**2. Modelling and design approach**

The design of the viaduct was performed according to Italian code [2, 3] taking into account the dissipative behaviour of the structural elements, material and geometric non-linearity, following a high-ductility approach.

The dissipative zones are concentrated in the seismic-restraint devices while non-dissipative elements are designed according to capacity design criterion [2, 3]. The structure is conceived and designed with the goal to create a stable dissipative mechanism under the seismic action at the life-safety limit state (ultimate limit state—ULS).

The plastic deformations of the base of the piers generate displacement demand that is requested to be lower than the capacity of the system. The comparison between ductility demand and available ductility was carried out on the basis of the instructions provided in [4]. In detail, deck, vertical support devices, foundation structures and abutments are designed to remain elastic.

'Over-strength' factors were considered for the verification of the pier sections outside the plastic hinge region and for the foundations as well. The fixed constraint devices were dimensioned according to the capacity design criterion. The dissipative devices were designed to support, without collapse, earthquake displacements caused at the collapse limit state (CLS).

The computational analysis has been carried out using finite element method (FEM). The model of the viaduct is of a three-dimensional type (**Figure 4**) and recreates the stiffness of the structural elements, constituting it, the non-linear features are concentrated within the restraint devices and at the base of the piers (plastic hinges). It takes into account geometric and material non-linearity.

**Figure 4.** Finite element model.

**Figure 2.** Viaduct plan and view.

152 Structural Bridge Engineering

**Figure 3.** Viaduct cross section.

remain elastic.

**2. Modelling and design approach**

at the life-safety limit state (ultimate limit state—ULS).

following a high-ductility approach.

The design of the viaduct was performed according to Italian code [2, 3] taking into account the dissipative behaviour of the structural elements, material and geometric non-linearity,

The dissipative zones are concentrated in the seismic-restraint devices while non-dissipative elements are designed according to capacity design criterion [2, 3]. The structure is conceived and designed with the goal to create a stable dissipative mechanism under the seismic action

The plastic deformations of the base of the piers generate displacement demand that is requested to be lower than the capacity of the system. The comparison between ductility demand and available ductility was carried out on the basis of the instructions provided in [4]. In detail, deck, vertical support devices, foundation structures and abutments are designed to For the piers, a moment-curvature constitutive law has been adopted by using the Takeda model [5]. The model represents the hysteretic features of reinforced concrete structures by means of the trilinear relation force displacement (**Figure 5a**), where the non-linearity is modelled by using concentrated plastic hinges and takes into account cracking and yielding. From the force-displacement law, it is possible to retrieve the corresponding momentcurvature link. Such assessment has been carried out distinctly for each pier. With this choice, it is possible to assess in the transient state the stiffness change of the substructures, checking the consequent redistribution of the stresses among the following piers of the viaduct.

The OP and OTP viscous dissipation devices are devices provided with a cylinder and a piston, depending on the velocity, where the lamination of a silicone fluid by means of a suitable hydraulic circuit allows energy dissipation. The typical constitutive law force-velocity, simulating the behaviour is of non-linear type (**Figure 5b**), is

$$F = C \cdot V^a \tag{1}$$

where *C* represents the damping constant and *α* is assumed to be equal to 0.15.

The dissipation devices have been modelled by means of spring-damper elements with the association of the law force-velocity (Eq. (1)). There are two types of devices, a plastic OP hydraulic one and the other one is an OTP thermal-plastic hydraulic one.

**Figure 5.** (a and b) Flow chart moment curvature according to Takeda and force-velocity law (*F* = *C* × *Vα*).

The OT devices, with a dynamic restraint (shock transmitter), represent a very stiff restraint against a dynamic action, whereas they allow slow displacements of the structures (e.g. due to thermal changes). Owing to their features, they have been modelled as truss elements with a high stiffness.

#### **2.1. Design choices relating to restraint diagram**

The restraint system is outlined in **Figure 6**. Longitudinally, the fixed piers (P7 and P8) absorb static stresses due to braking and play the role of a thermal centre point by means of a transversal one-way restraint; from a seismic point of view, the shorter piers (P–P3 and P14 with a height of less than 25 m) are free to oscillate, piers P3–P13 (about 30 m high) are provided with OTP-type dissipation devices, able to check the stress value, given by the deck; for the remaining piers (with height exceeding 35 m), the application of temporary-restraint devices (shock-transmitter OT) is provided. Transversally, the piers (P1–P4, P10, P13, P14) are provided with a multidirectional support, associated to an OP plastic hydraulic type device, whereas P5, P6, P9, P11, P12 are associated with a DEF\* (\*with shoes, able to accept longitudinal displacements) fixed type restraint; piers P7 and P8 are provided with a transversal one-way


**Figure 6.** Restraint diagram.

support device, associated with a DEF-type device. From a seismic point of view, only the piers, being less than 35 m high (P1–P4, P10, P13, P14), have been isolated. The classification diagram of the piers is represented in **Figure 7**.



**Figure 7.** Piers classification diagram.
