1.
Workshop
CERI, UCD, Dublin
Wednesday 29th August 2018

2.
Daniel Martinez Otero, Eugene OBrien, Abdollah
Malekjafarian, Yahya M. Mohammed, Nasim Uddin
Sensitivity of SHM Sensors to
Bridge Stiffness

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.
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.
Sensitivity of SHM Sensors to Bridge Stiffness
Sensor Installation at mid span

6.
Sensitivity of SHM Sensors to Bridge Stiffness
Situation Analysed

7.
Sensitivity of SHM Sensors to Bridge Stiffness
2nd Axle
Load
1st Axle
Load

8.
Sensitivity of SHM Sensors to Bridge Stiffness
Envelope of the sensitivities

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.
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.
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.
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.
Calculating Bridge Stiffness from deflection
measurements

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.
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