# An Equivalent Non-Uniform Beam-Like Model for Dynamic Analysis of Multi-Storey Irregular Buildings
## 2 The Beam-Like Model
### Shear and Torsional Beam Stiffness Optimization  
The proposed non-uniform beam-like approach, in spite of its simplicity, can be enhanced to reproduce accurately the dynamic behaviour of 3D frame structures by coupling an iterative procedure aiming at optimizing the calibration of the shear and torsional stiffness of the equivalent beam, as described in what follows. Precisely, in order to consider all the unknown or improperly identified structural properties, three coefficients \(k_{x},k_{y},k_{c}\) are applied to correct the shear and the torsional stiffness as Equation (4) shows. It is worth noting that the eigen-properties of the non-uniform beam-like model are strictly dependent on the values of the correction coefficients \(k_{x},k_{y},k_{c}\). In order to identify the best values of \(k_{x},k_{y},k_{c}\), appearing in Equation (4), the following objective function \(\mathcal{O}\) is introduced:  
\[\mathcal{O}\Big{(}k_{x},k_{y},k_{c}\Big{)}=\underbrace{\sqrt{\sum_{j=1}^{n} \left(\frac{\omega_{j}\Big{(}k_{x},k_{y},k_{c}\Big{)}-\widetilde{\omega}_{j} \Big{)}}{\widetilde{\omega}_{j}}}}_{frequencies}+\underbrace{\beta\Bigg{[}1- \sum_{j=1}^{n}\sum_{k=1}^{m}\frac{\widetilde{\boldsymbol{\varphi}}_{jk}\cdot \widetilde{\boldsymbol{\varphi}}_{jk}^{T}}{\sqrt{\left(\boldsymbol{\varphi}_{ jk}\cdot\boldsymbol{\varphi}_{jk}^{T}\right)\left(\widetilde{\boldsymbol{\varphi}}_{jk} \cdot\widetilde{\boldsymbol{\varphi}}_{jk}^{T}\right)\right]}}}_{modal\,shapes} \quad, \tag{17}\]  
where \(n<3N\) and \(\beta\) is a Boolean parameter.  
The objective function defined in Equation (17), dependent on the stiffness coefficients \(k_{x},k_{y},k_{c}\), represents a measure of the deviation with respect to target values of a given number \(n\) of natural frequencies \(\omega_{j}\) and modal displacement vector \(\boldsymbol{\varphi}_{jk}=\left[\varphi_{u,j}(\zeta_{k};k_{x},k_{y},k_{c}) \ \ \varphi_{v,j}(\zeta_{k};k_{x},k_{y},k_{c})\ \ \ \varphi_{\delta,j}(\zeta_{k};k_{x},k_{y},k_{c})\right]\), evaluated at suitably chosen \(m\) floor levels \(\zeta_{k},\ k=1,\ldots,m\). The target values, denoted in Equation (17) with a superimposed tilde \"-\", represent experimental values that could be obtained by means of dynamic identification methods. Bearing in mind that the present work aims at assessing the reliability of the proposed beam-like model and the suitability of the selected values for the stiffness coefficients \(k_{x},k_{y},k_{c}\), rather than testing a proper execution of dynamic identification methods, the target frequencies \(\widetilde{\omega}_{j}\) and modal shapes \(\widetilde{\boldsymbol{\varphi}}_{j}\) have been numerically obtained by means of detailed FEM models.  
It is worth highlighting that the Boolean parameter \(\beta\) allows to include or neglect the modal shapes in the definition of the objective in Equation (17). As it will be shown in the applicative section, for buildings with only one irregularity (either planar or vertical) the knowledge of the first natural frequencies is sufficient for the determination of reliablf r69o8896(deviation)-896(wi0 7.5716 Tf 4.423 -1u.618(is)-317(wo  
Figure 3: Scheme of (a) the eigen-vector matrix and (b) the j-th eigen-vector.  
Lastly, the dynamic response in the geometric space is obtained by means of the following expressions:  
\[\begin{array}{l}u_{x}(z,t)=\sum\limits_{j=1}^{S}\sum\limits_{i=1}^{N}\psi_{i}(z )\hat{\Psi}_{ij_{k}}\cdot z_{j}(t)\\ u_{y}(z,t)=\sum\limits_{j=1}^{S}\sum\limits_{i=1}^{N}\psi_{i}(z)\hat{\Psi}_{ij _{y}}\cdot z_{j}(t)\\ \vartheta(z,t)=\sum\limits_{j=1}^{S}\sum\limits_{i=1}^{N}\psi_{i}(z)\hat{\Psi }_{ij_{y}}\cdot z_{j}(t)\end{array}\, \tag{22}\]  
where \(S\) is the number of modes of vibration and \(N\) is the number of shape functions.