Influence of the Objective Function in the Dynamic Model Updating of Girder Bridge Structures
Abstract
1. Introduction
- Modeling errors, which stem from inaccurate or simplifying assumptions in the model definition; these can include uncertainties in the mathematical equations governing the model (model–structure errors), such as geometric simplifications or kinematic and static assumptions; or approximate boundary conditions and model characteristics (model–parameter errors), like simplified restraint and load definitions or erroneous modeling of structural members; or approximate discretization and simplification of the modeled system (model–order errors), such as truncation errors or mesh discretization.
- Parameter value errors, arising from inaccurate assumptions regarding the values of the parameters that govern the structural response, such as material properties as well as member dimensions and stiffnesses; these may not be precisely known or may exhibit spatial variability that are too significant to be accurately described.
- Numerical software/hardware errors, which stem from the algorithms used to solve the model governing equations, including rounding errors and iterative convergence issues.
2. Dynamic Model Updating and Definition of the Objective Functions
2.1. General Concepts
2.2. Model Updating Procedure
- Model class selection;
- Initial model definition;
- Output selection;
- Objective function definition;
- Sensitivity parameter selection;
- Parameter calibration (updating).
2.3. Adopted Objective Functions
- Natural frequency, ;
- Vibration mode shape, ;
- Modal flexibility matrix, .
3. Case Studies
3.1. Bridge A
3.2. Bridge B
4. Experimental Results
4.1. Bridge A
4.2. Bridge B
- Case 1: all equal (unitary) weighting coefficients ;
- Case 2: weighting coefficients inversely proportional to the modes order and normalized to obtain a unitary sum ;
- Case 3: weighting coefficients proportional to the OMA power spectral density (PSD) ordinates and normalized to obtain a unitary sum , with being the ordinate of the PSD at the i-th identified frequency.
- The position of the minimum of function f is sensitive to the weighting scheme; this is located at −5% of E0 for Case 1, at +5% of E0 for Case 2, and at E0 for Case 3;
- The position of the minimum of function MAC2 is insensitive to the weighting scheme, although its identification is not straightforward, due to very small variation for values of the parameter lower than −5% of E0;
- The position of the minimum of functions MFM1 and MFM3 is also insensitive to the weighting scheme, and is located at −5% of E0;
- The position of the minimum of function MFM2 slightly depends on the weighting scheme; this is located at +10% of E0 for Cases 1 and 3, and at +15% of E0 for Case 2.
- As already observed for Figure 13, functions MAC1, MAC2, MFM1, and MFM3 capture the coupling in the mode shapes that occurs at a parameter value of 1.1 E0 (+10%), which results in a peak of the normalized objective function.
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
FE | finite element |
MAC | modal assurance criterion |
MFM | modal flexibility matrix |
OMA | operational modal analysis |
RC | reinforced concrete |
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Objective Function | Considered Outputs | Literature | Pros | Cons |
---|---|---|---|---|
f | Frequencies | [26,27] | - Easy to apply - RMS measure | - Does not capture mode shape variation |
MAC1 | Mode shapes | [26] | - Captures mode shape variation | - Does not capture frequency variation - Mode shapes harder to be determined - Not an RMS measure |
MAC2 | Mode shapes | Proposed | - Captures mode shape variation - RMS measure | - Does not capture frequency variation - Mode shapes harder to be determined |
MFM1 | Frequencies and mode shapes | [13] | - Captures frequency and mode shape variations | - Mode shapes harder to be determined - Not an RMS measure |
MFM2 | Frequencies and mode shapes | Proposed | - Captures frequency and mode shape variations - RMS measure | - Mode shapes harder to be determined |
MFM3 | Frequencies and mode shapes | Proposed | - Captures frequency and mode shape variations - RMS measure | - Mode shapes harder to be determined |
Objective Function | Optimum Value of E [% w.r.t. E0] | Optimum Value of E [GPa] | Second-Order Derivative |
---|---|---|---|
f | −4.2% | 30.66 | 6.29 |
MFM1 | −3.5% | 30.88 | 6.14 |
MFM2 | −3.3% | 30.95 | 11.89 |
MFM3 | −2.9% | 31.06 | 7.66 |
Obj. Fun. | Case 1 | Case 2 | Case 3 | ||||||
---|---|---|---|---|---|---|---|---|---|
Opt. E [% of E0] | Opt. E [GPa] | II Ord. Der. | Opt. E [% of E0] | Opt. E [GPa] | II Ord. Der. | Opt. E [% of E0] | Opt. E [GPa] | II Ord. Der. | |
f | −3.2% | 33.02 | 0.34 | 6.1% | 36.17 | 0.24 | −1.1% | 33.74 | 0.34 |
MAC2 | −16.4% | 28.50 | 0.15 | −7.2% | 31.65 | 0.49 | −6.6% | 31.84 | 0.76 |
MFM1 | −5.2% | 32.33 | 3.35 | −5.1% | 32.37 | 8.12 | −5.1% | 32.37 | 11.23 |
MFM2 | 9.6% | 37.39 | 0.48 | 15.8% | 39.48 | 0.33 | 9.0% | 37.16 | 0.49 |
MFM3 | 5.3% | 35.90 | 2.55 | −5.1% | 32.37 | 5.90 | −5.1% | 32.37 | 8.58 |
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Di Re, P.; Vangelisti, I.; Lofrano, E. Influence of the Objective Function in the Dynamic Model Updating of Girder Bridge Structures. Buildings 2025, 15, 341. https://doi.org/10.3390/buildings15030341
Di Re P, Vangelisti I, Lofrano E. Influence of the Objective Function in the Dynamic Model Updating of Girder Bridge Structures. Buildings. 2025; 15(3):341. https://doi.org/10.3390/buildings15030341
Chicago/Turabian StyleDi Re, Paolo, Iacopo Vangelisti, and Egidio Lofrano. 2025. "Influence of the Objective Function in the Dynamic Model Updating of Girder Bridge Structures" Buildings 15, no. 3: 341. https://doi.org/10.3390/buildings15030341
APA StyleDi Re, P., Vangelisti, I., & Lofrano, E. (2025). Influence of the Objective Function in the Dynamic Model Updating of Girder Bridge Structures. Buildings, 15(3), 341. https://doi.org/10.3390/buildings15030341