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# "Sensitivity of SHM sensors to bridge stiffness" presented at CERI2018 by Daniel Martinez

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Bridges play an important role in transport infrastructure and it is necessary to frequently monitor them. Current vibration-based bridge monitoring methods in which bridges are instrumented using several sensors are sometimes not sensitive enough. For this reason, an assessment of sensitivity of sensors to damage is necessary. In this paper a sensitivity analysis to bridge flexural stiffness (EI) is performed. A discussion between the use of strain or deflections is provided. A relation between deflection and stiffness can be set by theorem of virtual work, expressing the problem as a matrix product. Sensitivity is obtained by deriving the deflection respect to the reciprocal of the stiffness at every analysed location of the bridge. It is found that a good match between the deflection and the bridge stiffness profile can be obtained using noise-free measurements. The accuracy of sensors is evaluated numerically in presence of damage and measurement noise. Field measurements in the United States are also described to identify the potential issues in real conditions.

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### "Sensitivity of SHM sensors to bridge stiffness" presented at CERI2018 by Daniel Martinez

1. 1. Workshop CERI, UCD, Dublin Wednesday 29th August 2018
2. 2. Daniel Martinez Otero, Eugene OBrien, Abdollah Malekjafarian, Yahya M. Mohammed, Nasim Uddin Sensitivity of SHM Sensors to Bridge Stiffness
3. 3. Sensitivity of SHM Sensors to Bridge Stiffness 𝛿 t×1 = P t×n J n×1 𝛿 = න 𝑀𝑀 𝑢 𝐸𝐼 𝑑𝑥Theorem of Virtual Work Matrix representation of Theorem where 𝐏 is the matrix representing 𝑴𝑴 𝒖 values and 𝐉 is vector of 𝟏 𝑬𝑰 components (reciprocal of stiffness) valid for statics with sensors on the bridge 𝐸𝐼 𝑛 = 1 𝐽 𝑛
4. 4. Sensitivity of SHM Sensors to Bridge Stiffness Sensitivity 𝑆 𝑖, 𝑗 : Sensitivity of the deflection respect to the reciprocal of the flexural stiffness 𝑆 𝑖, 𝑗 = 𝜕𝑢𝑖 𝜕𝐽𝑗 𝐽𝑗: reciprocal of the flexural stiffness at element 𝑗
5. 5. Sensitivity of SHM Sensors to Bridge Stiffness Sensor Installation at mid span
6. 6. Sensitivity of SHM Sensors to Bridge Stiffness Situation Analysed
7. 7. Sensitivity of SHM Sensors to Bridge Stiffness 2nd Axle Load 1st Axle Load
8. 8. Sensitivity of SHM Sensors to Bridge Stiffness Envelope of the sensitivities
9. 9. Calculating Bridge Stiffness from deflection measurements 𝛿 t×1 = P t×n J n×1 𝛿1 𝛿2 𝛿3 𝛿4 𝛿5 = 𝑃1𝐴 𝑃1𝐵 𝑃1𝐶 𝑃2𝐴 𝑃2𝐵 𝑃2𝐶 𝑃3𝐴 𝑃3𝐵 𝑃3𝐶 𝑃4𝐴 𝑃4𝐵 𝑃4𝐶 𝑃5𝐴 𝑃5𝐵 𝑃5𝐶 × 𝐽 𝐴 𝐽 𝐵 𝐽 𝐶 We can solve to find the stiffnesses only if the matrix is square, i.e., only if the number of unknown J’s equals the number of measurements 𝑏 = 𝐴𝑥 𝑨 IS A NON SQUARED MATRIX AND NOT INVERTIBLE
10. 10. Calculating Bridge Stiffness from deflection measurements The system of equations is also ill conditioned, i.e., even when the matrix is square, accuracy is very sensitive to noise in the measurement 𝛿1 𝛿2 𝛿3 𝛿4 𝛿5 = 𝑃1𝐴 𝑃1𝐵 𝑃1𝐶 𝑃2𝐴 𝑃2𝐵 𝑃2𝐶 𝑃3𝐴 𝑃3𝐵 𝑃3𝐶 𝑃4𝐴 𝑃4𝐵 𝑃4𝐶 𝑃5𝐴 𝑃5𝐵 𝑃5𝐶 × 𝐽 𝐴 𝐽 𝐵 𝐽 𝐶
11. 11. Calculating Bridge Stiffness from deflection measurements Moses algorithm + Penalty function Moses algorithm minimises the differences between measured deflections and the theoretical response. It solves the problem of the non-square P matrix Penalty function ensures that there is no sudden change of neighbouring stiffness values, avoiding, for example, negatives. It is based on a Blackman window. It solves the ill conditioning problem. Two problems: (i) Matrix not square, (ii) System is ill-conditioned
12. 12. Calculating Bridge Stiffness from deflection measurements Moses algorithm Penalty Function 𝐸 = 𝑚𝑖𝑛 𝛿 − 𝑃 𝐽 2 ෍ 𝑖=1 𝑛−1 𝑘 (−𝑐1 𝐽𝑖−𝑏 − ⋯ − 𝑐 𝑏 𝐽𝑖−1 + 𝐽𝑖 − 𝑐 𝑏+1 𝐽𝑖+1 − ⋯ − 𝑐2𝑏 𝐽𝑖+𝑏)2 Where k is a set constant, b is the number of points used in the filter each side and the sum of constants 𝑐1 to 𝑐2𝑏 equals 1
13. 13. Calculating Bridge Stiffness from deflection measurements
14. 14. Conclusions • Deflection sensitivity to stiffness is greater at mid- span. • Deflection data with loading conditions closer to mid-span should improve stiffness calculations. • Moses and penalty function can potentially calculate stiffness decreasing the influence of noise.
15. 15. The TRUSS ITN project (http://trussitn.eu) has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 642453 Thanks for your attention