## 2 Methodology
### FE Model Updating Using Cso
#### 2.2.2 Objective Function Formulation and Parameter Selection  
The procedure starts with the formulation of an objective function and the selection of parameters. The effectiveness of any optimization technique greatly depends on the formulation of this function. In this paper, the damage-sensitive quantities, namely, natural frequencies, the maximum amplitude of frequencies and shape correlation between experimental and numerical models of a planer frame structure are considered to formulate the objective function.  
The Newmark-Beta method was used for the calculation of dynamic responses. The displacement time-histories are calculated to generate strain time responses based on the strain-displacement relationship:  
\[\varepsilon_{n}(t)=-yx_{n}(t)^{\prime\prime} \tag{14}\]  
where the extreme fiber distance of a section calculated from the neutral axis is \(y\) and a double dash over \(x(t)\) indicates the double derivative of the displacement at that section. Suffix \(n\) represents the numerical FE model. Further, the strain time history is transferred to the frequency domain using the following expression:  
\[\varepsilon_{n}(\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\varepsilon_{n}( t)e^{-i\epsilon t}dt \tag{15}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Quantity** & **Pin Joint** & **Semi-Rigid Joint** & **Rigid Joint** & **Quantity** \\ \hline FF & 0–0.143 & 0.143–0.891 & 0.891–1 & FF \\ \hline \hline \end{tabular}
\end{table}
Table 1: Designation of connections concerning FF as per Eurocode-3.  
where \(\omega\) stands for frequency and \(\varepsilon_{n}(\omega)\) indicates amplitude corresponding to the frequency. For the experimental study, the strain dataset can be directly registered and transformed to the frequency domain using Fourier transform:  
\[\varepsilon_{\text{exp}}(\omega)=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty} \varepsilon_{\text{exp}}(t)e^{-i\omega t}dt \tag{16}\]  
In Equation (16) given above, \(exp\) stands for the experimental case. For \(i\)th strain sensor, the frequency shape co-relation utilizing experimental and numerical strain data are obtained as:  
\[\psi_{i}=\frac{\sum\varepsilon_{n,i}(\omega)\varepsilon_{\text{exp},i}(\omega )}{\left[\varepsilon_{n,i}^{2}(\omega)\right]^{1/2}\left[\varepsilon_{\text{ exp},i}^{2}(\omega)\right]^{1/2}} \tag{17}\]  
Equation (17) clearly states that if numerical and experimental structures are in the same condition then the shape co-relation value is about \(1\) and if they are not in the same condition and the value is deviating from \(1\). Finally, the objective function is formulated as:  
\[OF(\alpha_{Lr_{c}},\alpha_{Rr_{c}})=\sum\limits_{j=1}^{NT}\left[\sum\limits_{i =1}^{NS}W_{i}[\left\{\varepsilon(\omega)_{num}-\varepsilon(\omega)_{\text{ exp}}\right\}^{2}+\left\{\omega(\text{AMP}_{\text{max}})_{num}-\omega(\text{ AMP}_{\text{max}})_{\text{exp}}\right\}^{2}]_{i}\right]_{j} \tag{18}\]  
In the objective function (OF) presented by Equation (18), 'num' is numerical, 'exp' is experimental, 'NT' is number of trials, 'NS' is number of sensors, \(\alpha_{Lr}\) and \(\alpha_{Rr}\) represent the fixity factors for the left and right of each member. \(\omega(AMP_{max})\) indicates the natural frequency identified from the maximum amplitude of the frequency domain representation of the strain signal. A weight factor for each sensor, '\(W_{i}\)' is defined as:  
\[W_{i}=\frac{\psi_{i}}{\sum\psi_{i}^{2}} \tag{19}\]