## 3 Numerical Model
### Interaction Model  
In order to ensure that correct coupling is established between flexural, slip and shear behavior, a control is performed at element level through equilibrium and compatibility conditions. In particular, thanks to the series arrangement of the three model components (the two zero length springs and the force-based beam column element) representing the pier under lateral load, the shear force acting in each element will be the same, but the deformation development of each element will be different for different cases. In particular is discussed here the interaction between the flexural and shear behavior, as schematized in Figure 10a. In the proposed model all the shear deformation is concentrated in the translational shear spring whilst the flexural deformation is modelled by the beam-column element. Figure 8: The factor \(\beta\) after [PERSON] and [PERSON] [30] (in Equation (14)). Figure 7: The factor \(\alpha\) after [PERSON] and [PERSON] [30] (in Equation (14)). Figure 9: The factor \(\gamma\) after [PERSON] and [PERSON] [30] (in Equation (14)). OpenSES introduced, as an interaction model, the LimitStateMaterial command, based on [PERSON] works (2004), used to construct a uniaxial hysteretic material object with, among others, damage and post-damage unloading stiffness based on ductility. This command overcomes the pitfall of a simplistic shear-flexure interaction model represented by having a shear spring in series with a beam column element. In a very simple serial model, indeed, if the shear strength (the maximum of the shear spring response) is less than the bending yield strength (the shear corresponding to the development of the plastic hinges), the total response is correctly dominated by a brittle failure occurring in the elastic branch. If, on the other hand, the shear strength is higher than the bending yield strength, then the model is not able to capture any shear degradation, in contrast with theoretical and experimental evidence. Actually, in the latter case, when the initial shear strength is higher than yielding strength, but close enough to it, when degrading with the increase of inelastic deformation could be lower. In this case the activation of shear damage in the plastic branch is expected, as shown in Figure 4 (case B), this leads the structural response of the column to follow the degrading branch of the shear-strength-domain of Figure 5. In order to improve the series model, the LimitStateMaterial associated with LimitCurve command can be used to define a limit shear surface defined by the drift capacity model proposed by [PERSON], with the use of the shear-failure domain given by Equation (17). The behavior of the shear spring and bending beam-column element are illustrated schematically in Figure 10.  
If the shear capacity \(\mathrm{V_{n}}\) of the pier is lower than its flexural strength \(\mathrm{V_{f}}\), the pier will have a response controlled by the shear behavior as shown schematically in Figure 10b. Before reaching the shear capacity \(\mathrm{V_{n}}\), shear response and flexural response will develop simultaneously in accordance with the solid line in the figure and once the shear demand reaches the shear strength, \(\mathrm{V_{n}}\), a shear failure occurs and the shear response enters into descending range where significant deterioration behavior occurs, conditioning the overall response, that cannot develop forces higher than the shear strength. Figure 10: Combination of the single response components in shear-flexure interaction model: **(a)** Two-component model; **(b)** Response controlled by shear failure and **(c)** Response controlled by flexural failure. If the shear capacity \(\mathrm{V_{n}}\) of the column is higher than flexural strength \(\mathrm{V_{f}}\), the pier will have a response controlled by the flexural behavior as shown schematically in Figure 10c.