**2. Modelling**

An alternate approach to mitigate the hazardous effects of earthquakes is based on a consid‐ eration of the distribution of energy within a structure. The input energy from the ground acceleration is transformed into both kinetic and potential (strain) energy that must be either absorbed or dissipated trough heat. A large portion of the input energy, instead of being absorbed by hysteretic action (i.e. damage of the structure) can be dissipated with supple‐

This approach to seismic energy dissipation is made clear by considering the following time‐

where *E* is the absolute energy input from the earthquake motion, *E*k is the absolute kinetic energy, *E*s is the elastic (recoverable strain energy), *E*h is the irrecoverable energy dissipated by the structural system through inelastic or other forms of action (hysteretic or viscous), *E*d is the

The chapter analyses the seismic retrofit of an existing viaduct, by assessing an actually designed case study. The viaduct at issue, built during the Seventies, serving a fast‐moving thoroughfare in the city of Naples (Italy), features reinforced concrete piers and a mixed steel‐ concrete box deck. Three solutions, corresponding to the most common types of interventions for earthquake protection, have been compared: the first by using hydraulic dissipation devices, the second using sliding pendulum isolator devices and the third by using elastomeric isolators. The differences have been compared according to traffic interferences, interferences with existing structures, operational needs and to time and costs necessary to achieve a final

The viaduct analysed, which already has an earthquake intervention in place, features a length of 1360m and a continuous deck on 18 spans with variable clearance from 62 to 92m, separated by four longitudinal joints, implemented with Gerber saddles to assure structural continuity (**Figures 1** and **2**). The reinforced concrete piers have a hollow circular section, with a variable base diameter ranging from 2.65 to 3.30m. They are vertically compressed and feature a variable height ranging from 7.90 to 38.80m. The foundations are deep, well and raft based with piles. The structural joints are located at the piers 4, 6, 11 and 14. One particular aspect of the viaduct is that there is only one support per each pier, placed at the central part of the box. Hence, the deck is balanced, since the two adjacent runways are connected by means of

Intervention on an existing work always requires special attention and a meticulous study. For example, with the viaduct under examination, the base of the piers is inaccessible for any type of reinforcement intervention. Additionally, it is not possible to stop traffic in order to replace the supports, and therefore, it is not possible to work on the abutments, to draw back the ballast retainers if a larger gap than the present one (measuring 15 cm) should be required. Moreover,

energy dissipated by a supplemental damping system and *t* represents the time.

( ) () () () () *Et E t E t E t E t* =+++ *k zhd* (1)

mental systems.

130 Structural Bridge Engineering

project solution.

**1.1. The viaduct main features**

transverse girders at each support section (**Figure 3**).

dependent conservation of energy relationship:

The analyses have been carried out using a finite‐element analysis software on a model, consisting in mono‐dimensional elements; they are non‐linear with step‐by‐step integration of the equations of motion.

For seismic purposes, the following parameters (Italian Standard DM 14.01.2008 [1, 2]) have been taken into account:


**•** Topographic coefficient *T*1=1 that correspond to a flat surface.

**Figure 4.** Artificial accelerogram.

For the analysis, artificial spectrum‐compatible accelerograms have been used (**Figure 4**). It is noted that the use of real accelerograms and spectrum‐matching techniques, together with records selection tools, tends to be recommended for the derivation of suits of records for use in non‐linear dynamic analysis of structures but in this case, the access to real accelerograms was challenging. In literature [3], it is shown that the structural response estimated by using simulated records generally matches the response obtained using recorded motions. The software can generate artificial time‐histories of ground acceleration compatible with the target spectrum and gives a comparison between its response spectrum and the target spectrum itself (**Figure 5**).

The computational analysis has been carried out using finite‐element method. The model of the viaduct is three‐dimensional (3D) type (**Figure 6**), representing the stiffness features of the structural elements. The composing elements are all of linear elastic type, except for the restraint devices and for the base of the piers, where it is assumed that plastic hinges may form. It takes into account geometric and material non‐linearity.

**Figure 5.** Compatibility between response spectrum and target spectrum.

**Figure 6.** Finite‐element model.

**•** Rated life of the work: *V*N = 50 years, relative to bridges and infrastructures;

munication ways available;

132 Structural Bridge Engineering

**Figure 4.** Artificial accelerogram.

itself (**Figure 5**).

*S* wave in the first 30 m) range is 180—360 m/s;

**•** Topographic coefficient *T*1=1 that correspond to a flat surface.

**•** Use coefficient: *C*U = 2, corresponding to Use Class IV about strategic construction in case of calamity and bridges of particular importance after seismic events in order to keep com‐

**•** Reference life of the work equals to *V*R=100 years; it comes from the product *V*R = *V*N·*C*U;

**•** Soil type C: deposits of medium thickened soil, with layers of more than 30m where mechanical properties gradually increase with depth; VS30 (average equivalent velocity of

For the analysis, artificial spectrum‐compatible accelerograms have been used (**Figure 4**). It is noted that the use of real accelerograms and spectrum‐matching techniques, together with records selection tools, tends to be recommended for the derivation of suits of records for use in non‐linear dynamic analysis of structures but in this case, the access to real accelerograms was challenging. In literature [3], it is shown that the structural response estimated by using simulated records generally matches the response obtained using recorded motions. The software can generate artificial time‐histories of ground acceleration compatible with the target spectrum and gives a comparison between its response spectrum and the target spectrum

The computational analysis has been carried out using finite‐element method. The model of the viaduct is three‐dimensional (3D) type (**Figure 6**), representing the stiffness features of the structural elements. The composing elements are all of linear elastic type, except for the restraint devices and for the base of the piers, where it is assumed that plastic hinges may form.

It takes into account geometric and material non‐linearity.

For the piers, a moment‐curvature constitutive law has been adopted by using the Takeda model [4]. The model represents the hysteretic features of reinforced concrete structures by means of the trilinear relation force‐displacement, 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 moment‐curvature 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 re‐ distribution of the stresses among the following piers of the viaduct.

The support and restraint devices are represented by special elements where it is possible to specify stiffness in any of six degrees of freedom between two nodes. They do not behave like a standard beam element; the degrees of freedom are user specified and are independent of each other. For non‐linear analysis, each stiffness value may be defined via a non‐linear table: force‐velocity type for hydraulic devices and force‐displacement type for isolators.

The foundations of the piers and of the abutments are modelled with punctual restraints.
