OBSERVER/KALMAN AND SUBSPACE IDENTIFICATION OF THE UBC BENCHMARK STRUCTURAL MODEL

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1 OBSERVER/KALMAN AND SUBSPACE IDENTIFICATION OF THE UBC BENCHMARK STRUCTURAL MODEL Dionisio Bernal, Burcu Gunes Associate Proessor, Graduate Student Department o Civil and Environmental Engineering, 7 Snell Engineering Center, Northeastern University, Boston MA 5, U.S.A bernal@neu.edu, bvuran@lynx.neu.edu Introduction A two-bay by two-bay our story steel rame owned by the University o British Columbia has been selected as a benchmark structural model to test damage identiication techniques. The structure is approximately in a : scale and has member sizes and details that may be ound in Ventura et al. (997). Testing to generate data or damage identiication is scheduled to begin in the summer o. Beore addressing the ull complexity associated with real experimental data, however, it was considered appropriate to examine a number o cases using simulated data. For this purpose, mathematical models o the test structure with various degrees o reinement have been ormulated and used to generate data or various damage scenarios. This paper documents the damage identiication approach used by the authors and presents the results obtained or the cases considered thus ar. Cases Considered In all the cases examined the structure in both the undamaged and damage states is symmetric and loaded in such a way that the loor slabs undergo pure translation. Boundary accelerations are not imposed and since the weight o the loors is large in comparison to the weights o the columns and bracing elements, the dynamics are dominated by the our horizontal translations o the loors in each direction. Two damaged scenarios are considered: ) removal o bracing elements on the irst story and, ) removal o bracing elements rom the irst and the third stories. Sensor noise is simulated by contaminating the analytically computed response with white noise having a RMS equal to % o that o the response. Two scenarios regarding available inormation are considered, namely: a) the input is measured and b) the input is not measured. Finally, two levels o reinement on the mathematical model used to generate the data are examined: ) the data is generated using a shear building model (-DOF) and, ) the data is generated using a DOF model (includes rotations and axial extensions). Complete details o the cases considered can be ound in the paper by Johnson et. al. that appears in these proceedings. Damage Identiication Framework The methodology presented in this section is ormulated on the assumption that linearity holds in the pre and post-damage states. For the simulated data used here this condition is exactly realized.

2 Module Identiication o Modal Characteristics a) When the input is available the Eigensystem Realization Algorithm with a Kalman Observer is used (Juang 99). b) When the input is not measured Sub-ID, one o several subspace identiication algorithms currently available (Van Overschee and Moor 996) is applied. Module Location o Damage Regions This module contains two parts: a) computation o the lexibility matrix at the sensor locations and b) identiication a subset o elements that contain the damaged elements. a) Flexibility Matrix at Sensor Locations Three techniques that apply when ull sensor data is available are described in Bernal and Gunes (a). A brie description o one o them is presented next. We begin by assuming that a realization has been obtained and we designate the eigenvalues and eigenvectors o the transition matrix A (in continuous time) as Λ and ψ, respectively. Given that the basis o the realization is unspeciied, the entries in the eigenvectors relect an undetermined combination o displacements and velocities and its necessary to perorm a transormation to obtain the eigenvectors in the standard displacement-velocity orm. The details o the transormation are well known and may be ound in Bernal and Gunes (a), inal result is; p p Cψ [ Λ ] Φ d = Cψ Λ, or Φ = p+ Cψ [ Λ ] (,) where p =, or or displacement, velocity or acceleration sensing respectively, Φ d stands or the displacement partition o the eigenvector matrix and the matrix C is the state-to-output inluence matrix obtained in the realization. The contribution o the identiied modes to the lexibility matrix is; d T = φ φ () F ω where φ is the matrix o undamped mode shapes, the term in parenthesis is a diagonal matrix where ω is the undamped natural requency and d is a actor such that φ i d i is a mass normalized mode. I we assume that modal complexity is small then arbitrarily normalized undamped modes are obtained simply by rotating the identiied complex modes o the realization. The mass normalization actor can be computed as ollows (Bernal and Gunes a): where and Y = [ I I ] S Λψ S φ [ I I ] B z ~ Y = [ diag( Y ) diag( Y )... diag( )] ~ T ~ ~ T ~ d = ( Y Y ) Y F () Φ T d F φ b Y r T ~ F ] = (5,6,7) T = [ F F... F r (8,9)

3 The previous mass normalization is restricted to cases where the input is measured. For stochastic input the normalization is carried out by assuming that the mass matrix is known. b) Localization o Damaged Regions As one anticipates, examination o the truncated lexibility can provide inormation on the spatial distribution o damage. In the cases considered here the model o the structure and the conditions considered are suiciently simple that the region o damage can be identiied by inspection. In general, however, this is not possible a systematic approach is necessary. A general approach to extract spatial inormation on the distribution o damage rom changes in lexibility has been recently developed by the irst author and is outlined next. Consider a system or which the identiied lexibility matrices in the pre and post damage states are Fu and Fd. Assume that there are a number o load distributions that produce identical deormations when applied to the undamaged and damaged systems. I we collect all the distributions that satisy this requirement in the matrix L it is evident that one can write; ( Fu Fd) L = () which makes it apparent that L is the null let space o the change in lexibility resulting rom damage (assuming F d F u ). Perorming a singular value decomposition o the incremental lexibility one can write; ~ s T Fd Fu [ q~ ( ) = L] q () where we have accounted or the act that in practical applications the singular values associated with the let null space will not be exactly zero due to modal truncation or approximations in the identiied eigenproperties. From a physical perspective one appreciates that the load distributions that induce no stress in the damaged elements belong to L. These type o vectors are here designated as Damage Locating Vectors (DLV) since they identiy regions that contain damaged elements as regions o zero stress. It is important to emphasize that while elements with stresses rom a DLV can be exclude rom the ree parameter space the elements that show zero stress may or may not be damaged. The number o elements in the subset o potentially damaged elements identiied by the DLV s decreases as the number o sensors increases. It is possible to show that the number o vectors in L equals the number o sensors minus the rank o a stress resultant inluence matrix Q where Q i,j is the stress resultant at damaged location j due to a unit value o a load at the ith sensor. Needless to say, the matrix Q is unknown when the identiication is carried out since the damaged locations are undetermined. Nevertheless, the inormation is useul since it points out that there is a limit to the number o damaged elements that can be spatially identiied with a given number o sensors. For example, in a truss where the damaged elements lead to a Q matrix o ull rank there will be no DLV s when the number o damaged elements equals

4 or exceeds the number o sensors. An examination o the eect o the number o identiied modes and a complete documentation o other theoretical issues associated with the DLVs, as well as an illustration o their perormance in a number o structural conigurations will appear in a orthcoming paper. We note in closing this brie introduction that in the case o the shear buildings considered in this portion o the benchmark study the number o DLVs equals the number o sensors minus the number o damaged loors. Module Quantiication o Damage (Model Updating) The results rom modules and provide a subset o elements that contain the damaged ones. At this stage one may need to urther discriminate between damaged and undamaged elements and to quantiy the damage in the damaged components using a model update strategy. In the particular case o the benchmark structure, at the level o complexity considered thus ar, one is attempting to identiy damage in terms o story stiness values and not at the level o elements. It is possible, thereore, to perorm the model update by simply itting a shear building model to the identiied lexibility (using a least square criterion). In the damaged cases only the levels where the damage was identiied are treated as ree parameters. Results a) Table lists the identiied requencies and damping ratios or all the cases considered. A comparison with the exact results (Johnson et. al.) shows excellent accuracy in all cases. b) The absolute value o the story shears induced by the DLVs are added and plotted in Figs. and or the cases considered. As can be seen rom an inspection o this igure, the damaged levels are clearly identiied. c) The inter-story stiness values obtained rom the lexibility it are presented in Table together with the percent reduction. Final Remarks A multi-stage approach or damage identiication, location and quantiication has been presented and applied to a benchmark structure. The results obtained showed that the ERA-OKID technique and the Sub-Id approach were able to identiy the modal characteristics accurately in all cases. A new technique based on a singular value decomposition o the change in lexibility has been described and utilized to identiy the levels o the structure where the damage occurred. In the particular application o the decomposition o the change in lexibility has been described and utilized to identiy the levels o the structure where the damage occurred. In the particular application o the benchmark structure the damage identiication was restricted to the speciication o the levels where damage was detected and to a quantiication o the percent reduction in the inter-story stiness resulting rom the damage. In case there is no modeling error because the data used in the identiication was generated using a shear building. The results o the ID are in this case virtually exact. In case there is modeling error in the sense that the data was generated using a model that has DOF while the model update

5 o the last stage is carried out assuming a shear building model. Accuracy in the computed Table Natural requencies and damping ratios CASE : Known Input CASE : Known Input No Damage Damage Damage No Damage Damage Damage Mode CASE : Unknown Input CASE : Unknown Input No Damage Damage Damage No Damage Damage Damage Mode CASE : Unknown Input No Damage Damage Damage No Damage Damage Damage Mode Mode Table Interstory stinesses and % reductions CASE (known input) CASE (known input) No Damage Damage Damage No Damage Damage Damage Floor K ( e7) K ( e7) % K ( e7) % K K ( e7) % K ( e7) % e e e e CASE (unknown input) CASE (unknown input) No Damage Damage Damage No Damage Damage Damage Floor K ( e7) K ( e7) % K ( e7) % K( e7) K ( e7) % K ( e7) % CASE x-direction CASE y-direction No Damage Damage Damage No Damage Damage Damage Floor K ( e8) K ( e7) % K ( e7) % K( e8) K ( e7) % K ( e8) %

6 (a) (b) (c) (d) Figure Story shears induced by DLVs or known input cases: (a) Case -Damage (b) Case - Damage (c) Case - Damage (d) Case - Damage (a) (b) (c) (d) (e) () (g) (h) Figure Story shears induced by DLVs or unknown input cases: (a) Case -Damage (b) Case - Damage (c) Case - Damage (d) Case - Damage (e) Casex-Damage () Casex-Damage (g) Casey-Damage (h) Casey-Damage reduction in inter-story stiness can not in this case be compared to an exact result since there is no exact inter-story stiness (i.e, the ratio o shear to drit is dependent on the lateral load distribution). Notwithstanding, it is interesting to note that the procedure correctly identiies the inter-story stiness o the irst loor, which has a ull rotational restraint at the base, as substantially larger than the value o the other stories. Reerences. Bernal, D. and Gunes, B. () 'Extraction o second order system matrices rom state-space realizations', th ASCE Engineering Mechanics Conerence (EM), Austin, Texas, May -,.. Johnson, E.A., Lam, H. F., and Kataygiotis, L. A Benchmark Problem or Structural Health Monitoring and Damage Detection, th ASCE Engineering. Juang, J. (99). Applied System Identiication, Prentice-Hall, Englewood Clis, NJ.. Mechanics Conerence (EM), Austin, Texas, May -,. 5. Overschee, P. and Moor B. L. R. (996) Subspace Identiication or Linear Systems: Theory, Implementation, Applications, Kluwer Academic Publishers, Boston 6. Ventura C. E, Prion, H.G.L., Black, C., Rezai K, M., Latendresse, V. (997) Modal properties o a steel rame used or seismic evaluation studies, 5th International Modal Analysis Conerence in Orlando, Florida, February -6, 997.