**3. Damage Indices based on Image Analysis of Cracks**

The quantification of cracks in this work is based on a Fractal Analysis of Cracks (FAC) [23]. The quantification of the total length of cracks within a measurement region can be scale dependent; the smaller the scale and the more refined the crack pattern, the more likely it is that a longer total length of cracks would be measured. A typical scale-dependent example is the measurement of a coastline, which depends on the measurement scale. This method aims to quantify the number of cracks in a more objective and scale-invariant manner, rather than directly measuring the total lengths of cracks. The FAC method adopts a fractal analysis as a benchmark method to quantify a crack by estimating its fractal dimension. While mathematically, a line is one-dimensional and a filled rectangle

is two-dimensional, the dimensions of a crack distribution over a measurement region are typically a real number between 1 and 2, and do not need to be an integer. The FAC method quantifies a crack by its fractal dimension. The details of FAC can be found in [23].

The crack analysis method proposed in this paper modifies the FAC method. The main modifications made in this work include the following:


In this work, a framework for determining the damage indices by image analysis is proposed. In this framework, the damage indices include a flexural damage index *dF* and a shear damage index *dS*. The modified FAC method to determine these damage indices is composed of seven steps. All steps have been implemented in a public software implementation developed by the authors [35].


$$d = f - 1\tag{4}$$


the summation of all values in the flexural crack opening field multiplied by the width of each cell. *AS* is calculated in the same manner.

g. Calculate the flexural damage index *dF* and a shear damage index *dS* using Equations (5) and (6).

$$d\_F = d \cdot \frac{A\_F}{A\_S + A\_F} \tag{5}$$

$$d\_S = d \cdot \frac{A\_S}{A\_S + A\_F} \tag{6}$$

**Figure 6.** Demonstration of the proposed modified fractal analysis of cracks method. (**a**) crack opening; (**b**) binary crack pattern; (**c**) FAC analysis of crack; (**d**) linear regression; (**e**) flexural cracks; (**f**) shear cracks; (**g**) calculation of shear and flexural damage indices.

In most RC structures or components, crack orientation is a typical factor used to classify a crack as either flexural or shear. For RC columns or components that are subjected to bending and horizontal shear forces, horizontal cracks are typically classified as flexural, while the remaining cracks are classified as shear. This classification method is followed here. Furthermore, since the displacement field-based image analysis method provides not only the positions, opening widths, and sliding displacements of cracks, but also their orientations, it is practical to classify cracks according to their orientations. It should be noted that the classification of flexural and shear cracks by orientation is one of several classification methods, and is not necessarily applicable to all structure types. More details can be found in [10,12].

The proposed method not only integrates the previous crack image analysis [24] and FAC methods [23], but also makes some modifications. While the previous crack image analysis method requires analyzers to assign a crack orientation, the proposed method determines the crack orientation of each analyzed cell by finding the orientation that leads to the largest opening crack. While this is a conservative way to estimate crack orientation and opening width, it makes this method automatic, and does not require the orientation to be input manually. In addition, while the FAC method was originally designed for manually plotted cracks, this method uses automatically analyzed crack data for the FAC method. In the proposed method, the analyzed damage index is further separated into shear and flexural parts, providing more information on the failure mode for further safety evaluation. The integration of these methods and modifications makes it possible to carry out structural health monitoring based on crack information in practical applications.

#### **4. Experiments**

The proposed image-based shear and flexural damage indices were tested using two RC structural experiments [22]. The specimens were reduced-scale RC containment vessels (RCCVs), i.e., relatively short and wide tubular structures. They are denoted as RCCV #1 (Figure 7a) and RCCV #2 (Figure 7b), respectively. The specimens were identical in terms of geometry. The specimens were subjected to

a constant vertical force of 160 kN, and a cyclic horizontal displacement history imposed through hydraulic controlled actuators, as shown in Figure 7c. The outer and inner diameters were 2500 mm and 2200 mm, respectively. The height of the structures was 2250 mm. The concrete strengths of the two specimens were 37.0 and 43.4 MPa, respectively. The yields and ultimate strength of steel rebars were 379 MPa and 572 MPa, respectively. Four cameras were set up to take photos of the measurement regions, as shown in Figure 7d. The photos from the two northern cameras were used in this work.

**Figure 7.** Experimental configuration and photos of both RCCV #1 and RCCV #2. (**a**) Photo of RCCV #1; (**b**) Photo of RCCV #2; (**c**) Elevation; (**d**) Plan.

The two RCCVs had slightly different rebar designs. Four cylindrical layers of rebars were constructed in the concrete tubular structures. Each layer contained up to 90 rebars. The steel ratio of RCCV #1 was 0.02 with reinforcement extending into the top and bottom for strong interfaces between the roof, the specimen, and the foundations. RCCV #2 had gradually increasing vertical steel ratios ρ*<sup>v</sup>* near the top and bottom, as shown in Figure 8. The increased vertical steel reinforcement in RCCV #2 was designed to prevent sliding shear failure at the boundaries between the tubular structures and the top/bottom of the RC blocks, which occurred in the RCCV #1 test.

**Figure 8.** Steel rebar ratios in RCCV # 1 and RCCV # 2.

Four cameras were set up in both experiments; two were positioned to the north side and two to the south, as shown in Figure 7d. Two cameras were set up for each image measurement region, because stereo image analysis was employed, as described in the previous section. The measurement regions were painted with randomly striped patterns that provided image features for the displacement fields. The lightening conditions at the top and button regions of the specimens were not as good as those in the middle regions. In addition, the middle regions had better focal conditions in the experiments. Figure 9 shows the initial photos taken by the north cameras in both experiments.

**Figure 9.** Initial photos of the two RC containment vessels (RCCV) taken from the north cameras. (**a**) RCCV #1 left photo; (**b**) RCCV #1 right photo; (**c**) RCCV #2 left photo; (**d**) RCCV #2 right photo.

The experimental results show that the shear strength of the RCCV #2 was slightly higher than that of the RCCV #1 (see Figure 10a,b). The shear strengths of RCCV #1 and RCCV #2 were 5805 kN and 5580 kN, respectively. In addition, RCCV #1 and RCCV #2 had different ductilities. While both vessels reached their shear strengths for a displacement cycle of 16.9 mm (i.e., a drift ratio of 0.75% with respect to the specimen height of 2250 mm), RCCV #1 rapidly lost its shear strength after the 16.9 mm displacement cycle. In contract, RCCV #2 retained its shear capacity to 22.5 mm (i.e., a drift ratio of 1%), which was significantly higher because of the increased reinforcement at the top and bottom, as shown in Figure 10. The hysteresis loops of these specimens (see Figure 10c,d) show that the tangential stiffness did not significantly change until the cyclic displacements reached +/−3 mm. Details of the experimental results and explanations can be found in [22].

**Figure 10.** Shear/drift histories and hysteresis of RC containment vessels (RCCV) RCCV#1 and RCCV #2. (**a**) Shear and displacement history of RCCV #1; (**b**) Shear and displacement history of RCCV #2; (**c**) Hysteresis of RCCV #1; (**d**) Hysteresis of RCCV #2.

There were 163 and 1399 pairs of photos taken by the north cameras in the RCCV #1 and RCCV #2 experiments, respectively. Each pair of photos included a photo taken by the left camera and a photo taken by the right camera. The cameras were Canon EOS 5D Mark III with photo resolution of 3840 × 5760 pixels. Measurement regions were illuminated using a 100 W light-emitting-diode (LED). Figure 11 shows several north left camera photos of RCCV #1 and RCCV #2. The u in Figure 11 is the horizontal displacement at the top of the specimen. The displacements are so minor that the deformations are difficult to visually recognize in the figure. Since the RCCVs are shear-critical structures, a small displacement can cause significant shear failure. In addition to the experimental facilities and measurement devices, such as the load cells, the major way that we could observe the damage and the failure of the structure was to inspect the cracks on the surface. Diagonal (45-degree) shear cracks appeared on the north and south sides of the specimens, while the horizontal flexural cracks appeared at the top and bottom on the east and west sides. These cracks could be observed by human eyes only when we paused the testing, allowing people to get closer to the specimen to inspect the cracks. Details of the comparison of the manually plotted cracks and image analyzed cracks can be found in [24].

**Figure 11.** Selected experimental photos of RC containment vessels (RCCV) RCCV #1 and RCCV #2. (**a**) RCCV #1; (**b**) RCCV #2.

While both specimens underwent shear failures, different shear failure modes were observed for each vessel. RCCV #1 had a sliding shear mode at the top of the specimen, as shown in Figure 12a. A horizontal crack occurred at the top, where the shear stiffness dramatically changes, typically inducing a stress concentration. The red lines in Figure 12 represent the locations of the cracks. Sliding shear did not occur in RCCV #2 due to the gradual change in rebar density (the steel ratio was from 2% to 4%). RCCV #2 had a web shear failure in which the major shear crack passed through the specimen at a diagonal (45-degree) angle, as shown in Figure 12b.

**Figure 12.** Failure modes of RC containment vessel (RCCV) RCCV#1 and RCCV #2. (**a**) Sliding shear failure of RCCV #1; (**b**) Web shear failure of RCCV #2.

The crack patterns of the experimental photos, as shown in Figure 11, can be obtained by displacement field-based crack analysis. By using the displacement-based analysis, cracks as thin as 0.03 mm (approximately 0.06 pixels wide in the photos) that appeared at the very beginning of the failure could be detected. The crack patterns of the selected displacement peaks are shown in Figure 13. The crack patterns were analyzed and presented in a field discretized with a grid containing 90 × 60 cells. The size of each cell is equivalent to a sub-image with 27 × 27 pixels. In both cases, from the beginning of the tests, the cracks were distributed over almost the entire measurement region. The widths of the cracks then gradually increased from 0.03 mm (for the 2.3-mm displacement cycle) to up to 0.4 mm (for the 11.3-mm displacement cycle).

**Figure 13.** Displacement field-based crack analysis of RC containment vessel (RCCV) RCCV#1 and RCCV #2. (**a**) RCCV #1; (**b**) RCCV #2.

The proposed crack-based damage indices are calculated on the basis of the crack pattern obtained by the displacement field-based analysis (see Figure 14). In both experiments, the shear damage increased from 0 to approximately 0.75 for the displacement cycle of 8.4 mm (i.e., drift ratio of 0.375%), and did not significantly increase after that. The shear damage indices present a warning index that is capable of capturing the early stages of shear failure. Since both RCCVs #1 and #2 are shear critical, the cracks were mostly either at 45 degrees or 135 degrees, or typical shear cracks, with relatively fewer horizontal cracks observed in the measurement regions.

**Figure 14.** Image based damage index analysis of RC containment vessel (RCCV) RCCV#1 and RCCV #2. (**a**) RCCV #1; (**b**) RCCV #2.

This work examined the linear regression plots of several selected actuator control steps when analyzing the fractal dimension (as shown in Figure 6d). The plots showed that these points were very close to the line, and that the residual values were small. A selected plot of the linear regression of each specimen is shown in Figure 15. The crack pattern is a grid containing 90 × 60 cells, and is converted to different refinement of meshes with ε of 1, 2, 4, 8, 16, 32, 64, and 128 (while the most refined one is slightly more refined than the crack pattern), seven points were calculated in each of the fractal analyses.

**Figure 15.** Selected fractal analysis plots of RC containment vessel (RCCV) RCCV#1 and RCCV #2. (**a**) RCCV #1; (**b**) RCCV #2.

The computing speed of the proposed method is great enough for static structural health monitoring, but still not sufficient for non-stop real-time dynamic analysis. For each step of the analysis, including image rectification, displacement field analysis, crack opening and orientation analysis, fractal analysis of cracks, and damage indices calculation, it takes about 40 seconds of computing time using a laptop equipped with an Intel i5-7300HQ 2.5 GHz processor and 32GB main memory. Sufficient computing speed may allow us to carry out automatic, non-stop crack detection and health monitoring with a sampling rate of 0.025 Hz, that is, once or twice per minute. It is still insufficient for detecting dynamic responses during a vibration event such as an earthquake, which typically requires a sampling rate of 200 Hz to 1000 Hz. To achieve non-stop dynamic analysis for structural health monitoring, this method requires not only a significant improvement in camera and computing hardware, but also further optimization of the algorithms and programming code.

#### **5. Conclusions**

This work proposed a damage indexing method based on crack image analysis, with the aim of indicating the early stage failure of shear critical RC structures. This method is based on a displacement field-based crack image analysis method, which is capable of detecting early stage, thin cracks on concrete surfaces. It is especially practical when displacement sensors and load cells are not applicable in real structures. Early stage, thin cracks can be detected when they are as thin as 0.03 mm, which is considerably thinner than the width of a pixel in a digital photo, and cannot be visually seen as a dark line. Based on the crack image analysis, a previously proposed fractal analysis of cracks was employed to estimate the overall damage index. According to the crack orientations, this method separates the fractal analysis damage index into a shear damage index and a flexural damage index to distinguish between the different types of failure. The software implementation method is publicly available.

The results of two RCCV experiments were used to verify the proposed damage indexing method. Since both RCCV specimens were shear critical structures, the analyzed damage indices showed that the shear cracks dominated the major failure. The flexural crack indices were relatively low throughout the experiments. In both experiments, the shear damage indices reached a relatively high value (i.e., 0.7) at a displacement of only 8.4 mm on the top of the specimen (i.e., a drift ratio of 0.375%). Earlier damage could be detected when the displacement was only 3.4 mm (i.e., a drift ratio of 0.15%) or even earlier, while the stiffness was still unchanged. This indicates that the crack image analysis-based damage indexing method is capable of indicating early stage failure in shear critical structures.

While this method estimates the damage indices of a structure, damage indices obtained from different types of structures are not comparable. The safety of a structure depends on many factors, including complicated design details such as the design of ties and stirrups, which are not visually observable. A non-ductile structure having a lower damage index does not mean it is safer than a ductile structure with a higher damage index. The practical health monitoring application of this method to other structures still requires sufficient experiments and investigations based upon the specific structure type.

**Author Contributions:** Data curation, C.-H.C.; Funding acquisition, C.-l.W.; Investigation, Y.-S.Y. & C.-l.W.; Methodology, Y.-S.Y. & C.-H.C.; Writing—original draft, Y.-S.Y.; Writing—review & editing, Y.-S.Y.

**Funding:** This work is partially funded by the Ministry of Science and Technology [MOST 107-2625-M-027-003 and MOST 108-2625-027-001] and the National Center for Research on Earthquakes in Taiwan.

**Acknowledgments:** The authors would like to acknowledge the National Center for Research on Earthquake Engineering in Taiwan for providing partial measurement resources and data of the shake table tests presented in this paper. We would like to thank Uni-edit (www.uni-edit.net) for editing and proofreading this manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


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