4.3.1 Step 1: Selection of sensitive modes

The vibration modes for damage identification are selected based on the eigenvalue change ratio Δλ_i/λ_i. Before applying any damage identification algorithm to modal parameters obtained in modal testing, the sensitivity of the vibration modes to the existing damage should be evaluated to ensure the effectiveness and accuracy of damage localization and damage quantification. The eigenvalue change ratio Δλ_i/λ_i is adopted as a mode sensitivity indicator.

4.3.2 Step 2: Localization of damage

Damage is then localized using the DSCF and DLF. A DI based on the DSCF is proposed as a good indicator for damage location. The DI β*_ij at the jth element in the ith mode can be defined as:

${\beta }_{\mathit{ij}}^{*}=\{\begin{array}{ll}{\mathrm{DSCF}}_{\mathit{ij}}\hfill & \mathrm{if}{\mathrm{DLF}}_{\mathit{ij}}>\frac{5}{n}\%\hfill \\ 1\hfill & \mathrm{if}{\mathrm{DLF}}_{\mathit{ij}}\le \frac{5}{n}\%\hfill \end{array}$ [4.17]

[4.17]

where n is the total element number. Since DLF_ij is a damage location factor that reflects the sensitivity of the ith vibration mode to the damage at the jth element, a vibration mode is insensitive to damage at an element with extremely low DLF in this mode. Therefore, this mode is inadequate to detect damage at this element, even if the damage exists. The DI with extremely low DLF_ij is directly set to 1 correspondingly to avoid the potential numerical error. In this study, the criterion for extremely low DLF_ij is set at 5% of the average of DLF_ij in the ith mode.

Then, assuming that the damage indices β* at different elements is a normally distributed random variable, the DI β*_ij can be normalized as

$Z_{\mathit{ij}}^{*}=\frac{{\beta }_{\mathit{ij}}^{*}-{\overline{\beta }}_i^{*}}{{\sigma }_{\beta i}^{*}}$ [4.18]

[4.18]

where ${\overline{\beta }}_i$ and σ_βi represent the mean and standard deviation of the damage indices in the ith mode, respectively. A damage detection criterion can be set as the normalized DI Z*_ij larger than 2, to filter out small damage indices induced by measurement noise.

$Z_{\mathit{ij}}^{*}=0\mathrm{if}\left|Z_{\mathit{ij}}^{*}\right|<2$ [4.19]

[4.19]

Furthermore, in order to eliminate false positive indications of damaged element induced by the measurement and numerical errors at those elements with low strain energy, a modified damage indices β_j at the jth element can be derived from DLF_ij and Z*_ij as

${\beta }_j={\displaystyle \sum _{i=1}^m{Z}_{\mathit{ij}}^{*}\times {\mathrm{DLF}}_{\mathit{ij}}}$ [4.20]

[4.20]

Finally, a normalized DI Z_j can be generated from β_j to indicate the damaged elements,

$Z_j=\left|\frac{{\beta }_j-\overline{\beta }}{{\sigma }_{\beta }}\right|$ [4.21]

[4.21]

and a similar damage detection criterion can be set as in Equation [4.19].

$Z_j=0\mathrm{if}{Z}_j<2$ [4.22]

[4.22]

4.3.3 Step 3: Quantification of damage

Damage severity is finally quantified using the eigenvalue change ratio, DLF, and DSCF. Once the damaged element is detected in Step 2, the DSCF can be further used to calculate the severity of damage. If multiple modes are available, Equation [4.14] can be written in the matrix form as

$\begin{array}{l}RA=F\hfill \end{array}$ [4.23]

[4.23]

$R=\left[\begin{array}{cccc}{\mathrm{DLF}}_{11}\times {\mathrm{DSCF}}_{11}& {\mathrm{DLF}}_{12}\times {\mathrm{DSCF}}_{12}& \dots & {\mathrm{DLF}}_{1n}\times {\mathrm{DSCF}}_{1n}\\ {\mathrm{DLF}}_{21}\times {\mathrm{DSCF}}_{21}& {\mathrm{DLF}}_{22}\times {\mathrm{DSCF}}_{22}& \dots & {\mathrm{DLF}}_{2n}\times {\mathrm{DSCF}}_{2n}\\ \dots & \dots & \dots & \dots \\ {\mathrm{DLF}}_{m1}\times {\mathrm{DSCF}}_{m1}& {\mathrm{DLF}}_{m2}\times {\mathrm{DSCF}}_{m2}& \dots & {\mathrm{DLF}}_{mn}\times {\mathrm{DSCF}}_{mn}\end{array}\right],A=\left\{\begin{array}{c}{\alpha }_1\\ {\alpha }_2\\ \dots \\ {\alpha }_n\end{array}\right\},F=\left\{\begin{array}{c}-\frac{\mathrm{\Delta }{\mathit{\lambda }}_1}{{\mathit{\lambda }}_1}\\ -\frac{\mathrm{\Delta }{\mathit{\lambda }}_2}{{\mathit{\lambda }}_2}\\ \dots \\ -\frac{\mathrm{\Delta }{\mathit{\lambda }}_m}{{\mathit{\lambda }}_m}\end{array}\right\}$

where m is the total number of vibration modes, n is the total damaged element number, and DLF is the damage location factor γ_ij/γ_i.

For a unique solution, the number of the used vibration modes should be equal to the total number of unknown parameters (elemental damage severities). However, for the laboratory test or in situ application, the number of accessible vibration modes that are suitable for damage identification is very limited (typically the first few modes), and it is usually much less than the number of unknown parameters required. Furthermore, due to the existence of measurement errors, more measured modes than unknown parameters are required to construct a system of overdetermined equations to improve the accuracy of damage severities estimation. Therefore, to avoid non-uniqueness, appropriate assumptions on damage location or the damage severity distribution pattern have to be made to reduce the number of unknown parameters. For example, in damage identification with large sensor spacing, it is usually assumed that the damage only happens in a single element. For example, in the delamination identification case, it is reasonable to assume that the damage severity is the same for one delaminated area since delamination usually propagates between the same layers inside a composite plate.

When the number of measured modes is larger than the total number of unknown parameters, the least square fitting method is used to obtain the best solution.

$A={\left(R^{T}R\right)}^{-1}{R}^{T}F$ [4.24]

[4.24]

4.4.1 Description of the fiber-reinforced polymer (FRP) sandwich deck panel

The FRP sandwich deck panel is a plate-type structure with a length of 1.981 m and a width of 1.495 m. The sandwich panel was connected to two underlying concrete supports using steel bolted connectors, which are considered to provide a simply supported boundary condition. The FRP sandwich panel consists of two face laminates and a sinusoidal honeycomb core, as shown in Fig. 4.1. The thickness of face laminates and honeycomb core are t_f = 13.5 mm and t_c = 128 mm, respectively. The constituent materials used for the composite sandwich panel (both face laminates and core) consist of E-glass fibers and polyester resin, and their mechanical properties are listed in Table 4.1.

The geometry of the sinusoidal core is shown in Fig. 4.2. The wave function of corrugated core wall can be defined as

$x=\frac{b}{2}\left(1-\cos \frac{\pi y}{a}\right)$ [4.25]

[4.25]

The dimensions of the sinusoidal core are a = 50.8 mm, b = 50.8 mm, and t₁ = t₂ = 3.34 mm (as shown in Fig. 4.2).

4.4.2 Experimental setup

The modal testing of the FRP sandwich plate was conducted using a polyvinylidene fluoride (PVDF) sensor network bonded to the top surface in the longitudinal direction. The PVDF sensors used in the experiment are Model DT1–028K/L from the Measurement Specialties Company. A total of 285 nodes of PVDF sensors are uniformly distributed into 19 rows and 15 columns on the surface of the panel to build a sensor array. The plate is divided into 18 × 14 elements by the 19 × 15 sensors.

The panel is subjected to a dynamic pulse load applied at the central point using a modally tuned hammer (PCB 086C20). The response measurements are captured by the 19 × 15 sensor network to record the response of the structure subjected to the fixed point force excitation. A total of 19 × 15 node points are monitored corresponding to an actual spatial sampling distance of 101.6 mm. The coherent signals of the excitation and the responses are digitized and collected by the data acquisition system dSPACE CP1103 at a sampling frequency of 4000 Hz. Due to the fact that only 20 channels are available, only one column of nodes with 19 sensors is measured in each experiment. In this way, the 285 transfer functions between the hammer excitation and different PVDF sensors can still be measured for modal analysis in 15 experiments. The experimental setup using the distributed PVDF sensor array is shown in Fig. 4.3.

After the healthy FRP sandwich panel is tested, three stages of damage with increasing severity are artificially induced into the sandwich panel. In the first stage of induced damage (Damage Stage 1), a saw cut is induced between the top face sheet and the honeycomb core, to simulate debonding between the top face sheet and honeycomb core. A 203.2 mm × 203.2 mm area of face sheet-core debonding (cut) is shown in Fig. 4.4. In the second stage of increasing damage (Damage Stage 2), another saw cut is further induced between the bottom face sheet and the honeycomb core in the same area of the top face sheet-core debonding, to simulate debonding between the bottom face sheet and honeycomb core. Finally, in the third stage of the largest damage in this study (Damage Stage 3), the two vertical sides of the core between the two face sheet-core debonding areas are cut to simulate the core crashing scenario, as shown in Fig. 4.4.

4.4.3 Modal testing results

The measurements for each sensor node are repeated 16 times, and a pretrigger of 0.05 seconds at the level of 0.01 volt is set to synchronize the time-domain signal. The synchronized time history data from the excitation and response measurements are averaged to enhance the signal-to-noise ratio (SNR), as shown in Fig. 4.5. Then, the frequency-domain power spectral densities are generated from the excitation and response time-domain data using fast Fourier transform (FFT), as shown in Fig. 4.6. The frequency-response functions (FRFs) of these tested points are calculated from the excitation and response frequency-domain data. The FRF curve and its coherence curve, corresponding to time-domain data in Fig. 4.5 and frequency-domain data in Fig. 4.6, are illustrated in Fig. 4.7. Finally, these FRF curves are imported to the modal analysis program ME’Scope for curve fitting and modal extraction. The modal testing is conducted using a fixed point force excitation and a distributed PVDF sensor network, so the extracted mode shape should be the strain/curvature mode shape of the plate (Wang, 1998).

Since the plate is tested using a PVDF sensor network in the longitudinal direction (y-direction), only the curvature mode shapes in the longitudinal direction (κ_yy) are captured. Therefore, only the longitudinal bending modes, in which κ_yy is dominant, are suitable for damage identification. In the excitation frequency range of the modally tuned hammer, the first and third bending modes are captured. However, the second bending mode cannot be captured, because the fixed point force excitation is applied at the central point of the plate, which is at the nodal line of the second bending mode. Hence, the first and third bending modes in the longitudinal direction are investigated for damage identification. It should be noted that the modal frequency of the first bending mode lies closely around 60 Hz. Because the unshielded PVDF sensors are prone to electro-magnetic interference (EMI), their responses around 60 Hz could be noisy due to EMI from power system.

In the healthy stage and the three damage stages, the modal testing is conducted to extract the modal frequencies and curvature mode shapes of the FRP sandwich deck panel. The extracted curvature mode shapes and modal frequencies of the FRP sandwich deck panel for the first and third bending modes are shown in Plates I and II (in the color section between pages 172 and 173), respectively.

4.5.1 Step 1: Mode sensitivity

For the first longitudinal bending mode, the modal eigenvalue change ratios at three damage stages are 0.57%, 0.91%, and 3.59%, respectively. While for the third longitudinal bending mode, the modal eigenvalue change ratios at three damage stages are 0.71%, 1.55%, and 4.07%, respectively. It can be seen from the eigenvalue change ratio that both the modes are sensitive to the damage evolution (severity). Thus, both are used for damage identification.

4.5.2 Step 2: Damage localization

The modal strain energy of a plate can be derived from the curvature mode shapes of κ_xx, κ_yy, and κ_xy. However, one PVDF sensor can only be used to measure the strain response in one direction. Therefore, a sensor array positioned in one direction can only capture the strain/curvature mode shape in that direction at one time. Hence, for a longitudinal bending mode, the modal strain energy is approximated by its partial modal strain energy in the dominant direction. Assuming κ_xx = κ_xy = 0, the partial modal strain energy in the y-direction can be derived from Equation [4.5] as

${\gamma }_{\mathit{ij}}^y={\displaystyle \int {\displaystyle \underset{A_j}{\int }{\kappa }_{\mathit{yyi}}^2\mathrm{d}x\mathrm{d}y}}$ [4.26]

[4.26]

The DLF and DSCF matrices for damage identification can be generated as follows. First, a bivariate cubic spline interpolation function is constructed for the curvature mode shape. The bivariate cubic spline is constructed as the tensor product of two univariate cubic splines. It can be expressed as the weighted sum of products of two cubic spline functions:

$f\left(x,y\right)={\displaystyle \sum _i{\displaystyle \sum _{j}a\left(i,j\right)g_i\left(x\right)h_j\left(y\right)}}$ [4.27]

[4.27]

where g_i(x) and h_j(y) are the cubic spline functions in x- and y-directions, respectively. Then, using the bivariate cubic spline function, γ_ij and γ*_ij can be calculated from Equation [4.26]. Finally, the DLF and DSCF matrix can be calculated from γ_ij and γ*_ij.

The DSCF-based damage indices using the modal strain energy from the first and third longitudinal bending modes are shown in Plate III (in the color section between pages 172 and 173). Although the damaged area does not exactly match the sensor grids, the DSCF damage localization method correctly approximates the location and area of the damage. It can be noticed that in Damage Stage 3 a false positive indication is given near the damaged area. There might be two reasons: (a) the effect of damage might have spread out to surrounding elements (Choi et al., 2006); and (b) when cutting the two sides of the damaged area, the saw cut might have damaged the sinusoidal core of the neighboring element.

4.5.3 Step 3: Damage quantification

Assuming that each damaged element has the same damage severity, the DSCF method can be further applied for damage quantification using the DLF and DSCF derived in Step 2. The damage quantification results of the DSCF-based damage identification method are presented in Fig. 4.8. The damage quantification results, based on the four damaged elements indicated in Damage Stages 1 and 2, are marked in black in Fig. 4.8. The damage quantification result, based on the six damaged elements indicated in Damage Stage 3, is also shown in gray for comparison in Fig. 4.8. As shown in Fig. 4.8, the damage severity increases consistently with the Damage Stages 1 to 3.

4.6.1 Modeling of face sheets and the sinusoidal core

The lay-up of the face sheets and sinusoidal core is shown in Fig. 4.9, and corresponding layer material properties are listed in Table 4.2. The face laminate includes five ChopSM bonding layers (ChopSM308) and four bi-directional combination mat layers (CDM3208), while the microstructure of core walls consists of two layers of ChopSM (ChopSM308).

To predict the equivalent properties of the face laminates from the fiber and matrix material properties, a micro–macro mechanics approach (Davalos et al., 2001; Qiao and Wang, 2005) is adopted. First, the properties of each layer in the face sheet can be computed using periodic microstructure (PM) approaches (micromechanics) (Barbero, 1999). Then, based on ply properties and lay-up, the apparent stiffnesses of the face laminate can be predicted using classical lamination theory (macromechanics). The equivalent analysis of core is performed using the methods of homogenization and mechanics of materials approaches (Davalos et al., 2001; Qiao and Wang, 2005). The derived equivalent stiffness properties of face laminates and core walls are listed in Table 4.3.

4.6.2 Finite element modal analysis of FRP sandwich panel

The commercial finite element analysis package ABAQUS is used to conduct an eigenvalue analysis of the FRP sandwich deck panel. A detailed model of the full-size sandwich panel is constructed in ABAQUS, as shown in Plate IV (in the color section between pages 172 and 173). The face sheets and the sinusoidal core walls are modeled as four-node first-order plate elements S4 and three-node first-order plate elements S3. The corresponding stiffness properties of face sheets and core walls are given in Table 4.3. The boundary conditions are considered to be simply supported at both the transverse edges.

The FEA is conducted on both the health and damaged sandwich panel. Three damaged sandwich panel models are set up to simulate the FRP sandwich deck panels with artificially induced damage in the three damage stages (see Fig. 4.4c). The debonding between the face sheets and core is simulated by detaching the node/element connection between the face sheets and core elements. The cuts in the sinusoidal core are similarly simulated. The simulated mode shapes of the FRP sandwich panel are shown in Plate IV. The displacement mode shapes of the healthy and damaged panel are very close to each other; thus, only the displacement mode shapes of the healthy one are shown in Plate IV. The strain mode shape can be then extracted from the 19 × 15 nodes corresponding and identical to the 19 × 15 PVDF sensors in the experimental test, as shown in Plates V and VI (see in the color section between pages 172 and 173), respectively, for the first and third bending modes.

4.7.1 Step 1: Mode sensitivity

The modal frequencies of the healthy sandwich panel from the FEA are 62.983 and 418.6 Hz, which shows close agreement with the experimental test data of 61.08 and 403.00 Hz for the first and third bending modes, respectively. The result confirms that the boundary conditions at the two transverse supporting ends should be modeled as simply supported, and the free vibration modes of the FRP sandwich panel can be accurately predicted by the detailed finite element (FE) model. The modal eigenvalue change ratios at three damage stages from the FEA and experimental test are compared in Table 4.4. The FEA results also show that both the modes (first and third) are sensitive to the damage, so both can be used for damage identification. It can also be noticed that the FEA tends to underestimate the modal eigenvalue change ratios for the first mode but over-predict the ratios for the third mode.

4.7.2 Step 2: Damage localization

The DSCF-based damage identification method is applied to the curvature mode shapes for damage localization. The damage indices using the modal strain energy from the first and third longitudinal bending modes are shown in Plate VII (in the color section between pages 172 and 173). The results show the same indication of damaged elements in the first and second stages as the results from experimental test data (see Plates IIIa and IIIb). In the third stage, the FE results indicate damaged elements at the two sides of the damaged core, which corresponds to the saw cut in the third damage stage.

4.7.3 Step 3: Damage quantification

Assuming that the damaged elements have the same damage severity, the DSCF method can be further applied for damage quantification using the FEA data. The damage quantification results, based on the damaged elements indicated in Step 2, is shown in Fig. 4.10. As shown in Fig. 4.10, the damage quantification results from both the FEA and experimental data show the same trend, i.e., the damage severity increase consistently with the Damage Stages 1 to 3. The damage quantification results based on the experimental data are also shown for comparison. It can be seen that in Damage Stages 1 and 2, both the experimental and numerical damage quantification results show close agreement with each other; while in Damage Stage 3 the results show considerable discrepancy due to their differences in the damaged element indication and modal eigenvalue change ratios. Based on the FEA data, the core crashing can induce over 90% stiffness reduction in the damaged elements.