## 2 Theoretical Concepts
### Development of Hysteresis Models  
The mathematical laws commonly called hysteresis functions govern the nonlinear behavior of the structural elements. As the inelastic deformations increase, the structural response becomes more and more sensitive to the model used to account for cyclic behavior. The hysteresis relationships are highly dependent on the loading history and properties of the member and are usually determined from laboratory experimental tests. Although many complex models have been proposed, the bilinear hysteretic function has been largely used for both reinforced concrete and steel structures. Its simplicity resides in the fact that the behavior is described by only three rules: elastic, yielding, unloading, and load reversal slopes. The main characteristic of this model is that for high amplitude deformations a large amount of energy is dissipated, while for low amplitude cycles, zero hysteresis energy is consumed. This leads to the fact that early dissipation of the seismic energy does not allow an accurate build-up of the structural deformations. In term of reinforced concrete members, the usage of the bilinear model has the drawback of overestimating the energy dissipation capacity. Moreover, experimental tests conducted for beam-to-column connections showed that the behavior of reinforced concrete members under cyclic loading is complex and cannot be accurately described by simple bilinear hysteresis functions [17]. Although the numerical implementation of the dynamic nonlinear mathematical is usually a complex task, one of the main features that make the force analogy method attractive is the ability to account for the nonlinear behavior in a simple and direct way. This is also the case for the implementation of degrading hysteresis models that better suit the cyclic behavior of the RC structural elements. The FAM's main advantage is that it facilitates a simple path towards implementation of any polygonal hysteresis model. The procedure for the implementation of cyclic models in FAM 3D consists of two main steps. Firstly, a mathematical hysteresis model must be defined by establishing, for every possible case, the rules that govern the behavior. Following, the classical FAM equations are modified to account for complex cyclic of the structural elements and a new matrix \(\mathbf{K_{d}}\), that contains on the main diagonal the history variables defining for every plastic hinge the slopes of the branches characterizing the hysteresis model, is introduced. This matrix is utilized to determine the capacity function variation in time. Consequently, the key feature for the implementation of complex hysteresis models in the FAM is the usage of the capacity functions. They actually represent the branches of the cyclic model and are established using certain rules that define the hysteresis model. The general form of a capacity function is presented in Equation (9). \[\mathbf{f(\theta^{{}^{\prime\prime}}\mathbf{(t)})}=\mathbf{m}_{\mathbf{inter}}+ \mathbf{K_{d}}\cdot\mathbf{\theta^{{}^{\prime\prime}}\mathbf{(t)}} \tag{9}\]  
where \(\mathbf{m}_{\mathbf{inter}}\) is the vector containing the conveniently chosen intercepts of the hysteresis model. Moreover, the capacity function can be used to define constrains between consecutive bending moment values (Equation (11)). Coupling the Equations (5) and (6) leads to a system of equations in which \(\mathbf{m_{k+1}}\) and \(\boldsymbol{\theta}^{\prime\prime}\) are unknowns, \(\mathbf{u_{k+1}}\) being obtained from an explicit integration scheme using previous step information \(\mathbf{u_{k+1}}=f(\mathbf{u_{k}},\mathbf{a_{k_{k^{\prime}}}},\mathbf{u_{k}^{ \prime\prime}})\). Moreover, the values in \(\mathbf{K_{d}}\) depend on the prior history and are determined using the hysteresis model \((m-\theta^{\prime\prime})\) rules  
\[\left\{