PSI - Issue 78

Antonio Sánchez López-Cuervo et al. / Procedia Structural Integrity 78 (2026) 1791–1798

1797

(a)

(b)

(c)

Fig. 6: (a) Optimal solutions; (b) MAC between OMA and FEM displacement mode shapes; (c) MAC between OMA and FEM strain mode shapes.

• Elastic modulus of steel ( E ); • Steel density ( ρ ); • Bending sti ff ness of the column-slab joint (modelled as semi-rigid) for all intermediate levels ( k M y , k M z ); • Bending sti ff ness at the top columns ( k ′ M y , k ′ M z ); • Bending sti ff ness at the bottom columns ( k ′′ M y , k ′′ M z ); • Support sti ff ness in the three translational directions ( U x , U y , U z ), modeled as elastic springs.

In order to preserve the physical consistency of the structure, the following constraints were imposed:

k M y > k ′ M y

> k ′′ M

y ; U x

> U z ; U y > U z

(3)

Figure 6(a) shows a set of solutions obtained during the optimization process. Red plus markers ( + ) represent the optimal set, or Pareto front, while blue cross symbols (x) correspond to dominated solutions. The x and y axis show the value of the cost function defined in Eq. (1) using data from the acceleration- and strain-based modal identifications, respectively. Any solution belonging to the Pareto front can be used to update the FEM. In this work, the selected solution minimizes the sum of the squared values of the cost function (Eq. (1)) evaluated for each sensing system. Table 2 compares the identified natural frequencies (from accelerometers and SGs) with those obtained from the uncalibrated and calibrated FEMs. As shown, the calibration process significantly improves the accuracy, reducing the maximum error from − 27 . 61% (first mode) to − 3 . 96% (fifth mode). The reported error is computed as the average relative di ff erence between the FEM results and both sensor-based measurements.

Table 2: Comparison of experimental and FEM natural frequencies for both uncalibrated and calibrated models.

Acc.-OMA [Hz]

Strain-OMA [Hz]

FEM uncal. [Hz]

FEM cal. [Hz]

Mean error uncal. [%]

Mean error cal. [%]

-27.61% -20.03% -17.04% -16.66% -15.65%

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

2.608 8.078

2.611 8.086

3.330 9.701

2.644 7.922

-1.32% 1.98% 2.06% 0.32% -3.96%

12.874 16.393 18.704

12.881 16.409 18.720

15.072 19.134 21.641

12.612 16.349 19.453

Figure 6(b) presents the MAC values between the OMA-based displacement mode shapes and those obtained from the calibrated FEM. The fifth mode shows the largest deviation, with a MAC of 0.859. Similarly, Figure 6(c) shows the MAC values for the strain mode shapes. In this case, the fourth mode yields the lowest MAC value (0.858). The small reported errors in both frequency and mode shape correlation confirm the robustness of the proposed calibration approach.