**3. Analysis results and match with test data**

The results of the seismic computational analyses are shown in the following charts. They are presented as a comparison between theoretical and experimental data. The former are characterized by the law of materials and devices deriving from mathematical assumptions, the latter instead use the output results of the laboratory-type test (**Figure 8**). The constitutive diagram of the piers, shown in **Figure 9**, remains the same in both analyses.

**Figure 8.** Comparison between theoretical and type-test law for devices.

**Figure 9.** Constitutive diagrams of the piers.

**-** Linear step-by-step dynamic analysis for comparison with non-linear step-by-step analysis in order to establish the structure factor *q*, implicitly deriving from the hysteretic cycle of load-unload of the materials, making up the substructures. With such *q* value, then, the overstrength factor, required for the check of the sections, placed outside of the critical areas, has

**-** Modal analysis with design spectrum (using the estimated *q*) and approval of the non-linear

For seismic purposes, the following parameters (referred to DM 14/01/2008 [2]) have been taken into account: rated life of the works, *VN* = 50 years; use coefficient of the works, *CU* = 2. The result is a reference period equal to *V*R = 100 years. The soil has been considered of class C: deposits of medium thickened soil, with layers of more than 30m where mechanical properties

The results of the seismic computational analyses are shown in the following charts. They are presented as a comparison between theoretical and experimental data. The former are characterized by the law of materials and devices deriving from mathematical assumptions, the latter instead use the output results of the laboratory-type test (**Figure 8**). The constitutive

diagram of the piers, shown in **Figure 9**, remains the same in both analyses.

step-by-step analysis by comparison with the total forces at the base of the piers.

been calculated (capacity design).

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gradually increase with depth.

**3. Analysis results and match with test data**

**Figure 8.** Comparison between theoretical and type-test law for devices.

**Figure 10.** Representative bar charts of the forces and moments at the base of the piers—comparison between experimental and numerical results.

**Figure 11.** Bar charts representing forces and displacements on the level of the devices—comparison between experimental and numerical results.

The results of the non-linear analyses are represented here below; each chart shows by the most adverse effects of three accelerograms at the ULS applied as explained in Section 2.2 (**Figures 10** and **11**).

**Figure 10.** Representative bar charts of the forces and moments at the base of the piers—comparison between experi-

**Figure 11.** Bar charts representing forces and displacements on the level of the devices—comparison between experi-

mental and numerical results.

158 Structural Bridge Engineering

mental and numerical results.

**Figure 12.** Charts displacement versus time and force versus time of the OTP and OP devices—comparison between experimental and numerical results.


**Figure 13.** Diagrams longitudinal moment curvature at the base of the piers—comparison between experimental and numerical results.

It is easy to find the substantial congruence of the two analyses in terms of forces and displacements, which in the diagrams in **Figure 12** are compared in a time history analysis.

**Figure 14.** Sum of the forces at the base of the piers (in kN)—estimation of the structure factor.

From the moment-curvature diagrams of **Figure 14**, it is, moreover, possible to verify how the piers remain substantially elastic, dissipating energy only for the cracked range without yielding the metal rebars.

The non-linear analysis has been carried out by considering the constitutive law of concrete piers (**Figure 9**), so a comparison with a linear analysis is necessary in order to assess and confirm the structure factor, adopted in the verifications. The structure factor is equal to *q* = 1.5 and derives from the ratio of the summation of the shear force at the base of each pier (**Figure 13**).

The superstructure is requested to remain elastic during the seismic event. To control this requirement, the set of largest transversal displacements relating to the top of the piers has been applied to an accurate finite element model of the deck, where the girders and the slab are represented with shell elements, the bracing system are modelled with linear element and the devices are springs characterized by their stiffness.

The displacement set is the output of the non-linear analysis, with a peak of 0.33 m at the pier 7 (**Figure 15**).

It follows the calculation of stresses on the steel elements (**Figure 16**), where it is notable that the stress level is lower than the yielding limit.


**Figure 15.** Transversal displacement of the piers applied to the FEM model.

**Figure 16.** Stresses on the steel elements.

**Figure 13.** Diagrams longitudinal moment curvature at the base of the piers—comparison between experimental and

It is easy to find the substantial congruence of the two analyses in terms of forces and displacements, which in the diagrams in **Figure 12** are compared in a time history analysis.

**Figure 14.** Sum of the forces at the base of the piers (in kN)—estimation of the structure factor.

From the moment-curvature diagrams of **Figure 14**, it is, moreover, possible to verify how the piers remain substantially elastic, dissipating energy only for the cracked range without

The non-linear analysis has been carried out by considering the constitutive law of concrete piers (**Figure 9**), so a comparison with a linear analysis is necessary in order to assess and confirm the structure factor, adopted in the verifications. The structure factor is equal to *q* = 1.5 and derives from the ratio of the summation of the shear force at the base of each pier

The superstructure is requested to remain elastic during the seismic event. To control this requirement, the set of largest transversal displacements relating to the top of the piers has

numerical results.

160 Structural Bridge Engineering

yielding the metal rebars.

(**Figure 13**).
