A clear trend indicating that the non-negative constraint enhances the performance of the inverse eigenvalue approach was found using the simulated shear beam. In order to substantiate these observations, irreducible model error must be considered. For the purpose of validation, the authors tested the methods and observations on vibration data measured from the bolted fixed-moment steel frame shown in Fig. 4.4a.
The 1.22 m tall frame was comprised of 3.18 mm thick steel plates and steel angles, and assembled as shown in the elevation drawing Fig. 4.5. At each connection, the plate parallel to the floor is fastened to an orthogonal plate by two angles and six bolts as presented in detail drawing C-C’. Two damage scenarios were tested. In damage case 1 (DC1) a single steel angle connecting the fourth level post to the third level beam was removed (Fig. 4.4b), and in damage case 2 (DC2) the steel angle at the first level was removed. The damage introduced was intended to significantly impact local mass and stiffness. The modelled beam elements affected by the damages are
Figure 4.4: a) Photo of the bolted steel frame experiment before damage. b) Photo of damage case 1 where the third level steel angle was removed
Figure 4.5: Diagram of the physical system with dimensions. Locations of the hammer strikes and accelerometers used for system identification are shown. On the far right is the structural model and the locations damage case 1 (DC1) and damage case 2 (DC2)
shown in Fig. 4.5 along side an element naming convention that is adhered to in this paper.
The frame was instrumented with four uniaxial accelerometers (PCB 333B30) located at each level. To excite the structure’s modes, five hammer strikes from a modal impact hammer (PCB 086C03) were delivered at levels two, three, and four totaling 15 tests for the undamaged and damaged configurations. The sampling frequency was 400 Hz collecting 8192 points for each test. The ERA-OKID was used to extract modal information using the input measurements from the force transducer at the tip of the hammer and the outputted accelerations measured at the four degrees of freedom. The lowest four identified frequencies were selected. The average of the frequencies identified from the undamaged frame and the damaged frame are listed in Table 1. The coefficient of variation of the identified frequencies on average was below 0.001 due to the controlled lab space where the experiments were conducted. Because the variance was low, only the average of the identified frequencies were used to implement the algorithms.
Table 4.1: Comparison of Average Modal Frequencies
Mode Model- undamaged (Hz) Undamaged structure (Hz) Damage case one (Hz) Damage case two (Hz) 1 6.10 6.02 6.10 5.90 2 19.38 19.30 18.68 19.08 3 34.37 34.41 33.87 34.38 4 47.35 47.34 47.29 47.45
The finite element model used to derive the eigenvalue sensitivities (eq. 4.7) is comprised of 34 discrete beam elements and 30 translation degrees of freedom oriented parallel to the floor. The modulus of elasticity was 200 GPa. Coincident nodes were
fixed. At first, the derived natural frequencies were significantly larger than those identified from the undamaged frame indicating a need to update the model. The discrepancy between the model and the physical system was presumably dominated by an inadequate assumption on the rigidity of the connections. In order to match the model’s frequencies to those identified, two design parameters were selected for updating: the moment of inertia of all vertically oriented and all horizontally oriented beam elements at the connections. The natural frequencies derived from the updated model are listed under Model in Table 5.1.
Each stiffness parameter (δki, eq. 4.3) corresponds to the elasticity of a single beam element. Among the 34 total stiffness parameters, 11 are identical in terms of their influence on the identified eigenvalues due to the frame’s symmetry about its vertical axis. Hence, it is impossible to distinguish parameters on the left hand side of the frame from the right hand side. In order to clearly present the following analysis, the parameters of the sensitivity matrix are restricted to the 19 stiffness parameters demarcated in Fig. 4.6a. For the same reasons, the mass parameters (δmi) are restricted to the subset associated with the nodes demarcated in Fig. 4.6c.
Despite removing the frame’s global symmetry, the presence of local quasi-dependencies between stiffness parameters would complicate successful application of the inverse method. This challenge is ubiquitous to the damage identification IEP because the number of unique measurements is so limited. In order to locate local colinearity, the cosine angles between all 19 stiffness parameter sensitivities were calculated. Fig. 4.7a presents the cosine angles between each column of the stiffness sensitivity matrix
Figure 4.6: Parameter subset selections highlighted in red: a) the stiffness elements associ- ated with beams, posts, and connections, b) the stiffness elements associated with connections only, and c) the mass nodes
derived from the baseline model, computed as Ci,j =
|Si· Sj|
||Si||2||Sj||2, i, j = 1...p
k (4.13)
where i and j respectively indicate the row and column of C. If Ci,j = 1 then the
ith and jth elements are colinear, and if Ci,j = 0 then the ith and jth elements are orthogonal. The white colored blocks represent values of Ci,j ≥ 0.99. The clusters of Ci,j ≥0.99 indicate that the parameters representing the connections and the pa- rameters defining the adjacent beams or posts (e.g. El.1 and El.2; El.3 and El.4) are nearly colinear and indistinguishable. The model discretization created additional complications because of the disproportionate scaling of the colinear adjacent param- eters. Fig. 4.7b presents the normalized euclidean magnitudes of the columns of the sensitivity matrix indicating the disproportionate scaling. It is clear from Fig. 4.7b
that the beams and posts are more sensitive to damage than the connections. The proposed method, like any inverse based approach, cannot distinguish parameters among a colinear cluster. Instead inverse algorithms have a strong tendency to iden- tify the most sensitive parameter among a colinear cluster which in this case are the beams and posts.
Figure 4.7: Left hand side figure depicts the cosine angles between each column of the stiff- ness sensitivity matrix normalized between zero and one where zero indicates orthogonality. The right hand side figure presents the normalized Euclidean magnitude of each column of the stiffness sensitivity matrix
Since the inverse method cannot distinguish damage between the connections and the beams and posts, only the stiffness parameters associated with the connections are included in the columns of the stiffness sensitivity matrix in addition to the mass parameters for the remainder of this paper. The 11 stiffness parameters associated with connections are demarcated in Fig. 4.6b.