## 2 Formulation of the Effectiveness Criterion
### Input: Bridge Characteristics and Seismic Scenario  
The first step consists of collecting the geometrical and material characteristics of the bridge, including  
* The masses of the bridge elements, grouped in the mass of the deck (\(m_{deck}\)), the mass of the piers (\(m_{pier}\)), and the mass of the pier caps (\(m_{piercap}\));
* The properties of the pier materials: the modulus of elasticity and characteristic compressive strength of concrete (\(E_{c}\),\(f_{c\mathrm{k}}\)) and the modulus of elasticity and characteristic yield strength of steel reinforcement (\(E_{s}\),\(f_{\mathrm{yk}}\));
* Geometric properties of the piers (cross-sectional area A and area moment of inertia I; height of the column);
* Details of the steel reinforcement in reinforced concrete (RC) piers;
* Bearing layout.  
The mass of the deck includes the mass of both structural and non-structural components such as the slab, beams, parapets, lights, etc. The material and geometric properties are those required for evaluating the stiffness of the piers. It is worth noting that the same information will be needed for the design of the isolation system in the second phase of the method [23]. Information can be retrieved from the bridge documentation (design report, drawings, etc.), if available, or from site inspection and structural tests for geometric and mechanical properties, respectively.  
The assessment of effectiveness is performed by studying a simplified model, or \"elementary bridge unit\", which represents a single pier of the bridge, its pier cap, and a section of the deck with assigned mass (\(c\)) depending on the tributary loading area of the pier itself. Starting from this simplified physical model, an equivalent SDOF model of the elementary bridge unit is formulated (Figure 3).  
In accordance with the Italian Building Code, IBC [20], SS7.9.4.1, the effective mass of the elementary bridge unit is defined as the sum of the masses of the deck section, the pier cap and the upper third of the pier:  
\[m_{eff}=m_{deck}+m_{piercap}+m_{pier}/3 \tag{1}\]  
where the seismic mass of the deck \(m_{deck}\) includes masses of structural elements (\(m_{g1}\)) and nonstructural elements (\(m_{g2}\)), plus the mass relating to the traffic loads multiplied by the participation factor \(\psi_{2}=0.2\) (\(m_{0,2q}\)), according to IBC [20], SS2.5.3 and SS5.1.3.12:  
\[m_{deck}=m_{g1}+m_{g2}+m_{0,2q} \tag{2}\]  
The loads transmitted from the superstructure to the piers depend on the static scheme of the bridge.  
For simply supported bridges, the bearing layout provides, for each span of the deck, a pinned axis (the bearings can rotate around the transversal axis of the bridge but do not allow for displacement) and a rolling axis (the bearings can rotate around the transversal  
Figure 3: Equivalent SDOF of the elementary bridge unit: \(m_{eff}\) = effective mass; \(h\) = height.  
axis and move longitudinally). Therefore, the seismic mass acting on the single pier, in the longitudinal direction, is the mass of the span connected to the pier via the pinned axis. In the transverse direction, it is assumed that the displacements of both axes are constrained, and therefore, the seismic mass acting on the pier is equal to half the mass of each span supported on the pier.  
For continuous deck bridges, the mass distribution on the pier is also affected by the stiffnesses of the pier and the deck and in principle can be evaluated through an elastic analysis that considers these properties. However, to simplify the assessment, the bridge mass can be assumed as distributed over the pier according to the concept of the tributary loading area. The adequacy of this assumption for a continuous deck is demonstrated in reference [23] and in Appendix A of this paper.