# An Equivalent Non-Uniform Beam-Like Model for Dynamic Analysis of Multi-Storey Irregular Buildings
## 2 The Beam-Like Model
### The Equations of Motion of the Proposed Non-Uniform Beam-Like Model  
In order to evaluate the response of the non-uniform equivalent beam by introducing a limited number of degrees of freedom, a Rayleigh-Ritz discretization was performed. The discretization is based on the choice of an appropriate \(N\) number of modal shapes of a uniform shear cantilever beam, as displacement shape functions of the-non uniform beam, defined as follows:  
\[\psi_{m}(\zeta)=\sin\biggl{(}\frac{\pi}{2}(2m-1)\zeta\biggr{)}\hskip 14.226378 ptm =1,2,\ldots,\infty, \tag{5}\]  
where \(\zeta=z/h\) is the dimensionless abscissa of the beam, \(z\) being the along-axis abscissa and \(h\) the beam length. All the introduced displacements can therefore be expressed as  
\[ \begin{split} u_{x}(\zeta,t)&=\sum\limits_{i=1}^{N} \psi_{i}(\zeta)q_{ix}(t)\\ u_{y}(\zeta,t)&=\sum\limits_{i=1}^{N}\psi_{i}(\zeta)q _{iy}(t)\\ \delta_{z}(\zeta,t)&=\sum\limits_{i=1}^{N}\psi_{i}( \zeta)q_{iy}(t)\end{split} \hskip 5.690551 pt, \tag{6}\]  
\(q_{ix}(t),q_{iy}(t),q_{iy}(t)\) being the generalized \(i\)-th coordinates along the \(x,y,\delta\) directions, which represent the contribution of the single shape function to the total response. The equations of motion of the proposed beam-like model are derived through the application of [PERSON]'s principle. Displacements \(u_{gx}(t)=u_{x}(\zeta=0,t)\), \(u_{gy}(t)=u_{y}(\zeta=0,t)\) at the base \((\zeta=0)\) are considered in the formulation in order to take into account seismic excitations. In the presentformulation, primes and dots denote differentiation with respect to the normalized abscissa \(\zeta\) and time \(t\), respectively. In absence of non-conservative forces, [PERSON]'s principle can be reduced to the contribution of kinematic \(T\) and elastic energy \(U\) only, as follows:  
\[\int\limits_{t_{0}}^{t_{1}}[\delta(T-U)]dt=0\qquad\forall\,t_{0},t_{1}, \tag{7}\]  
where \(\delta(T-U)\) indicates the variation of the of the kinetic energy \(T\) and the elastic energy \(U\), whose relevant expressions are provided in the Appendix A. The equations of motion have been obtained imposing the appropriate variation \(\delta q_{j_{x}},\delta q_{j_{y}},\delta q_{j_{s}}\) of the generalized coordinates. The mathematical notation of [PERSON]'s principle provided by Equation (A3) can be simplified by omitting for convenience the dependency on \(t\) and \(\zeta\), and, furthermore, by rearranging all the terms in a more compact manner as follows:  
\[\delta\mathbf{q}^{T}\mathbf{M}\tilde{\mathbf{q}}+\delta\mathbf{q}^{T}\mathbf{ K}\mathbf{q}=\delta\mathbf{q}^{T}\mathbf{P}, \tag{8}\]  
The generalized mass matrix \(\mathbf{M}\), the stiffness matrix \(\mathbf{K}\), the load vector \(\mathbf{P}\) and the vector \(\mathbf{q}\), appearing in Equation (8), are defined as follows:  
\[\mathbf{M}=\left[ \begin{array}{ccc}\mathbf{M}_{x}&&\\ &\mathbf{M}_{y}&\\ &&\mathbf{M}_{\beta}\end{array} \right],\ \mathbf{K}=\left[ \begin{array}{ccc} \mathbf{K}_{x}&&\mathbf{K}_{x\beta}\\ &\mathbf{K}_{y}&\mathbf{K}_{y\beta}\\ \mathbf{K}_{\beta x}&\mathbf{K}_{\beta y}&\mathbf{K}_{\beta}\end{array} \right],\ \mathbf{P}=\left[ \begin{array}{c}\mathbf{P}_{x}\\ \mathbf{P}_{y}\\ \mathbf{P}_{\beta}\end{array} \right],\ \mathbf{q}=\left[ \begin{array}{c} \mathbf{q}_{x}\\ \mathbf{q}_{y}\\ \mathbf{q}_{\beta}\end{array}