## 3 Numerical Model
### Shear Behavior  
In the past, different shear-capacity models have been proposed to account for the flexure-shear coupling, leading to the strength degradation of columns with deformation. The first one was the formulation codified in the ATC seismic design guidelines, where a shear-capacity curve degrading with displacement ductility was proposed (Figure 4). In this study the phenomenological model illustrated in Figure 5 has been adopted for modelling the shear spring, accounting for both strength and deformation components due to shear action. As shown in Figure 5, the first two branches represent the backbone behavior of the shear component of the element before the intersection with the shear strength domain (marked with a red circle) and the start of the degrading branch. Figure 3: Properties of a RC section: (**a**) Fiber discretization of the section (y and z are the element local axis); (**b**) steel reinforcement and (**c**) cover (unconfined) and core (confined) concrete constitutive law (the meaning of the symbols is given in the text). The pre-cracking shear stiffness \(K_{S,uncracked}\) can be calculated through the elasticity theory:  
\[K_{S,uncracked}=\frac{GA_{v}}{H} \tag{3}\]  
\[G=\frac{E_{c}}{2(1+\
u)}\approx 0.4E_{c} \tag{4}\]  
where \(H\) is the column height, \(G\), \(E_{c}\), and \(\
u\) are respectively the shear, [PERSON]'s, and [PERSON]'s moduli of concrete, and \(A_{v}\) is the shear effective area of the column. In general, this stiffness is contributing to a minor displacement increase, since even in squat elements the flexural stiffness is significantly smaller, but it can be useful to modify the fiber element formulated within OpenSEES as an Euler-Bernoulli beam-column to a full Timoshenko element. The post-cracking shear stiffness \(K_{S,crracked}\) can be calculated considering the deformation of transversal steel through the diagonal cracks. [PERSON] and [PERSON] [27] proposed an equivalent strut-model:  
\[K_{S,crracked}=\frac{\rho_{w}sin^{4}(\theta)cot^{2}(\theta)}{sin^{4}(\theta)+1 0\rho_{w}}E_{s}A_{v} \tag{5}\]  
Figure 4: ATC model for shear-strength degradation (d\({}_{y}\): yielding displacement). Three cases are considered: (**A**) shear failure before flexural yielding (pure shear failure); (**B**) shear failure after flexural yielding (shear-flexural failure); (**C**) flexural failure. Figure 5: Phenomenological model for the shear spring. where \(\rho_{w}\) is the transversal steel reinforcement ratio, \(\theta\) is the angle between the diagonal cracks and the member axis and \(Es\) is the Young's modulus of steel. In Figure 5 the shear-strength domain represents the maximum shear that the column can sustain. As it is evident, that limit state curve is not constant, as in the usual formulation contained in most design codes (like Eurocode 2 or ACI-318), but is dependent with the displacement and the element once reached the maximum strength, will follow the degrading branch. In the literature, there are several shear-capacity models that have been proposed to account for the shear-strength degradation of columns under seismic loading [9; 28; 29; 30]. It is worth noting that all the aforementioned formulations are defined in terms of total displacement (i.e., the sum of single displacement components) so that the shear-strength domain has been placed in Figure 5 (where the displacement is defined just in terms of shear deformations) only for illustration purposes. As shown in Figure 3, the failure is activated when the shear capacity curve (bold black dash line) intercept the shear demand curve (black line) which represent the global behavior of the column gives by the summation of the flexure, slippage and shear behaviors.