**2.2. Non-linearity considered**

where μ is the friction coefficient, *N*Sdis the load acting on the isolator (quasipermanent load),

All the parameters, characterising the devices, are specified in **Chart 2** since the vertical load

These devices are characterised by low horizontal stiffness, high vertical stiffness and negli‐ gible damping capacity. These characteristics permit to increase the fundamental period of

The fundamental design parameters used to determine vertical and horizontal stiffness are the geometrical characteristics (overall dimensions and thickness) and the mechanical character‐ istics of the elastomeric compound that is characterised by an effective dynamic shear modulus

**Chart 3.** Features of the sliding isolators used in the analysis. *K*r is the restoring stiffness, *K*e is the equivalent stiffness (in case of linear analysis), ξe is the effective viscous damping, *F*0 is the friction force developed by the isolator and *F*max

The compounds contain suitable antiageing additives that guarantee limited variation of the

Their behaviour is modelled as linear by means of the equivalent stiffness. Five types of isolators have been defined. All the parameters, characterising the devices, are specified in

**Chart 3** since the vertical load on each support is different, being a continuous deck.

This kind of isolators consists in a rubber bearing made up of layers of elastomer.

vibration of the structure and to resist to vertical loads without appreciable settling.

*R* is the equivalent radius of curvature, and *d* is the displacement.

on each support is different, being a continuous deck.

*2.1.3. Elastomeric isolators (non-dissipating)*

*G* equal to 1.4 MPa.

138 Structural Bridge Engineering

is the maximum horizontal force.

physical and mechanical characteristics in time.

According to the standards [1, 2], the structure needs to be designed in order to develop a stable dissipative mechanism, where dissipation is attributed to the piers (plastic hinges) or to appropriate devices. The elements not involved in the dissipative process need to remain elastic similar to the substructure of isolated bridges.

In this retrofit, the dissipation is committed to devices, and so it is necessary to check the behaviour of the base section of the piers.

The hysteretic model of Takeda has been selected to represent the non‐linear behaviour of the concrete structures. It is a realistic conceptual model that recognises the continually varying stiffness and energy‐absorbing characteristics of the structure. Three linear segments define the primary curves (**Figure 9**), and the first break refers to cracking. The yield load (end point of the second line) is obtained assuming a parabolic compressive stress‐strain curve for the concrete. The yield deflection depends on the deflection caused by curvature based on cracked section, the shearing deflection and the deflection caused by slip of the reinforcement and depression of the concrete. The slope of the third segment is related to the strain‐hardening properties of the reinforcement. The response under load reversal, depending on the displace‐ ment ductility and the axial load ratio, is explained in the study of Takeda et al. [4].

The dissipating devices have been modelled as springs and they have been characterised by their constitutive law, described in the previous paragraphs, that is force‐velocity or force‐ displacement type (as shown in **Figure 10**).

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

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