PSI - Issue 44

Corrado Chisari et al. / Procedia Structural Integrity 44 (2023) 1100–1107 Corrado Chisari et al./ Structural Integrity Procedia 00 (2022) 000 – 000

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and masonry and concrete elements by node congruence. For masonry and concrete a density of 1250 kg/m 3 and 2500 kg/m 3 was considered, respectively, and Poisson’s ratio ν=0.2. In this preliminary analysis, the RC frame was neglected and the masonry material was assumed fully homogeneous, i.e., the limestone insertions were ignored. Base nodes were considered fully fixed, while the interaction between the tower and the nearby church was modelled as continuous springs with variable stiffness. In particular, the springs were directed orthogonally to the contact surface and no constraint was considered in the other directions. In this way, it is assumed that the façade and the longitudinal wall of the church, which are in contact with the tower respectively on the surfaces having normal -x and +y, produce in-plane reaction only. A linear variability of the spring stiffness with the height was considered, to model the different stiffness of the wall within the height.

4.2. Model updating

Considering the model described above, six parameters were assumed unknown and had to be calibrated. They are listed in Table 1, together with their assumed variation range.

Table 1. Unknown parameters, variation ranges and calibrated values.

Parameter

Symbol Min. value Max value Calibrated value

Masonry elastic modulus [MPa] Concrete elastic modulus [MPa]

500

2500

1640

E m E c

20,000

40,000

25800

Façade spring stiffness at ground [N/mm 3 ]

10 10

10 6 10 6 1.5 1.5

3.15∙10 1 1.40∙10 2

1

k k

f,g

Longitudinal wall spring stiffness at ground [N/mm 3 ]

1

l,g

Normalised façade spring stiffness at top [-]

0.0 0.0

0.627

r f,t r l,t

Normalised longitudinal wall spring stiffness at top [-]

1.10

Model updating, i.e., calibration of the unknown parameters collected in the vector x , was performed by using the software TOSCA-TS (Chisari & Amadio, 2018). In particular, the following multi-objective optimisation problem had to be solved: = ( , ) (1) where: • = ∑ ( , − , , ⁄ ) 2 =1 is the function measuring the discrepancy of the model from the experimental data in terms of frequencies; • = ∑ (1 − , ) 2 =1 is the function measuring the discrepancy of the model in terms of mode shapes • N is the number of modes considering in the comparison, N =4. • , , , are respectively the numerical and experimental frequencies, corresponding to the same mode i ; • , is the Modal Assurance Criterion index (Allemang, 2003) between the i -th experimental and numerical modes. Before evaluating ω f , ω MAC , the modes were previously rearranged based on the maximum MAC in order to solve possible mode switching (Chisari et al., 2015). The result of the multi-objective optimisation is represented by the Pareto Front, i.e., the set of nondominated solutions. Within it, it is possible to select a solution which is a compromise between the two objectives, i.e., mode shape consistency and frequency fitting (Chisari et al., 2018), which is shown in Fig. 6a as red point. The compromise solution is characterised by input parameter values shown in the last column of Table 1. It must be underlined, however, that the estimation of the input parameters is not affected by the same level of uncertainty, since the functions and have different sensitivity to each of them. The characterisation of the uncertainty in the calibration will represent an aspect to investigate in the future.