## 2 Theoretical Concepts  
### Force Analogy Method Theoretical Background  
The force analogy method distinguishes from classical nonlinear analysis methods implemented in commercial software through the fact that throughout the entire dynamic analysis, the initial stiffness matrix of the structure \(\mathbf{K}_{0}\) is used. The basic idea of the FAM is to use a change in displacement vector in order to obtain the same level of force one will obtain using a variable stiffness model. Therefore, the equation of motion considered by the FAM is:  
\[\mathbf{M}\cdot\mathbf{\dot{u}(t)+C\cdot\mathbf{\dot{u}(t)+K}_{0}\cdot\mathbf{ \dot{u}(t)}=-M\cdot h\cdot a_{g}(t)} \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}_{0}\) are the mass, damping, and stiffness matrices, respectively, \(\mathbf{u(t)}\) is the total displacement vector, \(\mathbf{h}\) is a influence vector, and \(\mathbf{a}_{g}(t)\) represents the horizontal components of the ground acceleration. The total displacement vector \(\mathbf{u(t)}\) is split into the elastic displacement \(\mathbf{\dot{u}(t)}\) and the plastic displacement \(\mathbf{u}^{{}^{\prime\prime}}\)**(t)**  
\[\mathbf{u(t)=\ddot{u}(t)+u^{{}^{\prime\prime}}(t)} \tag{2}\]  
\[\mathbf{M}\cdot\mathbf{\ddot{u}(t)+C\cdot\mathbf{\dot{u}(t)+K}_{0}\cdot u(t)=-M \cdot h\cdot a_{g}(t)+K_{0}\cdot u^{{}^{\prime\prime}}(t)} \tag{3}\]  
In Equation (3) the physical nonlinearity is treated as a perturbation of the excitation forces. The last term on the right-hand side of Equation (3) \(\mathbf{K}_{P}\cdot u^{{}^{\prime\prime}}(t)\) has the physical meaning of \"the forces producing inelastic displacement in all structural coordinates\". In the civil engineering common approach, the plasticity is concentrated in zero-length hinges with moment-rotation model parameters. Although many sophisticated models that use distributed plasticity formulations have been developed, simplified models approaches proved more practical due to their relatively condensed and efficient numerical formulations. In addition, the simplified models are able to capture more effectively the relevant features with same or insignificantly lower level of accuracy [16]. Therefore considering that the beam end rotation is the only source of nonlinearity, the term \(\mathbf{K}_{P}\cdot u^{{}^{\prime\prime}}(t)\) in Equation (3) can be replaced by unbalanced moments \(\mathbf{K}_{P}\cdot\boldsymbol{\theta}^{{}^{\prime\prime}}(t)\) generated by the inelastic beam end rotation \(\boldsymbol{\theta}^{{}^{\prime\prime}}(t)\),  
\[\mathbf{M}\cdot\mathbf{\dot{u}(t)+C\cdot\mathbf{\dot{u}(t)+K}_{0}\cdot u(t)=-M \cdot h\cdot a_{g}(t)+K_{P}\cdot\theta^{{}^{\prime\prime}}(t)} \tag{4}\]  
where \(\mathbf{K}_{P}\) represents the member force recovery matrix. For the sake of brevity, in the FAM, the classical nonlinear differential equation is changed with a linear equation (Equation (4)) and an associated inequality (Equation (5)) which mimic the original nonlinear differential equation. Analyzing Equation (4) it is obvious that a second equation must be added to determine the two unknown vectors \(\mathbf{u(t)}\) and \(\boldsymbol{\theta}^{{}^{\prime\prime}}(t)\). This equation is obtained by limiting the bending moment values using the cross-section capacity functions. Moreover, the bending moment \(\mathbf{m(t)}\) is calculated using Equation (6). \[\mathbf{m(t)\leq f(\theta^{{}^{\prime\prime}}(t))} \tag{5}\]  
\[\mathbf{m(t)=K}_{P}^{T}\cdot\mathbf{u(t)-K}_{R}\cdot\boldsymbol{\theta}^{{}^{ \prime\prime}}(t) \tag{6}\]  
where \(\mathbf{K}_{R}\) represents the member restoring force matrix. The member force recovery matrix \(\mathbf{K}_{P}\) and member restoring force matrix \(\mathbf{K}_{R}\) are assembled from beam element stiffness matrix in local axis coordinates. The detailed procedure is illustrated in the authors' previous work [10]. Moreover, the second-order differential equation (Equation (3)) can be reduced to a first-order differential equation using the state-space formulation.