PSI - Issue 84

1156 Domenico Liberatore et al. / Procedia Structural Integrity 84 (2026) 1151–1158 where 0 is a regularization factor of the convexity of the limit domain and  regulated the shape of the damage domain in compression. Moreover, to overcome mesh dependency problems, a nonlocal integral regularization technique is adopted allowing to evaluate the nonlocal equivalent strain ̅ . The damage variable is then evaluated as: = max ℎ {0, min { ̃, 1}} (3) with: ̃ =1+ ̅ 1 ( − ) 3 − ( ̅ − ) ( ̅ − ) 2 ( ̅ + ̅ −2 2 ) (4) where regulates the shape of the softening branch, and depends on two different parameters in tension and compression, and , respectively; is the tensile strain threshold and is the ultimate value of the equivalent strain, which corresponds to a fully damaged state. Due to the lack of information regarding the mechanical characteristics of the materials and the unfeasibility of performing destructive tests on the bridge, the values provided by the Italian Standard Code are considered for the numerical model. The mechanical parameters adopted for the different materials are given in Table 1, where the parameter is equal to 0.03 and  is equal to 80000 for all materials. Table 1: Mechanical parameters of masonry materials. Material E [MPa] [-] [MPa] [MPa] [-] [-] [-] [-] [-] [kN/m 3 ] Travertine 6020 0.3 8.2 0.18 5.45 10 -5 4.97 10 -3 5.00 10 -3 -10000 -30 22 Brick cladding 3290 0.3 4.3 0.2 1.11 10 -4 4.78 10 -3 5.00 10 -3 -7000 -10 18 Tuff 2960 0.3 3.2 0.16 9.88 10 -5 3.95 10 -3 5.00 10 -3 -5000 -30 16 4.2. Numerical model validation and modal analysis The dynamic characteristics of the numerical model, in terms of modal frequencies and shapes, are evaluated through a modal analysis. The initial elastic parameters of the material, particularly the Young’s modulus, are at first evaluated according to the Italian Standard Code, but are recalibrated through a back analysis in order to minimize the average error between the first five numerical frequencies and the results obtained from the ambient vibration tests. The obtained average error is equal to 8.6%. The first three numerical frequencies exhibit higher values than the experimental ones and higher error, while this decreases for the higher modes, as shown in Table 2, where the differences are probably related to the limited information available on the original materials. The modal shapes of the numerical model are also confirmed by the analysis of the mass participation factors.

Table 2: Modal frequencies of Pons Fabricius.

Mode

Numerical [Hz]

Experimental [Hz]

Error [%]

Direction Numerical

Direction Experimental

1 2 3 4 5

4.11 5.45 8.33 9.04

3.75 6.47 7.33 9.39

9.7

Transverse Transverse Longitudinal Transverse Longitudinal Longitudinal Transverse Transverse

-15.7 13.6

-3.7 -0.2

10.40

10.42

Vertical

Vertical

In general, within the framework of modal analysis, the values of the elastic moduli and the stiffness of the springs can be increased, since the analysis is performed in the elastic range and the values from the Standard Code refer to cracked material conditions. Such increase is generally in the range of 30-40%; in addition, the elastic moduli of masonry can be increased even more, given the uncertainty associated with the mechanical properties of the material.