## 2 The Novel Equivalent Beam Model Calibration
### The Sub-Structure Approach (e-SBM)
\begin{split}\frac{1}{2}\int\Big{(}\{u^{\prime\prime}\}^{T}[D_{I} ]\{u^{\prime\prime}\}+\{\gamma\}^{T}[S]\{\gamma\}+\{\chi^{\prime}\}^{T}[D_{0}] \{\chi^{\prime}\}+D_{I}\theta^{2}\Big{)}dx=\\ \sum_{k=1}^{N}\frac{1}{2}\int\Big{(}\{u^{\prime\prime}\}^{T}[D_{ k}]\{u^{\prime\prime}\}+\{\gamma\}^{T}[S_{k}]\{\gamma\}+\{\chi^{\prime}\}^{T}[D_{0k}] \{\chi^{\prime}\}+D_{ik}\theta^{2}\Big{)}dx,\end{split}  \tag{31}\]with  
\[\{u\}=\{v,w,\psi\}^{T},\quad\{\chi\}=\big{\{}\chi_{y},\chi_{z},\theta_{B}\big{\}}^ {T},\quad\{\gamma\}=\big{\{}\gamma_{y},\gamma_{z},\theta_{S}\big{\}}^{T}, \tag{32}\]  
and where \([D_{0k}],[D_{lk}]\) and \([S_{k}]\) are the stiffness parameters of each equivalent beam substructure in the global reference system \(x,y,z\); \(D_{t}\) and \(D_{lk}\) are the torsional stiffness of the equivalent beam and the \(k\)-th sub-structure, respectively. These matrices are computed starting from the stiffness values in the local reference system \(\eta,\zeta\) (\(\eta\) horizontal axis and \(\zeta\) vertical axis)  
\[\left[D_{0}^{\eta,\zeta}\right]_{k}=\left[ \begin{array}{ccc}D_{0k}&0&0\\ 0&0&0\\ 0&0&0\end{array} \right],\quad\left[D_{l}^{\eta,\zeta}\right]_{k}=\left[ \begin{array}{ccc}D_{lk}&0&0\\ 0&0&0\\ 0&0&0\end{array} \right],\quad\left[S^{\eta,\zeta}\right]_{k}=\left[ \begin{array} []{ccc}S_{k}&0&0\\ 0&0&0\\ 0&0&0\end{array} \right], \tag{33}\]  
and then changing the orientation and location of the lateral load-resisting sub-system by the transformation  
\[[X_{k}]=[T]_{E}^{T}[X^{\eta-\zeta}]_{k}[T]_{K}\qquad\text{with}\ [T]_{k}=\left[ \begin{array}{ccc}\cos\alpha_{k}&-\sin\alpha_{k}&r_{k\gamma}\\ \sin\alpha_{k}&\cos\alpha_{k}&r_{k\zeta}\\ 0&0&1\end{array} \right]\qquad k=1,\ldots,N, \tag{34}\]  
where \(X\) can be \(D_{0}\), \(D_{l}\) or \(S\), \(\alpha_{k}\) is the angle between \(\eta\) and \(y\), and \(r_{k\gamma}\) and \(r_{k\zeta}\) are the coordinates of the origin of the system \((\eta,\zeta)\) in the reference system \((x,y)\). Solving Equation (31), the following relations are obtained  
\[[S] =\pi^{2}[B][C]^{-1}[B][C]^{-1}[B] \tag{35}\] \[=[B][C]^{-1}[B]\Bigg{(}[I]\frac{1}{l_{0}^{2}}[B]^{-1}[C]\Bigg{)} ^{-1}\] (36) \[=[A]-[B][C]^{-1}[B]\] (37) \[D_{t} =\sum_{k=1}^{N}D_{lk}, \tag{38}\]  
where  
\[[A] =\sum_{k=1}^{N}\Bigg{(}\Bigg{(}[I]+\frac{\pi^{2}}{l_{0}^{2}}[D_{0k }][S_{k}]^{-1}\Bigg{)}^{-1}[D_{0k}]+[D_{lk}]\Bigg{)} \tag{39}\] \[=\sum_{k=1}^{N}\Bigg{(}[I]+\frac{\pi^{2}}{l_{0}^{2}}[D_{0k}][S_{k }]^{-1}\Bigg{)}^{-1}[D_{0k}][S_{k}]^{-1}\Bigg{(}[I]+\frac{\pi^{2}}{l_{0}^{2}}[ D_{0k}][S_{k}]^{-1}\Bigg{)}^{-1}[D_{0k}]\] (40) \[=\sum_{k=1}^{N}\pi^{4}\Bigg{(}[I]+\frac{\pi^{2}}{l_{0}^{2}}[D_{0k }][S_{k}]^{-1}\Bigg{)}^{-1}[D_{0k}][S_{k}]^{-1}\] \[\times\Bigg{(}[I]+\frac{\pi^{2}}{l_{0}^{2}}[D_{0k}][S_{k}]^{-1} \Bigg{)}^{-1}[D_{0k}][S_{k}]^{-1}\Bigg{(}[I]+\frac{\pi^{2}}{l_{0}^{2}}[D_{0k }][S_{k}]^{-1}\Bigg{)}^{-1}[D_{0k}], \tag{41}\]  
Approximate values for \(l_{0}\) are suggested when vibration analyses are of interest. For example, \(l_{0}=2H\), \(l_{0}=2/3H\), \(l_{0}=2/5H\) can be assumed for modeling the first, second and third vibration mode, respectively.