the frame, the strain coefficient of the plastic joint should be modified according to Equation (14)--see [54; 55; 56; 57]. \[\alpha_{s,spring}=\alpha_{s,mem}/\left(1+n\cdot(1-\alpha_{s,mem})\right) \tag{14}\]  
where \(\alpha_{s,mem}\) shows the real strain coefficient of the frame, and \(\alpha_{s,spring}\) is the strain hardening coefficient of the rotational spring. All beam elements are subjected to the uniformly distributed load equal to 21.89 kN/m. Additionally, the following relations (Equations (15)-(22)) are used to determine the parameters required in the definition of the behavioral curve of the rotational springs [54; 55; 56; 57]:  
\[\begin{array}{c}\theta_{p}=0.318\Big{(}\frac{h}{t_{w}}\Big{)}^{-0.55}\Big{(} \frac{b_{f}}{2t_{w}}\Big{)}^{-0.345}\Big{(}\frac{t_{w}}{t_{y}}\Big{)}^{-0.023} \Big{(}\frac{t_{w}^{0.09}}{533}\Big{)}^{-0.33}\Big{(}\frac{t_{w}^{2}r_{y}}{35 5}\Big{)}^{-0.13}\\ \qquad\qquad  
\[\Lambda=536\big{(}\frac{h}{l_{w}}\big{)}^{-1.26}\Big{(}\frac{b_{f}}{2l_{w}}\Big{)}^ {-0.525}\Big{(}\frac{l_{b}}{l_{y}}\Big{)}^{-0.13}\Big{(}\frac{c_{\text{min}}^{2} \cdot F_{y}}{355}\Big{)}^{-0.291} \tag{17}\]  
\[\theta_{pc}=7.5\big{(}\frac{h}{l_{w}}\big{)}^{-0.61}\Big{(}\frac{b_{f}}{2l_{w}} \Big{)}^{-0.71}\Big{(}\frac{l_{b}}{r_{y}}\Big{)}^{-0.11}\Big{(}\frac{c_{\text{ min}}^{1},d}{533}\Big{)}^{-0.161}\Big{(}\frac{c_{\text{min}}^{2},F_{y}}{355} \Big{)}^{-0.32} \tag{18}\]  
\[\theta_{p}=0.0865\Big{(}\frac{h}{l_{w}}\Big{)}^{-0.365}\Big{(}\frac{b_{f}}{2l_ {w}}\Big{)}^{-0.14}\Big{(}\frac{l}{l}\Big{)}^{0.34}\Big{(}\frac{c_{\text{min}} ^{1},d}{533}\Big{)}^{-0.721}\Big{(}\frac{c_{\text{min}}^{2},F_{y}}{355}\Big{)} ^{-0.23} \tag{19}\]  
\[\Lambda=495\big{(}\frac{h}{l_{w}}\big{)}^{-1.34}\Big{(}\frac{b_{f}}{2l_{w}} \Big{)}^{-0.595}\Big{(}\frac{c_{\text{min}}^{2},F_{y}}{355}\Big{)}^{-0.36} \tag{20}\]  
\[d<21 in\]  
\[M_{y}=1.17\cdot Z\cdot F_{y} \tag{21}\]  
\[\theta_{y}=\Big{(}1.17\cdot Z\cdot F_{y}/6E\Big{)}/L \tag{22}\]  
where \(h\) is the web depth; \(b_{f}\) is the width flange of a beam; \(L_{b}\) is the distance from the column face to the nearest lateral brace; \(d\) is the beam depth. Additionally, \(r_{y}\) shows the radius of gyration about the \(y\)-axis of the beam, and \(t_{w}\) is the web thickness of the beam. The parameter \(Z\) is the plastic section modulus, and \(F_{y}\) shows the expected yield strength.