PSI - Issue 78

Shahin Sayyad et al. / Procedia Structural Integrity 78 (2026) 277–284

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standalone model of the tower. Five distinct linearly elastic isotropic materials for masonry were defined and assigned to different parts of the tower, corresponding to the bell chamber, shaft, basement (battered wall), barrel vaults and arches, as well as stair ramps. The FEM model of the tower, with the aforementioned parts highlighted, is depicted in Fig. 2, along with its translation to OpenSeesPy. The mechanical properties of the materials assigned to different parts of the structure, including the el astic modulus E, Poisson’s ratio ν, and unit weight γ, were previously calibrated through a manual finite element model calibration process. These calibrated parameters were subsequently implemented in OpenSeesPy as input parameters (Table 2) to reconstruct the FEM model of the tower. To ensure the accuracy of the OpenSeesPy model, both models were compared in terms of the natural frequencies obtained from eigenvalue analysis. This comparison resulted in an error of effectively 0% across all modes, demonstrating the model's accuracy in reconstruction using OpenSeesPy. The translation of the model into OpenSeesPy was performed using exported CSV files from Midas FEA NX that included node, element, and material information. To handle this data, the pandas library was adopted, enabling an efficient translation of the model into OpenSeesPy, where nodes and tetrahedral elements with one-point Gauss integration were programmatically defined using Python scripts.

Nodal nformation Element Connectivity Material and Section Properties

Midas EAN

OpenSeesPy

Fig. 2. Finite element model of the bell tower in Midas FEA NX.

2.4. Model calibration The calibration or updating problem is defined as an optimization problem, with the primary objective of minimizing the sum of the relative differences between the experimental and numerical modal properties. This is achieved by selecting the most relevant physical parameters, such as the mechanical properties of the model, as design variables. In the present study, two formulations of the finite element model calibration problem were considered using a single objective approach: one based solely on natural frequencies (Eq. 1), and the other incorporating the weighted residuals of both natural frequencies and mode shapes (Eq. 2).

2

exp

num −  f

   

   

n

f

(1)

i

i

F

w i

1 1 = 



num

i

= 

f i

2

exp

num −  f

    

   

n  =

n  =

(

)

f

exp   ,

num

(2)

i

i

1 ( w diag MAC −

(

))

F

w

=

+

2 1

i

m

i

i



num

1

i

i

f i

where w iω and w m are the weighting factors corresponding to natural frequencies and mode shapes, respectively. These weighting coefficients in a single objective approach are used to account for the relative contributions and uncertainties associated with experimental estimations of natural frequencies and mode shapes. Given that natural frequencies can be obtained experimentally with ease and high reliability, they are typically assigned higher weighting factors than mode shapes, which are less sensitive to stiffness changes and approximately ten times more influenced by noise