# Damage Identification in Cement-Based Structures: A Method Based on Modal Curvatures and Continuous Wavelet Transform

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- Evaluating the changes in terms of modal parameters of scaled concrete beams subjected to loading tests leading to cracking phenomena.
- Analyzing the modal curvatures also through continuous wavelet transform.
- Proposing damage indices considering both the curvature change and the CWT-based analysis.
- Evaluating the sensitivity of the results with respect to data processing parameters.

## 2. Materials and Methods

- -
- Mixing of sand and intermediate/coarse gravels (2 min).
- -
- Addition of cement and further mixing (2 min).
- -
- Addition of BCH and further mixing (7 min).
- -
- Addition of RCF and further mixing (2 min).
- -
- Water addition and further mixing (10 min).
- -
- Pouring of fresh mix in moulds.

- -
- n. 6 sensorized specimens: sensors for the measurement of electrical impedance and free corrosion potential were embedded in the specimens for SHM purposes (both beyond the scope of this article, but very important to continuously monitor the health status of the material). Plastic tubes were employed for easing the cable routing; they require particular attention since they inevitably contribute to the determination of the element dynamic behaviour. The layout of the specimens is reported in Figure 1.
- -
- n. 6 non-sensorized specimens: these were manufactured to evaluate the effect of the embedded sensors on the dynamic behaviour of the elements (rigidity should be affected by sensors, representing discontinuities in the material) and the consequent modal parameters. Half of them were dedicated to the assessment of flexural strength according to the EN 12390-5 standard [57]; the obtained value was relevant for the design of the loading tests to be performed on the concrete beams.

- 90% of the fracture load assessed on dedicated specimens (t1).
- Fracture load (i.e., the load at which the first crack forms), specific for the specimen under test (t2).
- The load at which the crack aperture is approximately 1 mm (t3).

#### 2.1. Modal Analysis, Modal Curvatures Computation, and Damage Indices Definition

- ${S}_{fv}\left(\omega \right)$ is the cross-spectrum between the vibration acceleration (i.e., accelerometer signal) and the force (i.e., load cell signal).
- ${S}_{ff}\left(\omega \right)$ is the auto-spectrum of the input force.

_{1}function in correspondence with the structure resonances. The Polymax algorithm was used for this aim; it estimates the modal parameters in the frequency domain depending on the interpolation of FRFs through fractional polynomial functions.

_{n}) and loss factor (η) were computed, as reported in (2) and (3), respectively.

- ${f}_{n}^{t0}$ is the natural frequency at t0 (intact specimen).
- ${f}_{n}^{td}$ is the natural frequency at different test times (damaged specimen, i.e., t1, t2, and t3).

- ${\eta}^{t0}$ is the loss factor at t0 (intact specimen).
- ${\eta}^{td}$ is the loss factor at different test times (damaged specimen, i.e., t1, t2, and t3).

- $\left|{\phi}^{\u2033}\right|$ is the modulus of the modal curvature (${\phi}^{\u2033}$)
- $\angle {\phi}^{\u2033}$ is the phase of the modal curvature (${\phi}^{\u2033}$)

^{®}(R2023b) was employed; it uses the Morse wavelet. The symmetry (γ) and the time-bandwidth product parameters were set at 2 and 2.5, respectively, empirically selecting them to highlight the curvature formation in the mode shape. The absolute value obtained from this process can be plotted in function of time and frequency, obtaining the so-called scalogram. Then, the image was binarized using the global threshold method; hence, a binary image was created exploiting Otsu’s method [59], which creates a global threshold minimizing the intraclass variance of black/white pixels subjected to the threshold. Then, the area of the high-valued pixels was computed, and the values obtained at the different test times were compared.

_{curv}, as reported in (5), is the sum of the absolute differences in modal curvature between each test time related to the damaged specimen with respect to the intact specimen (assessed at t0), where each difference is normalized with respect to the absolute value of the curvature maximum value at t0.

- ${{\phi}^{\u2033}}_{tx}$ is the modal curvature computed at the t
_{x}test time (i.e., t1, t2, and t3). - ${{\phi}^{\u2033}}_{t0}$ is the modal curvature computed at t0 (intact specimen).

_{CWT}, was defined based on CWT-based analysis. At first, the image obtained from CWT computation was binarized according to an automated threshold defined according to Otsu’s method [59]. Then, the area of the high CWT coefficients in the binarized image was computed and the value was normalized with respect to the area obtained for t0, as reported in (6).

_{curv}is expected to increase with damage, since changes in modal curvature will be more significant, DI

_{CWT}should decrease (i.e., smaller high pixel area).

_{global}, was defined, by combining the previous ones, as reported in (7).

#### 2.2. Sensitivity Analysis to Data Processing Parameters

- Interpolation smoothing factor.
- Oversampling factor.

## 3. Results and Discussion

#### 3.1. Modal Parameters

#### 3.2. Modal Curvatures and CWT-Based Analysis

#### 3.3. Damage Related Indices

_{curv}, DI

_{CWT}, and DI

_{global}(see Section 3 for details). The results obtained on all the tested specimens for each test time are reported in Table 7 and are summarized in Figure 17; it can be observed that the values obtained for t2 and t3 test times fall in compatible measurement ranges, meaning that the two conditions cannot be distinguished. This is probably linked to the fact that the vibrational analyses were performed after the load removal, letting the crack partially close. On the contrary, there is a significant variation between the indices obtained at t1 (i.e., 90% of fracture load) and the values after damage occurrence (i.e., at t2 and t3). The first crack formation (i.e., t2) can be promptly detected, signalling an alteration of the dynamic behaviour of the element; this proves that the proposed assessment strategy is effective and adequately sensitive to the occurrence of damage. Hence, the results confirm the possibility of discriminating between intact (i.e., t1) and damaged (i.e., t2 and t3) conditions, but further investigations are needed to distinguish among different levels of damage (e.g., between t2—crack formation—and t3—crack aperture of 1 mm—conditions).

#### 3.4. Sensitivity Analysis

_{curv}, with an almost constant sensitivity, varying in the range of 2.2–2.6 depending on the test time, as it can be deduced from Table 8); when it is halved (passing from 0.4 to 0.2), DI

_{curv}decreases of approximately 26%, DI

_{CWT}increases of about 16%, and DI

_{global}decreases of about 36% at t3 test time. The DI

_{CWT}index trend is no longer monotonically decreasing as expected since the smoothing operation is too aggressive and the cuspid is no more identifiable. On the other hand, considering the softest smoothing operation (i.e., smoothing factor equal to 0.9), it is possible to notice that this is not sufficient to obtain a good quality baseline curvature at t0 (Figure 18). The damage is no longer identifiable through none of the indices (their trends are opposed compared to the expected ones)—even if considering only t3 test time the cuspid would be more evident (but at the expense of signal quality, especially at t0). Therefore, a smoothing factor equal to 0.40 can be considered adequate for our purposes (a comparison can be observed in Figure 19). Concerning the oversampling factor, its effect is slightly on DI

_{CWT}and, hence, DI

_{global}indices. No significant changes can be evidenced, hence a ×5 oversampling factor can be considered a good compromise to have a good quality signal and limited computational load at the same time.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Sequence of measurements with 5 accelerometers; 4 measurements are necessary to cover the whole specimen.

**Figure 7.**Sum FRFs obtained on the different specimens at t0 (intact beams). The three blue bands indicate the three considered mode shapes, namely rigid-body mode shape (

**left**), I mode shape (

**centre**), and II mode shape (

**right**). The circular markers are related to their natural vibration frequencies.

**Figure 8.**Natural vibration frequencies are reported as mean ± standard deviation at each test time for all the specimens.

**Figure 9.**Example of I mode shape with cuspid formation due to crack (t2 and t3 test times)—specimen A.

**Figure 11.**Loss factor values are reported as mean ± standard deviation at each test time for all the specimens.

**Figure 12.**MAC values are reported as mean ± standard deviation at t1, t2, and t3 test times with respect to t0 for all the specimens.

**Figure 13.**Modal curvature computation for I mode shape: mode shape (raw and smoothed) (

**top**), and modal curvature (

**bottom**)—(Specimen A, test time: t3).

**Figure 14.**Comparison of I modal curvature computed at each test time (i.e., t0, t1, t2, and t3—Specimen A)—2000 points obtained through interpolated curvatures.

**Figure 15.**Comparison of the CWTs scalograms of I modal curvature computed at each test time (i.e., t0, t1, t2, and t3—Specimen A).

**Figure 16.**Comparison of the binarized CWTs scalograms of I modal curvature computed at each test time (i.e., t0, t1, t2, and t3—Specimen A).

**Figure 17.**Damage-related indices were reported as mean ± standard deviation at each test time. Note: The left vertical axis refers to DI

_{curv}and DI

_{global}, whereas the right vertical axis is related to DI

_{CWT}for all the specimens.

**Figure 18.**Modal curvature computation for I mode shape: mode shape (raw and smoothed) (

**top**), and modal curvature (

**bottom**)—(Specimen A, test time: t0), smoothing factor: 0.9.

**Figure 19.**Modal curvature computation for I mode shape: mode shape (raw and smoothed) (

**top**), and modal curvature (

**bottom**)—(Specimen A, test time: t7), comparison among different smoothing factors.

Cement [kg/m^{3}] | Water [kg/m^{3}] | Air [%] | Sand [kg/m^{3}] | Intermediate Gravel [kg/m^{3}] | Coarse Gravel [kg/m^{3}] | RCF [kg/m^{3}] | BCH [kg/m^{3}] |
---|---|---|---|---|---|---|---|

470.0 | 235.0 | 2.5 | 795.0 | 321.0 | 476.0 | 0.9 | 10.0 |

Variable | Values |
---|---|

Smoothing factor | 0.20 |

0.40 | |

0.90 | |

Oversampling factor | ×2 |

×5 | |

×10 |

Specimen | Fracture Load [kN] |
---|---|

A | 13.1 |

B | 13.1 |

C | 15.2 |

G | 13.8 |

H | 13.5 |

I | 12.3 |

**Table 4.**Natural vibration frequency values for each tested specimen (A, B, C, G, H, and I) at each test time (t0, t1, t2, and t3) and related percentage variations with respect to undamaged condition (t0).

Specimen | Test Time | f_{n} (Δf_{n} [%]) [Hz] | ||
---|---|---|---|---|

Mode Shape | ||||

Rigid-Body | I | II | ||

A | t0 | 341 (-) | 1487 (-) | 3371 (-) |

t1 | 237 (−30.44) | 1488 (−0.07) | 3383 (−0.35) | |

t2 | 218 (−36.12) | 796 (−46.47) | 3265 (−3.14) | |

t3 | 210 (−38.48) | 715 (−51.92) | 2888 (−14.34) | |

B | t0 | 329 (-) | 1446 (-) | 3273 (-) |

t1 | 294 (−10.64) | 1442 (−0.34) | 3295 (−0.67) | |

t2 | 249 (−24.20) | 748 (−48.30) | 3172 (−3.09) | |

t3 | 258 (−21.64) | 481 (−66.73) | 2871 (−12.28) | |

C | t0 | 343 (-) | 1467 (-) | 3315 (-) |

t1 | 285 (−16.91) | 1462 (−0.33) | 3331 (−0.48) | |

t2 | 203 (−40.82) | 794 (−45.88) | 3170 (−4.37) | |

t3 | 259 (−24.52) | 597 (−59.33) | 2884 (−13.00) | |

G | t0 | 289 (-) | 1519 (-) | 3338 (-) |

t1 | 235 (−18.69) | 1487 (−2.10) | 3330 (−0.26) | |

t2 | 199 (−31.04) | 767 (−49.50) | 2687 (−19.52) | |

t3 | 212 (−26.48) | 870 (−42.73) | 2861 (−14.31) | |

H | t0 | 303 (-) | 1441 (-) | 3280 (-) |

t1 | 159 (−47.47) | 1425 (−1.10) | 3319 (−1.18) | |

t2 | 160 (−47.19) | 951 (−33.98) | 3127 (−4.66) | |

t3 | 181 (−40.36) | 786 (−45.41) | 2942 (−10.31) | |

I | t0 | 300 (-) | 1403 (-) | 3224 (-) |

t1 | 215 (−28.52) | 1412 (−0.63) | 3173 (−1.58) | |

t2 | 206 (−31.41) | 1002 (−28.59) | 3142 (−2.54) | |

t3 | 175 (−41.79) | 708 (−49.55) | 2829 (−12.25) |

**Table 5.**Loss factor (η) values for each tested specimen (A, B, C, G, H, and I) at each test time (t0, t1, t2, and t3) and related percentage variations (Δη) with respect to undamaged condition (t0).

Specimen | Test Time | η (Δη [%]) [%] | ||
---|---|---|---|---|

Mode Shape | ||||

Rigid-Body | I | II | ||

A | t0 | 10.76 * (-) | 1.70 (-) | 0.69 (-) |

t1 | 2.48 (−76.99 *) | 1.15 (−32.35 *) | 0.86 (24.35) | |

t2 | 3.99 (−62.96 *) | 3.69 (117.09) | 1.17 (69.42) | |

t3 | 12.35 (14.80) | 3.98 (134.12) | 2.23 (223.37) | |

B | t0 | 9.45 (-) | 1.98 (-) | 0.70 (-) |

t1 | 10.31 (9.05) | 2.67 (35.35) | 0.95 (34.86) | |

t2 | 17.45 (84.57) | 9.23 (367.19) | 1.42 (102.19) | |

t3 | 20.85 (120.52) | 11.44 (479.27) | 2.25 (219.69) | |

C | t0 | 5.56 (-) | 2.03 (-) | 0.91 (-) |

t1 | 6.15 (10.55) | 2.27 (11.97) | 1.41 (54.93) | |

t2 | 10.40 (87.05) | 6.30 (210.58) | 2.54 (179.82) | |

t3 | 11.36 (104.25) | 9.14 (350.61) | 4.11 (353.01) | |

G | t0 | 9.34 (-) | 1.59 (-) | 0.78 (-) |

t1 | 9.34 (0.00) | 2.00 (26.14) | 0.78 (0.00) | |

t2 | 14.75 (57.96) | 7.76 (389.43) | 2.22 (186.60) | |

t3 | 19.12 (104.79) | 9.56 (502.87) | 3.05 (293.02) | |

H | t0 | 1.16 * (-) | 1.46 * (-) | 1.01 (-) |

t1 | 0.22 (−81.45 *) | 0.84 (−42.12 *) | 1.55 (52.82) | |

t2 | 0.85 (−26.72) | 1.73 (18.82) | 2.67 (163.97) | |

t3 | 2.38 (105.33) | 3.11 (113.36) | 2.30 (127.47) | |

I | t0 | 9.87 (-) | 1.79 (-) | 1.17 (-) |

t1 | 10.31 (4.51) | 2.68 (49.76) | 1.99 (70.08) | |

t2 | 15.23 (54.29) | 3.45 (92.79) | 2.05 (75.21) | |

t3 | 16.56 (67.78) | 7.90 (341.55) | 2.95 (152.13) |

**Table 6.**MAC values for each tested specimen (A, B, C, G, H and I) considering different test times (i.e., t1, t2, and t3) with respect to t0 (i.e., undamaged conditions).

Specimen | Test Time | MAC [%] | ||
---|---|---|---|---|

Mode Shape | ||||

Rigid-Body | I | II | ||

A | t0 | 93.81 | 90.97 | 29.19 |

t1 | 93.14 | 78.19 | 59.84 | |

t2 | 87.50 | 72.85 | 57.59 | |

t3 | 98.63 | 98.63 | 89.64 | |

B | t0 | 94.44 | 64.21 | 84.20 |

t1 | 85.31 | 66.86 | 89.10 | |

t2 | 97.47 | 99.67 | 82.71 | |

t3 | 93.60 | 50.40 | 86.58 | |

C | t0 | 92.14 | 60.88 | 82.65 |

t1 | 97.13 | 99.31 | 84.85 | |

t2 | 84.74 | 64.44 | 34.35 | |

t3 | 94.96 | 74.40 | 83.07 | |

G | t0 | 46.34 | 99.36 | 81.35 |

t1 | 50.94 | 80.74 | 72.58 | |

t2 | 72.75 | 64.52 | 75.62 | |

t3 | 94.78 | 99.41 | 57.18 | |

H | t0 | 97.81 | 76.38 | 75.40 |

t1 | 95.61 | 65.20 | 66.77 | |

t2 | 93.81 | 90.97 | 29.19 | |

t3 | 93.14 | 78.19 | 59.84 | |

I | t0 | 87.50 | 72.85 | 57.59 |

t1 | 98.63 | 98.63 | 89.64 | |

t2 | 94.44 | 64.21 | 84.20 | |

t3 | 85.31 | 66.86 | 89.10 |

**Table 7.**Damage-related indices for each tested specimen (A, B, C, G, H, and I) at each test time (t1, t2, and t3).

Specimen | Test Time | Damage Indices | ||
---|---|---|---|---|

DI_{curv} | DI_{CWT} | DI_{global} | ||

A | t1 | 4.98 | 1.01 | 4.94 |

t2 | 8.74 | 0.83 | 10.53 | |

t3 | 8.66 | 0.82 | 10.59 | |

B | t1 | 3.93 | 1.17 | 3.36 |

t2 | 7.49 | 0.89 | 8.44 | |

t3 | 7.82 | 0.90 | 8.69 | |

C | t1 | 2.61 | 0.96 | 2.73 |

t2 | 8.35 | 0.81 | 10.33 | |

t3 | 8.90 | 1.00 * | 8.92 * | |

G | t1 | 2.13 | 0.87 | 2.44 |

t2 | 6.03 * | 0.82 | 7.37 * | |

t3 | 5.03 | 0.85 * | 5.93 | |

H | t1 | 3.61 | 1.08 | 3.36 |

t2 | 7.07 | 0.92 | 7.65 | |

t3 | 9.13 | 0.90 | 10.16 | |

I | t1 | 3.64 | 1.17 | 3.10 |

t2 | 6.57 | 0.86 * | 7.65 * | |

t3 | 7.49 | 0.89 | 8.44 |

Variable | Values | Test Time | Damage Indices | ||
---|---|---|---|---|---|

DI_{curv} | DI_{CWT} | DI_{global} | |||

Smoothing factor (with oversampling factor ×5) | 0.20 | t1 | 2.88 | 1.01 | 2.85 |

t2 | 6.82 | 0.94 | 7.29 | ||

t3 | 6.39 | 0.95 | 6.75 | ||

0.40 | t1 | 4.98 | 1.01 | 4.94 | |

t2 | 8.74 | 0.83 | 10.53 | ||

t3 | 8.66 | 0.82 | 10.59 | ||

0.90 | t1 | 8.09 | 0.61 | 13.22 | |

t2 | 11.26 | 1.01 | 11.17 | ||

t3 | 11.36 | 1.28 | 8.68 | ||

Oversampling factor (with smoothing factor 0.4) | ×2 | t1 | 4.98 | 1.02 | 4.85 |

t2 | 8.74 | 0.85 | 10.34 | ||

t3 | 8.66 | 0.81 | 10.69 | ||

×5 | t1 | 4.98 | 1.01 | 4.94 | |

t2 | 8.74 | 0.83 | 10.53 | ||

t3 | 8.66 | 0.82 | 10.59 | ||

×10 | t1 | 4.98 | 1.01 | 4.92 | |

t2 | 8.74 | 0.84 | 10.44 | ||

t3 | 8.66 | 0.82 | 10.51 |

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## Share and Cite

**MDPI and ACS Style**

Cosoli, G.; Martarelli, M.; Mobili, A.; Tittarelli, F.; Revel, G.M.
Damage Identification in Cement-Based Structures: A Method Based on Modal Curvatures and Continuous Wavelet Transform. *Sensors* **2023**, *23*, 9292.
https://doi.org/10.3390/s23229292

**AMA Style**

Cosoli G, Martarelli M, Mobili A, Tittarelli F, Revel GM.
Damage Identification in Cement-Based Structures: A Method Based on Modal Curvatures and Continuous Wavelet Transform. *Sensors*. 2023; 23(22):9292.
https://doi.org/10.3390/s23229292

**Chicago/Turabian Style**

Cosoli, Gloria, Milena Martarelli, Alessandra Mobili, Francesca Tittarelli, and Gian Marco Revel.
2023. "Damage Identification in Cement-Based Structures: A Method Based on Modal Curvatures and Continuous Wavelet Transform" *Sensors* 23, no. 22: 9292.
https://doi.org/10.3390/s23229292