# Parameter Identification of Structures with Different Connections Using Static Responses
## 5 Parameter Identification for a Frame with Semi-Rigid Connections  
### Member Stiffiness Matrix of Semi-Rigid Frame  
The connection type of joint plays a key role in the structural analysis. The analysis and design of steel frames are usually performed under the assumption that the beam-column connections are rigid or pin. In actual engineering, there are some frame structures with rigid connections, which have a certain rotational stiffness owing to damage and other factors. The effect of nodal rotational stiffness on the structural analysis should be considered in the damage identification of frame structures. This study considers frame element connections, including specific nodal rotational stiffness. Semi-rigid connections are modeled by zero-length attaching the rotational springs [28,29] with rotational stiffnesses \(R_{1}\) and \(R_{2}\) at the ends of the member, as shown in Figure 7.  
Figure 5: Cross-sectional area and moment of inertia as a function of the number of iterations. (**a**) Cross-sectional area; (**b**) Moment of inertia.  
Figure 6: Function value for rigid frame damage identification.  
The nodal rotational stiffness represents the degree of rotation capacity between the beam and the column. Parameters \(\beta_{1}\) and \(\beta_{2}\) are defined as the fixity factors at ends 1 and 2 of the element, respectively, and are related to the corresponding rotational spring stiffnesses \(R_{1}\) and \(R_{2}\), respectively. The member stiffness matrix of a frame element with semi-rigid connections at the ends is given [30; 31] in Equation (8). By setting both the parameters \(\beta_{1}\) and \(\beta_{2}\) equal to 1, Equation (8) becomes equivalent to the member stiffness matrix of the rigid frame element; this means that the nodal rotational stiffnesses are infinite. The unknown parameters, i.e., cross-sectional area \(A_{u}\) and moment of inertia \(I_{u}\), of the damaged member are included in the following matrix. Additionally, the joint damage can be represented as a reduction in the connection rigidity [20; 32], and this study used the rotational stiffness \(R_{u}\) to indicate the joint damage condition. The rotational stiffness \(R_{u}\) is also included in the following matrix.  
\[\mathbf{k^{\prime}}=\frac{EI_{u}}{L}\begin{bmatrix}\frac{A_{u}}{I_{u}}\\ 0&\frac{12}{L^{2}}\left(\frac{n_{1}}{n_{7}}\right)\\ 0&\frac{6}{L}\left(\frac{n_{2}}{n_{7}}\right)\\ -\frac{A_{u}}{I_{u}}&0&0\\ 0&-\frac{12}{L^{2}}\left(\frac{n_{1}}{n_{7}}\right)-\frac{6}{L}\left(\frac{n_ {2}}{n_{7}}\right)\\ 0&\frac{6}{L}\left(\frac{n_{2}}{n_{7}}\right)&2\left(\frac{n_{2}}{n_{7}} \right)&0&-\frac{6}{L}\left(\frac{n_{2}}{n_{7}}\right)\\ \end{bmatrix} \tag{8}\]  
where parameters \(n\) are defined using joint fixity factors \(\beta_{1}\) and \(\beta_{2}\) as follows:  
\[n_{1}=\beta_{1}+\beta_{2}+\beta_{1}\beta_{2},\ n_{2}=2\beta_{1}+\beta_{1}\beta _{2},\ n_{3}=3\beta_{1}\beta_{2},\ n_{5}=3\beta_{1}\beta_{2},\ n_{6}=2\beta_{2 }+\beta_{1}\beta_{2}\mu_{7}=4-\beta_{1}\beta_{2} \tag{9}\]  
Joint fixity factors \(\beta_{1}\) and \(\beta_{2}\) can be defined using the nodal rotational stiffness of the end springs as follows:  
\[\beta_{1}=\frac{1}{1+\left(\frac{3 EL_{u}/L}{R_{u1}}\right)}\ \ \ \ \ \beta_{2}=\frac{1}{1+\left(\frac{3 EL_{u}/L}{R_{u2}}\right)} \tag{10}\]  
where \(R_{u1}\) and \(R_{u2}\) are the unknown nodal rotational spring stiffnesses at different ends of the element, and \(E\) and \(L\) are the modulus of elasticity and length of the member, respectively.