**3. Results**

The results of the analyses performed have been presented as a comparison among the restraint diagrams proposed, in terms of stresses at the base of the piers (**Figures 11** and **12**), displace‐ ments on the top of the pier (**Figure 13**), displacements and forces acting on the devices (**Figures 14** and **15**) and dissipated energy (**Figures 16** and **17**). The following charts relate to the maximum stresses for each base section of the piers. These results were assessed as the square root of the instant‐by‐instant stresses, since the pier is circular; the charts also show the values of the "yield moment" of the examined sections.

**Figure 11.** Bar chart of the bending stresses ULS (max XY) at the base of the piers.

**Figure 10.** Constitutive laws of dissipative devices, dampers and sliders.

140 Structural Bridge Engineering

**Figure 5** and described in Sections 2.1.1 and 2.1.2 (**Chart 4**).

**Chart 4.** Parameters of non‐linear material characterising the constitutive law of the pier bases.

The results of the analyses performed have been presented as a comparison among the restraint diagrams proposed, in terms of stresses at the base of the piers (**Figures 11** and **12**), displace‐ ments on the top of the pier (**Figure 13**), displacements and forces acting on the devices

**3. Results**

The following charts recall the features of each device of the restraint diagrams outlined in

**Figure 12.** Bar chart of the shear stresses ULS at the base of the piers.

As asserted in Section 2.2, the main requirements are to ensure that the substructure behaves elastically.

It can be deduced from **Figure 11** how all the solutions lead to stress levels that are lower than the yield limit of the piers. Furthermore, the solution with viscous devices better exploits the bending‐resistant capacity and the shear‐resistant capacity of the reinforced concrete section.

The following chart refers to the displacement of pier top. This parameter is linked to the control of the local ductility at the plastic hinges [5] by the chord rotation. The latter is measured over the length of the pier, between the end section of the plastic hinge and the section of zero moment.

The verification that deformation demands are safely lower than the capacities of the plastic hinges should be performed by comparing plastic hinge rotation demands, θp,E, to the relevant design rotation capacities, θp,d, as follows: θp,E<θp,d.

Again, it is necessary to ensure that the displacements are lower than the yield displacement limit (**Chart 5**):

$$D\_{\gamma} = \phi\_{\gamma} H^{2} / 3\tag{4}$$

The following charts refer to devices, and in particular, they show displacements and the conveyed forces.

The chart in **Figure 14** regards the displacements of the devices and to be understood it needs to be read together with the chart in **Figure 15**, which concerns about the stress level reached by the devices.

In general, the amount of the displacements increases when the height of the pier is low, due to the ductility of the element. Considering **Figure 13**, the displacement of the pier is greater for higher piers.

It is possible to affirm that for lower piers (from pier 11 to 17), the behaviour of sliders and elastomeric isolators in terms of strength is similar; while the displacement is different, the latter request more displacement than the former in order to guarantee the same level of force.

The displacements of OP‐type dampers are null, and from the bar chart shown in **Figure 13**, it is possible to appreciate that the displacement of the top of the piers with fix restraint or OP restraint is greater.

**Figure 13.** Bar charts representing the displacements on the level of the heads of the piers.

**Figure 14.** Bar charts representing the displacements on the level of the devices.

<sup>2</sup> · /3 *D H y y* = f

conveyed forces.

142 Structural Bridge Engineering

by the devices.

for higher piers.

restraint is greater.

The following charts refer to devices, and in particular, they show displacements and the

The chart in **Figure 14** regards the displacements of the devices and to be understood it needs to be read together with the chart in **Figure 15**, which concerns about the stress level reached

In general, the amount of the displacements increases when the height of the pier is low, due to the ductility of the element. Considering **Figure 13**, the displacement of the pier is greater

It is possible to affirm that for lower piers (from pier 11 to 17), the behaviour of sliders and elastomeric isolators in terms of strength is similar; while the displacement is different, the latter request more displacement than the former in order to guarantee the same level of force.

The displacements of OP‐type dampers are null, and from the bar chart shown in **Figure 13**, it is possible to appreciate that the displacement of the top of the piers with fix restraint or OP

**Figure 13.** Bar charts representing the displacements on the level of the heads of the piers.

(4)

**Figure 15.** Bar charts representing the forces on the devices.

The following diagrams contain the hysteretic energy dissipation processes of the following two main opposing mechanisms of the earthquake (devices and piers): the devices in terms of force‐displacement (there is dissipation only in case of dampers and sliders) and the material of the piers, relating to the base section, in terms of moment‐curvature. The area underlying the curves is an indicator of the dissipated energy.

**Figure 16.** Comparison of energy dissipation (restraint device and section at the base of the pier n.1 – earthquake ULS of Collapse).

The following chart (**Figure 17**) summarises the endeavour performed by the piers during the duration of the earthquake: the same is obtained as the sum of the products of forces at the base of the pier and the displacement at the top of the same pier for each moment of the seismic event. In particular, it is clear how the piers are implicated in the total dissipation process of the seismic energy, partly carried out also by the devices in terms of displacement. Moreover, it can be deduced how the solution with the sliders allows transferring only a reduced portion of energy to the piers.

**Figure 17.** Bar charts representing energy dissipation in the earthquake expressed by the piers.


**Chart 5.** Displacements correspondent to yield limit curvature of the pier.
