M Z ( s ) + C Z ( s ) + K Z ( s ) = ( s ) FE - PowerPoint PPT Presentation

Handling the temperature effect Introduction in vibration monitoring of civil structures: • Usefulness of global vibration-based SHM methods a combined subspace-based and nuisance rejection approach • Limitations due to temperature effects on the dynamics of civil engineering structures ´ Etienne Balm` es • Wanted: discriminate between changes in modal parameters MssMat , Ecole Centrale Paris, France due to damages and changes due to temperature effects Mich` ele Basseville, Laurent Mevel, Houssein Nasser • A statistical subspace-based damage detection algorithm: null space of a matrix built on reference modes/modeshapes IRISA (CNRS & INRIA & Univ.), Rennes, France at a known temperature • Proposed solution to temperature handling: National Computer & Security project Constructif no temperature measurement, thermal effect modeling, statistical nuisance rejection houssein.nasser@irisa.fr -- http://www.irisa.fr/sisthem/ 1 2 Modeling Content  M ¨ Z ( s ) + C ˙ Z ( s ) + K Z ( s ) = ν ( s )         FE model:  Parametric subspace-based damage detection     Y ( s ) = L Z ( s )      ( Mµ 2 + Cµ + K ) φ µ = 0 , The temperature effect - Examples ψ µ = Lφ µ The temperature effect - Modeling  X k +1 = F X k + V k         State space:  The temperature effect - Rejection     Y k = HX k      ∆ Experimental results F Φ λ = λ Φ λ , ϕ λ = H Φ λ Conclusion e τµ = λ ψ µ = ϕ λ , � �� � � �� � modes modeshapes 3 4

  Λ modes θ ∆ Parametric subspace-based damage detection Canonical parameter : =     vec Φ mode shapes    X k +1 = F X k + V k F Φ λ = λ Φ λ   Φ           Φ∆       Observability in modal basis : O p +1 ( θ ) =   .  .  .        ∆     Φ∆ p Y k = H X k ϕ λ = H Φ λ      θ 0 : reference parameter for safe structure   R 0 R 1 R 2 . . . Left null space: S T S = I s , S T O p +1 ( θ 0 ) = 0       R 1 R 2 R 3 . . .   ∆   ∆  Y k Y T  ,   R i = E =   H   k − i   R 2 R 3 R 4 . . .       . . ... . Y k : N -size sample of new measurements   . . .  . . .    Residual for SHM: R i = H F i G = ⇒ H = O C √ N vec( S T ( θ 0 ) ˆ ζ N ( θ 0 ) ∆ = H )   � G H C ∆ � F 2 G = F G . . .    HF  O ∆   J ( θ 0 ) : sensitivity of ζ w.r.t. modal changes; Σ( θ 0 ) : covariance = ,   HF 2     � � G ∆   X k Y T . = E  .  . k N Σ − 1 J ( J T Σ − 1 J ) − 1 J T Σ − 1 ζ N χ 2 -test: ζ T → ( H, F ) − → ( λ, φ λ ) H − → O − ≥ h 5 6 Focussed monitoring Fisher information F( θ 0 ) ∆ = J ( θ 0 ) T Σ( θ 0 ) − 1 J ( θ 0 ) The temperature effect - Example 1 Z24 bridge       J T a Σ − 1 J a J T a Σ − 1 J b   F aa F ab  θ a θ =    , J = ( J a J b ) , F =    =         J T b Σ − 1 J a J T b Σ − 1 J b F ba F bb θ b  • A benchmark of the BRITE/EURAM project SIMCES and of = F aa − F ab F − 1 ∆ F ⋆ F ba a bb the European COST action F3 Sensitivity approach - Partial residual • Response to traffic excitation under the bridge measured over one year in 139 points ζ a ∆ a Σ − 1 ζ, ∆ = J T χ 2 ζ T a F − 1 ˜ ˜ aa ˜ ˜ = ζ a a • Two damage scenarios (DS1 and DS2): Min-max approach - Robust residual pier settlements of 20mm and 80mm. ∆ ζ a − F ab F − 1 ζ ⋆ = ˜ ˜ χ ⋆ 2 = ζ ⋆T F ⋆ − 1 ζ ⋆ ζ b , a bb a a a a 7 8

Identified first four natural frequencies / Test values (Results with four sensors) χ 2 Mode 1 2 3 4 Evolution of the test values over nine months (log-scale). Undamaged Freq.(Hz) 3.88 5.01 9.80 10.30 8.80 · 10e2 Damaged (1) Freq.(Hz) 3.87 5.06 9.79 10.32 8.00 · 10e5 Damaged (2) Freq.(Hz) 3.76 4.93 9.74 10.25 3.96 · 10e6 Distribution of the test values for each of the nine months. 9 10 1.72 The temperature effect - Example 2 safe 1.715 damaged 1.71 Simulated bridge deck 1.705 Frequency 1.7 1.695 Finite elements toolbox OpenFEM (with Matlab or Scilab). 1.69 60 m span, 9600 volume elements, 13668 nodes. 1.685 −1 −2 −3 −4 −5 −6 −7 −8 ∆ T ° Temperature variations: either a uniform temperature elevation Decreasing temperature effect on the first frequency. or a linear variation with z . 4000 safe damaged 3500 3000 2500 χ 2 2000 1500 1000 500 0 −1 −2 −3 −4 −5 −6 −7 −8 ∆ T ° Partial χ 2 -test values. Linear thermal field (Left) and induced axial stress (Right). Safe and damaged The warmer deck expands while the cooler bottom contracts. 11 12

The temperature effect - Modeling Computing the residual sensitivity w.r.t. T • Thermal field → materials expansion → thermal stress field. Small deviations ∆ • Temperature modification = external load. E( ζ N ) = J ( θ 0 ) δθ + J T δT, J T = J ( θ 0 ) J θT • Static equilibrium under thermal loading ↔ pre-stress. J θT involves computing δµ and δφ for δK = K T,i , δM = δC = 0 , based on the differentiation of: • Stiffness K , and thus modal frequencies, affected. Mass M and damping C assumed not affected. ( Mµ 2 + Cµ + K ) φ µ = 0 • Assuming thermal loads inducing small perturbations: that is: stiffness = linear function of the thermal field T ( x ) . δµ = − φ T ( µ 2 δM + µ δC + δK ) φ φ T (2 µ M + C ) φ • If T ( x ) = linear combination of constant thermal fields, and temperature effect on K : ( µ 2 M + µ C + K ) δφ = − δµ (2 µ M + C ) φ − ( µ 2 δM + µ δC + δK ) φ ∆ K = K 0 + K T = K 0 + � i α i K T,i with φ T δφ = 0 . 13 14 Implementation issues The temperature effect - Rejection • θ 0 : reference modal parameter for safe structure • Compute the key matrices: Compute the null space S ( θ 0 ) , sensitivities J ( θ 0 ) and J T and covariance Σ the sensitivities J ( θ 0 ) and J T , on a long data sample for the safe structure. the covariance Σ( θ 0 ) and Fisher matrix F ∆ = F( θ 0 , T ) In case of nonstationary excitation, computing Σ • Y k : N -size sample of new measurements on current data might be preferable. √ • Compute the residual ζ N ( θ 0 ) ∆ N vec( S T ( θ 0 ) ˆ = H ) ∆ ζ θ 0 − F θ 0 ,T F − 1 • Σ computed with QR (Zhang, 2003). • Compute the robust residual ζ ⋆ = ˜ T,T ˜ ζ T θ 0 Null space S computed with QR (Nasser, 2006). T F ⋆ − 1 • Compute the χ 2 -test : ζ ⋆ ζ ⋆ θ 0 θ 0 θ 0 15 16

Conclusion Example - Back to the simulated bridge deck 1.705 safe 1.7 damaged Temperature effect in vibration-based SHM 1.695 1.69 Frequency Statistical parametric model-based approach 1.685 1.68 1.675 Subspace-based damage detection algorithm 1.67 1 2 3 4 5 6 7 8 ∆ T ° Increasing temperature effect on the first frequency. Local rejection of the temperature seen as nuisance 18 x 10 4 safe without rejection 16 damaged without rejection safe with rejection 14 Example: simulated bridge deck damaged with rejection 12 10 χ 2 8 Ongoing: empirical null space merging data at # temperatures, 6 4 analytical temperature-adjusted null space 2 0 1 2 3 4 5 6 7 8 ∆ T ° Global (dotted) and minmax (solid) χ 2 -test values. Future: in-operation examples, Minmax operating range: 8 C × 2 extension to 3D temperature fields, Safe and damaged thermal model parameterization 17 18