# An Experimental Study of the Feasibility of Identifying the Impact Damages of Reinforced Concrete Piers Using a Modal Frequency Method

Authors: Xiwu Zhou; Wenchao Zhang; Yushen Gao; Guoxue Zhang; Mengdan Wen
Journal: Advances in Civil Engineering
 (2020)

---

## Abstract

In this research study, horizontal impact tests were carried out on five reduced scale pier models using China’s most advanced multifunctional ultrahigh heavy drop hammer impact test system and DHDAS dynamic signal acquisition and analysis system. Due to the fact that the traditional measurement method can only be used for local measurement damage, and the volatility is high, this paper proposes a test method for the modal frequency identification of the overall damage of reinforced concrete pier and applies the ultrasonic damage measurement method to verify the results. The tests analyzed the modal frequencies and ultrasonic velocity identifications for the purpose of evaluating the impact damages of bridge piers, as well as the relationship between them. The results showed that the modal frequencies were consistent with the ultrasonic waves in identifying and evaluating the damages to the piers. Also, the modal frequency damage factors were determined to be functions of the ultrasonic wave velocity damage factors. Therefore, the results of this study confirmed that it was feasible to characterize the impact damages of piers using a modal frequency method.


---

## Body

## 1. Introduction

Pier structures may become damaged, cracked, or aged to varying degrees for a variety of reasons during their utilization. With China’s recent economic and infrastructure development, many types of river-crossing and sea-crossing bridges have been rapidly developed, which have resulted in ship-bridge collision accidents occurring more frequently. Therefore, in order to ensure the safety and durability of pier structures, it is of great significance to examine the impact damages of reinforced concrete piers [1–3].In recent years, researchers around the world have carried out studies regarding the identification of RC structural damages and have proposed a variety of identification methods [4–6]. These methods mainly include static, dynamic, and dynamic-static combination methods. The static method is to analyze the static test data (force and displacement) to determine the change of structural element stiffness, which can then be used for damage assessments and the determination of structural bearing capacities [7]. Dynamic identification methods mainly use vibration modes, transfer functions, curvature modes, frequency response functions, and relative acceleration to identify damage [8–10]. Dynamic-static combination methods seek to identify structural damage by using the static deflection, vibration mode, and output error method [11,12]. At present, the most commonly used damage identification methods are traditional experimental methods, such as acoustic or ultrasonic methods, electromagnetic methods, and radiography methods [13–15]. These methods require the damage areas to be known beforehand, and it must be easy for the damage detection sensor to reach the tested part. In addition, these sensors are only able to detect small areas of the structural or adjacent surfaces. Therefore, these methods have very limited application potential. It is known that different types of structures have different damage forms. In regard to reinforced concrete piers, the main damages which occur are the results of impacts with ships and vehicles [16–19]. At the present time, the research studies regarding the impact damages of reinforced concrete piers have mainly been based on traditional experimental and numerical simulation methods. For example, Zhang and Zhou et al. applied ultrasonic detectors to identify the damages of concrete piers and evaluate the damage of piers based on the change of ultrasonic wave velocity [20, 21]. Gholipour and Ren et al. used the numerical simulation method to identify the damages of concrete piers [22, 23]. Some researchers have also performed identifications of pier impact damages using modal parameters. For example, Zhan et al. proposed an impact vibration test method for measuring the natural frequencies of existing piers and carried out numerical research examinations [24, 25]. The previous studies conducted by Yuen and Pandey et al. showed that the vibration mode ratio, characteristic parameters, and curvature modal difference indexes could be directly obtained from the measured frequencies and modes. This information could then be directly used for damage identifications without requiring finite element models of the target structures. Therefore, this method possessed the characteristics of a model-free indexing method [26, 27]. You-Sheng completed a comparative analysis of several calculation methods for the natural frequencies of pile foundation piers and pointed out their respective advantages and disadvantages [28]. However, these studies mainly use impact vibration test methods and numerical research methods.Currently, the most commonly accepted and promising damage identification method is a cross-disciplinary experimental modal analysis method. This method combines system identification, vibration theory, vibration testing technology, signal acquisition, and analysis. It has been found that the results of identifications of structural damages using modal frequencies to assess the overall damages of structures and study the overall damage information of the structures have shown high antinoise and identification accuracy abilities. Also, these types of methods have been observed to have convenient measurement adaptabilities [29–31]. Furthermore, this method has been widely used in aviation, aerospace, precision machine tools, and other fields in developed countries, for fault diagnosis, load identification, and dynamic modification. However, the applications of this method for pier impact damage detections have seldom been reported. Therefore, this study proposed to use a modal frequency method to identify the impact damages of reinforced concrete piers. A multifunctional ultrahigh heavy drop hammer impact test system and a DHDAS dynamic signal acquisition and analysis system were utilized in this study to carry out experimental horizontal impact tests on five reduced scale pier model specimens. The feasibility and accuracy of identification in the evaluations of the pier impact damages using modal frequencies were examined and analyzed, and a traditional ultrasonic velocity method was used for verification purposes.


## 2. Experimental Impact Tests

### 2.1. Test Equipment

#### 2.1.1. Horizontal Impact Equipment

This study’s impact tests adopted a multifunctional ultrahigh heavy drop hammer testing machine system, which consisted of a vertical drop hammer driving system and a horizontal impact system. The experimental equipment was able to obtain vertical and horizontal two-way impact data, as shown in Figure1(a). The kinetic energy of the horizontal impact test cart (hereinafter referred to as the cart, as detailed in Figure 1(b)) was provided by the vertical drop hammer driving system (Figure 1(c)). The dimensions of the car were L × W × H = 2.08 m × 1.15 m × 0.64 m. The front end of the cart was equipped with a rigid impact hammer measuring L × B = 590 mm × 220 mm. There were four specially designed steel cylinder installation pressure sensors located between the hammer and the cart. At the end of the track, a laser velocity measurement system was used to measure the instantaneous speeds of the impacts, as shown in Figure 1(d).Figure 1
(a) Multifunctional ultrahigh heavy drop hammer testing machine system, (b) horizontal traction impact testing cart, (c) vertical drop hammer device, and (d) laser velocimetry.
(a)
(b)
(c)
(d)The impact velocity (kinetic energy) of the cart was related to the lifting height and mass of the drop hammer, as well as the mass of the cart. Among these, the maximum lifting height of the hammer was 18 m; the mass of the hammer was 166 kg; and the mass of the cart wasm1 = 1,200 kg. The mass totals of the drop hammer and cart could be adjusted using weight blocks.


#### 2.1.2. Modal Frequency Acquisition Equipment

During the modal analysis, an example of the cohesive hammer used is shown in Figure2(a). A DHDAS (Dong-Hua Test Real Time Data Measurement and Analysis Software System) dynamic signal acquisition and analysis system (Figure 2(b)) was adopted for the acquisition of the excitation signals, and the pier response signals were measured using an acceleration sensor (Figure 2(c)). Then, a spectral analysis was conducted on the measured signals in order to obtain the transfer functions, implement the fitting of the transfer functions, and determine the natural frequencies of the piers in combination with the vibration mode. The whole test system is shown in Figure 3.Figure 2
(a) Cohesive hammer; (b) DHDAS dynamic signal acquisition and analysis system; (c) acceleration sensor.
(a)
(b)
(c)Figure 3
Modal test system.


#### 2.1.3. Ultrasonic Wave Velocity Acquisition Equipment

The concrete damage of the main damaged area of the test piece was detected by a ZBL-U520 nonmetal ultrasonic detector (as shown in Figure4). In the test, the measuring points in the high damage area such as the impact point area and the bottom area of the test piece were examined.Figure 4
(a) ZBL-U520 host; (b) ultrasonic transmitter.
(a)
(b)


### 2.2. Test Specimens

Five reinforced concrete circular pier specimens were designed for this study’s experimental tests. The column section radius of each specimen was 170 mm, and the height of each specimen was 2,200 mm. The selected stirrup was I-grade ordinary steel bar HPB300 with a diameter of 8 mm, which adopted a welding construction. The model is shown in Figure5. The longitudinal reinforcements were HRB400 reinforcements. The pier model numbers and design parameters are shown in Table 1. The longitudinal reinforcements and distributions of the specimens are detailed in Figure 6.Figure 5
Schematic diagram of the specimens’ reinforcements.Table 1
Test specimen numbers and design parameters.
Specimen no.
Longitudinal reinforcement (mm)
Stirrup (mm)
Concrete strength grade
Z16-1
10Φ16
Φ8@50
C40
Z16-2
10Φ16
Φ8@50
C40
Z16-3
10Φ16
Φ8@50
C40
Z20-1
10Φ20
Φ8@50
C40
Z20-2
10Φ20
Φ8@50
C40Figure 6
Longitudinal reinforcements and distributions of the specimens.Reinforcement samples were taken from the specimens for the testing process. Also, loading-tension tests were carried out on an electrohydraulic servo universal testing machine, which obtained the yield strengths and ultimate strengths of the steel bars. At the same time, six test cubes (150 mm × 150 mm × 150 mm) were constructed according to requirements of the experiments, and compression tests were carried out on the specimens. The basic parameters of the steel bars and concrete are shown in Table2.Table 2
Basic parameters of the steel bars and concrete.
Steel diameter
Elastic modulus of reinforcement (N/mm2)
Yield strength (N/mm2)
Ultimate strength (N/mm2)
Elastic modulus of concrete (N/mm2)
Concrete strength (N/mm2)
Φ16
2.0 × 105
440
580
3.28 × 104
45.63
Φ20
486
640


### 2.3. Test Design

Horizontal impact tests were carried out on the five reinforced concrete column specimens. Each column was planned to be impacted four times, in which the Z16-1 and Z20-1 specimens followed Scheme 1, while the Z16-2, Z16-3, and Z20-2 specimens adopted Scheme 2. The scheme details are shown in Tables3 and 4. Then, based on the principle of the conservation of energy, the theoretical speeds of the impacts were calculated as follows:(1)m2gh=12m1+m2v12+μm1gh,where m1 and m2 denote the masses of the cart and drop hammer, respectively; g=9.81m/s2 is the gravitational acceleration; h represents the drop height of the hammer before release; v1 is the instantaneous velocity of the test cart before impact; and μ denotes the friction coefficient between the rail and the test cart, which was calibrated and determined by the preliminary impact test results, and for the different m1 and h. The friction coefficient was approximately equaled to 0.12. A laser velocity measurement system was installed at the end of the track for the purpose of measuring the instantaneous velocities of the impacts. As detailed in Tables 3 and 4, the measured velocities were approximately equal to the theoretical velocities obtained using formula (1) (the rate of the velocity variation was ±5%), which verified the reliability and accuracy of the impact test system.Table 3
Impact scheme 1.
Impact times
Drop height (m)
Car quality (kg)
Axial pressure (kN)
Drop hammer quality (kg)
Friction coefficient
Theoretical speed (m/s)
Measured average velocity (m/s)
Speed change rate (%)
1
4
1200
250
166
0.12
1.124
1.114
−0.890
2
6
1200
250
166
0.12
1.376
1.335
−2.980
3
8
1200
250
166
0.12
1.589
1.563
−1.636
4
10
1200
250
166
0.12
1.777
1.770
−0.56Table 4
Impact scheme 2.
Specimen no.
Impact times
Drop height (m)
Car quality (kg)
Axial pressure (kN)
Drop hammer quality (kg)
Friction coefficient
Theoretical speed (m/s)
Measured average velocity (m/s)
Speed change rate (%)
Z16-2
4
6
1200
250
166
0.12
1.376
1.337
−2.834
Z16-3
4
8
1200
250
166
0.12
1.589
1.513
−4.783
Z20-2
4
8
1200
250
166
0.12
1.589
1.588
−0.0629Prior to the testing, the pier specimens were fixed on rigid bases with four prestressed bolts, which were considered to be approximately equivalent to fixed end constraints. The upper ends of the columns were subjected to axial pressure of 250 kN by an oil jack mounted on the cross beam of the reaction frame, as shown in Figure7. After installing each of the reinforced concrete column specimens, the drop hammer was raised to set heights to carry out the impact tests. For all of the tests, the impact points were located 1,100 mm from the bottom of each column, i.e., at the center of each specimen.Figure 7
Axial pressure device.Seven acceleration sensors were evenly arranged on the back of each specimen during the tests (as shown in Figure8). Prior to and following each impact, a cohesive hammer was used to impact the top of the impact surface of each specimen. At the same time, a DHDAS dynamic signal acquisition and analysis system was used to collect the excitation signals and pier response signals measured by the acceleration sensors for the purpose of obtaining the natural vibration frequencies of the pier specimens and carry out damage identification processes of the concrete piers. A ZBL-U520 nonmetallic ultrasonic testing instrument was used to detect the damages to the concrete pier specimens in the main damage areas. In this way, the feasibility and accuracy of identifications and evaluations of the pier impact damages using modal frequencies were successfully verified.Figure 8
Arrangements of the restraint device and acceleration sensors.


## 2.1. Test Equipment

### 2.1.1. Horizontal Impact Equipment

This study’s impact tests adopted a multifunctional ultrahigh heavy drop hammer testing machine system, which consisted of a vertical drop hammer driving system and a horizontal impact system. The experimental equipment was able to obtain vertical and horizontal two-way impact data, as shown in Figure1(a). The kinetic energy of the horizontal impact test cart (hereinafter referred to as the cart, as detailed in Figure 1(b)) was provided by the vertical drop hammer driving system (Figure 1(c)). The dimensions of the car were L × W × H = 2.08 m × 1.15 m × 0.64 m. The front end of the cart was equipped with a rigid impact hammer measuring L × B = 590 mm × 220 mm. There were four specially designed steel cylinder installation pressure sensors located between the hammer and the cart. At the end of the track, a laser velocity measurement system was used to measure the instantaneous speeds of the impacts, as shown in Figure 1(d).Figure 1
(a) Multifunctional ultrahigh heavy drop hammer testing machine system, (b) horizontal traction impact testing cart, (c) vertical drop hammer device, and (d) laser velocimetry.
(a)
(b)
(c)
(d)The impact velocity (kinetic energy) of the cart was related to the lifting height and mass of the drop hammer, as well as the mass of the cart. Among these, the maximum lifting height of the hammer was 18 m; the mass of the hammer was 166 kg; and the mass of the cart wasm1 = 1,200 kg. The mass totals of the drop hammer and cart could be adjusted using weight blocks.


### 2.1.2. Modal Frequency Acquisition Equipment

During the modal analysis, an example of the cohesive hammer used is shown in Figure2(a). A DHDAS (Dong-Hua Test Real Time Data Measurement and Analysis Software System) dynamic signal acquisition and analysis system (Figure 2(b)) was adopted for the acquisition of the excitation signals, and the pier response signals were measured using an acceleration sensor (Figure 2(c)). Then, a spectral analysis was conducted on the measured signals in order to obtain the transfer functions, implement the fitting of the transfer functions, and determine the natural frequencies of the piers in combination with the vibration mode. The whole test system is shown in Figure 3.Figure 2
(a) Cohesive hammer; (b) DHDAS dynamic signal acquisition and analysis system; (c) acceleration sensor.
(a)
(b)
(c)Figure 3
Modal test system.


### 2.1.3. Ultrasonic Wave Velocity Acquisition Equipment

The concrete damage of the main damaged area of the test piece was detected by a ZBL-U520 nonmetal ultrasonic detector (as shown in Figure4). In the test, the measuring points in the high damage area such as the impact point area and the bottom area of the test piece were examined.Figure 4
(a) ZBL-U520 host; (b) ultrasonic transmitter.
(a)
(b)


## 2.1.1. Horizontal Impact Equipment

This study’s impact tests adopted a multifunctional ultrahigh heavy drop hammer testing machine system, which consisted of a vertical drop hammer driving system and a horizontal impact system. The experimental equipment was able to obtain vertical and horizontal two-way impact data, as shown in Figure1(a). The kinetic energy of the horizontal impact test cart (hereinafter referred to as the cart, as detailed in Figure 1(b)) was provided by the vertical drop hammer driving system (Figure 1(c)). The dimensions of the car were L × W × H = 2.08 m × 1.15 m × 0.64 m. The front end of the cart was equipped with a rigid impact hammer measuring L × B = 590 mm × 220 mm. There were four specially designed steel cylinder installation pressure sensors located between the hammer and the cart. At the end of the track, a laser velocity measurement system was used to measure the instantaneous speeds of the impacts, as shown in Figure 1(d).Figure 1
(a) Multifunctional ultrahigh heavy drop hammer testing machine system, (b) horizontal traction impact testing cart, (c) vertical drop hammer device, and (d) laser velocimetry.
(a)
(b)
(c)
(d)The impact velocity (kinetic energy) of the cart was related to the lifting height and mass of the drop hammer, as well as the mass of the cart. Among these, the maximum lifting height of the hammer was 18 m; the mass of the hammer was 166 kg; and the mass of the cart wasm1 = 1,200 kg. The mass totals of the drop hammer and cart could be adjusted using weight blocks.


## 2.1.2. Modal Frequency Acquisition Equipment

During the modal analysis, an example of the cohesive hammer used is shown in Figure2(a). A DHDAS (Dong-Hua Test Real Time Data Measurement and Analysis Software System) dynamic signal acquisition and analysis system (Figure 2(b)) was adopted for the acquisition of the excitation signals, and the pier response signals were measured using an acceleration sensor (Figure 2(c)). Then, a spectral analysis was conducted on the measured signals in order to obtain the transfer functions, implement the fitting of the transfer functions, and determine the natural frequencies of the piers in combination with the vibration mode. The whole test system is shown in Figure 3.Figure 2
(a) Cohesive hammer; (b) DHDAS dynamic signal acquisition and analysis system; (c) acceleration sensor.
(a)
(b)
(c)Figure 3
Modal test system.


## 2.1.3. Ultrasonic Wave Velocity Acquisition Equipment

The concrete damage of the main damaged area of the test piece was detected by a ZBL-U520 nonmetal ultrasonic detector (as shown in Figure4). In the test, the measuring points in the high damage area such as the impact point area and the bottom area of the test piece were examined.Figure 4
(a) ZBL-U520 host; (b) ultrasonic transmitter.
(a)
(b)


## 2.2. Test Specimens

Five reinforced concrete circular pier specimens were designed for this study’s experimental tests. The column section radius of each specimen was 170 mm, and the height of each specimen was 2,200 mm. The selected stirrup was I-grade ordinary steel bar HPB300 with a diameter of 8 mm, which adopted a welding construction. The model is shown in Figure5. The longitudinal reinforcements were HRB400 reinforcements. The pier model numbers and design parameters are shown in Table 1. The longitudinal reinforcements and distributions of the specimens are detailed in Figure 6.Figure 5
Schematic diagram of the specimens’ reinforcements.Table 1
Test specimen numbers and design parameters.
Specimen no.
Longitudinal reinforcement (mm)
Stirrup (mm)
Concrete strength grade
Z16-1
10Φ16
Φ8@50
C40
Z16-2
10Φ16
Φ8@50
C40
Z16-3
10Φ16
Φ8@50
C40
Z20-1
10Φ20
Φ8@50
C40
Z20-2
10Φ20
Φ8@50
C40Figure 6
Longitudinal reinforcements and distributions of the specimens.Reinforcement samples were taken from the specimens for the testing process. Also, loading-tension tests were carried out on an electrohydraulic servo universal testing machine, which obtained the yield strengths and ultimate strengths of the steel bars. At the same time, six test cubes (150 mm × 150 mm × 150 mm) were constructed according to requirements of the experiments, and compression tests were carried out on the specimens. The basic parameters of the steel bars and concrete are shown in Table2.Table 2
Basic parameters of the steel bars and concrete.
Steel diameter
Elastic modulus of reinforcement (N/mm2)
Yield strength (N/mm2)
Ultimate strength (N/mm2)
Elastic modulus of concrete (N/mm2)
Concrete strength (N/mm2)
Φ16
2.0 × 105
440
580
3.28 × 104
45.63
Φ20
486
640


## 2.3. Test Design

Horizontal impact tests were carried out on the five reinforced concrete column specimens. Each column was planned to be impacted four times, in which the Z16-1 and Z20-1 specimens followed Scheme 1, while the Z16-2, Z16-3, and Z20-2 specimens adopted Scheme 2. The scheme details are shown in Tables3 and 4. Then, based on the principle of the conservation of energy, the theoretical speeds of the impacts were calculated as follows:(1)m2gh=12m1+m2v12+μm1gh,where m1 and m2 denote the masses of the cart and drop hammer, respectively; g=9.81m/s2 is the gravitational acceleration; h represents the drop height of the hammer before release; v1 is the instantaneous velocity of the test cart before impact; and μ denotes the friction coefficient between the rail and the test cart, which was calibrated and determined by the preliminary impact test results, and for the different m1 and h. The friction coefficient was approximately equaled to 0.12. A laser velocity measurement system was installed at the end of the track for the purpose of measuring the instantaneous velocities of the impacts. As detailed in Tables 3 and 4, the measured velocities were approximately equal to the theoretical velocities obtained using formula (1) (the rate of the velocity variation was ±5%), which verified the reliability and accuracy of the impact test system.Table 3
Impact scheme 1.
Impact times
Drop height (m)
Car quality (kg)
Axial pressure (kN)
Drop hammer quality (kg)
Friction coefficient
Theoretical speed (m/s)
Measured average velocity (m/s)
Speed change rate (%)
1
4
1200
250
166
0.12
1.124
1.114
−0.890
2
6
1200
250
166
0.12
1.376
1.335
−2.980
3
8
1200
250
166
0.12
1.589
1.563
−1.636
4
10
1200
250
166
0.12
1.777
1.770
−0.56Table 4
Impact scheme 2.
Specimen no.
Impact times
Drop height (m)
Car quality (kg)
Axial pressure (kN)
Drop hammer quality (kg)
Friction coefficient
Theoretical speed (m/s)
Measured average velocity (m/s)
Speed change rate (%)
Z16-2
4
6
1200
250
166
0.12
1.376
1.337
−2.834
Z16-3
4
8
1200
250
166
0.12
1.589
1.513
−4.783
Z20-2
4
8
1200
250
166
0.12
1.589
1.588
−0.0629Prior to the testing, the pier specimens were fixed on rigid bases with four prestressed bolts, which were considered to be approximately equivalent to fixed end constraints. The upper ends of the columns were subjected to axial pressure of 250 kN by an oil jack mounted on the cross beam of the reaction frame, as shown in Figure7. After installing each of the reinforced concrete column specimens, the drop hammer was raised to set heights to carry out the impact tests. For all of the tests, the impact points were located 1,100 mm from the bottom of each column, i.e., at the center of each specimen.Figure 7
Axial pressure device.Seven acceleration sensors were evenly arranged on the back of each specimen during the tests (as shown in Figure8). Prior to and following each impact, a cohesive hammer was used to impact the top of the impact surface of each specimen. At the same time, a DHDAS dynamic signal acquisition and analysis system was used to collect the excitation signals and pier response signals measured by the acceleration sensors for the purpose of obtaining the natural vibration frequencies of the pier specimens and carry out damage identification processes of the concrete piers. A ZBL-U520 nonmetallic ultrasonic testing instrument was used to detect the damages to the concrete pier specimens in the main damage areas. In this way, the feasibility and accuracy of identifications and evaluations of the pier impact damages using modal frequencies were successfully verified.Figure 8
Arrangements of the restraint device and acceleration sensors.


## 3. Feasibility Analysis of Modal Frequency Identification of Pier Damage

### 3.1. Modal Frequency Acquisition of Pier

After the modal test method was used to obtain the pier response signal, the DHDAS dynamic signal acquisition and analysis system was applied for data analysis (specimen Z16-1 was selected as an example). This system mainly included the model building, data import, parameter identification, mode shape determination, and modal verification.


#### 3.1.1. Model Building

An analytical model corresponding to the distribution of the test pier and acceleration sensor was established (as shown in Figure9).Figure 9
Analysis model.


#### 3.1.2. Data Import

When importing data, the measurement method selects the “measurement method,” and the method selects single-point excitation. The frequency response function is used to analyze the data (as shown in Figure10).Figure 10
Force measurement data import.


#### 3.1.3. Parameter Identification

The parameter identification method applied in this experiment uses the internationally emerging transfer function-based Polylscf, which is able to obtain a very clear steady-state graph, and the recognition accuracy of each modal parameter is very high. The method mainly includes three parts: frequency band selection, steady-state graph calculation, and frequency response curve fitting.(1)
Frequency band selection: the vertical cursor on the left side is typically placed at the far left when the band is selected. When moving the horizontal cursor to determine the number of nodes, in most cases the number of nodes should be greater than 3. After determining the frequency range and the number of nodes, the steady-state graph calculation can be performed (as shown in Figure11).(2)
Steady-state graph calculation: in the calculation of the steady-state graph, the letterS indicates that all three modal parameters have reached a steady state, while V indicates that only the frequency and modal participation factors have been stabilized. We then select the pole corresponding to the frequency point of S to perform the vibration mode calculation. For the normalization method, the maximum value of the mode shape is selected to be normalized, after which the calculation of the modal parameters can be completed (as shown in Figure 12).(3)
Frequency response curve fitting: after the modal parameter calculation has been completed, the effect of fitting the frequency response curve of each measuring point can then be viewed, and the fitting degree shown in Figure13 is 0.94.Figure 11
Frequency band selection.Figure 12
Steady-state graph calculation.Figure 13
Frequency response curve fitting.


#### 3.1.4. Mode Shape Determination

After the modal parameters have been calculated, the modal shapes of each stage can be viewed in the mode animation display interface. As the test column of this paper is equivalent to the cantilever beam, the accuracy of the mode shape in the test can then be verified through comparison with the theoretical mode shape of the cantilever beam (as shown in Figure14). If the theoretical mode shape has not been met, then the test is repeated.Figure 14
(a) First mode shape; (b) second mode shape; (c) theoretical mode shape of the cantilever beam.
(a)
(b)
(c)


#### 3.1.5. Modal Verification

The verification results of each modal parameter file, mainly MAC, MSF, and MOV, are viewed. This modal verification is as follows.Figure15 shows a schematic diagram for the distribution of the modal assurance criterion (MAC) and modal scale factor (MSF) array. As can be seen in the diagram, the majority of the modal analysis results were found to be ideal, with low modal influence and good decoupling observed. Among the three modal validation parameters detailed in Table 5, the MAC, MSF, and MOV (modal complexity) of the first modal frequency were all 1. These results indicated that the first modal frequency obtained from the test was the true physical modal frequency of the pier.Figure 15
(a) Modal assurance criterion (MAC); (b) modal scale factor (MSF).
(a)
(b)Table 5
Parameters relating to the modal verifications.
Serial number
Frequency
MAC
MSF
MOV
First order
Second order
First order
Second order
First order
27.34
1.00
0.29
1.00
1.18
1.00
Second order
56.66
0.29
1.00
0.85
1.00
0.25


### 3.2. Experimental Modal Frequency to Identify Pier Damage

The first-order modal frequencies of five test pier models before and after impact are obtained by the abovementioned modal frequency acquisition and analysis method. The modal frequency is a function of pier stiffness. Therefore, when the pier is damaged, the modal frequency will change correspondingly. In this study, by analyzing the changes in the modal frequencies, the positions and degrees of the damages could be determined. It is known that the pier stiffness will change after damages occur. In this study, the change rates of component stiffness were adopted to represent the pier damages, and the damage factorD1 of pier damage was defined for the first time in this research study as follows:(2)D1=ΔKK=1−ωb2ω02,where ω0 is the frequency (Hz) before damage and ωb is the frequency (Hz) after damage.The experimentally measured data and the calculatedD1 are shown in the following table:It can be seen from Table6 that, with the increase of impact times, the first-order modal frequency gradually decreases, the damage factor gradually increases, and the change trend of modal frequency and damage factor remains consistent. This is due to the fact that the pier is gradually damaged with the increase of impact times, and the rigidity decreases. Therefore, the damage factor characterized by the modal frequency can be used as the basis for evaluating the damage of the pier.Table 6
Identification of the pier damages using the modal frequencies of the test columns.
Specimen no.
Impact time
Impact velocity (m/s)
Impact energy (J)
First-order frequency (Hz)
Damage factorD1
Before impactω0
After impactωb
Z16-1
1
1.181
836.8566
27.337
23.538
0.258626
2
1.376
1136.026
23.538
22.78
0.305606
3
1.589
1514.953
22.78
21.194
0.398931
4
1.717
1768.853
21.194
19.7
0.480685
Z20-1
1
1.047
657.7254
23.116
21.242
0.207939
2
1.354
1099.99
21.242
21.017
0.224629
3
1.603
1541.765
21.017
20.985
0.226989
4
1.806
1956.982
20.985
19.544
0.329506
Z16-2
1
1.378
1139.33
24.365
20.453
0.219390
2
1.336
1070.938
20.453
20.062
0.295337
3
1.316
1039.114
20.062
19.271
0.322022
4
1.316
1039.114
19.271
18.565
0.374430
Z16-3
1
1.496
1342.81
22.817
19.362
0.279916
2
1.496
1342.81
19.362
19.118
0.297950
3
1.471
1298.305
19.118
18.395
0.350046
4
1.549
1439.641
18.395
17.47
0.413769
Z20-2
1
1.587
1511.141
27.383
20.472
0.441069
2
1.587
1511.141
20.472
19.926
0.470485
3
1.587
1511.141
19.926
19.332
0.501585
4
1.587
1511.141
19.332
19.09
0.513985


### 3.3. Comparative Analysis of the Pier Crack and Modal Frequency Damage

After the cumulative impact, the final damage degree of the high damage area (the bottom area of the pier and the vicinity of the impact point) of each pier is shown in Table7.Table 7
Cracks in high damage area of the pier.
Specimen no.
Left side of impact point
Bottom area of impact point
Right side of impact point
Bottom back of impact point
Z16-1
Z20-1
Z16-2
Z16-3
Z20-2The left and the right side areas in the table are the areas below the side of the impact point. It can be seen that the horizontal bending cracks appear in the front bottom region of each member and that the crack pattern and failure mode are the same. These cracks are similar in nature to typical curved cracks. The number, length, and maximum width of the main cracks in each area are shown in Table8.Table 8
Comparison of the numbers, lengths, and widths of the main cracks.
Specimen no.
Left side area
Bottom area of impact point
Right side area
Number
Length (mm)
Width (mm)
Number
Length (mm)
Width (mm)
Number
Length (mm)
Width (mm)
Z16-1
0
0
0
6
500
2.25
0
0
0
Z20-1
4
590
0.39
2
455
0.39
3
470
0.32
Z16-2
0
0
0
1
560
0.72
0
0
0
Z16-3
0
0
0
2
600
2.16
0
0
0
Z20-2
0
0
0
4
720
3.54
0
0
0It can be seen from Table8 that the development of pier crack width is consistent with the change of damage factor proposed by the pier modal frequency, and thus, it is feasible to identify the impact damage of reinforced concrete pier by modal frequency, and the modal frequency of the pier can be used to evaluate the damage degree.


## 3.1. Modal Frequency Acquisition of Pier

After the modal test method was used to obtain the pier response signal, the DHDAS dynamic signal acquisition and analysis system was applied for data analysis (specimen Z16-1 was selected as an example). This system mainly included the model building, data import, parameter identification, mode shape determination, and modal verification.


### 3.1.1. Model Building

An analytical model corresponding to the distribution of the test pier and acceleration sensor was established (as shown in Figure9).Figure 9
Analysis model.


### 3.1.2. Data Import

When importing data, the measurement method selects the “measurement method,” and the method selects single-point excitation. The frequency response function is used to analyze the data (as shown in Figure10).Figure 10
Force measurement data import.


### 3.1.3. Parameter Identification

The parameter identification method applied in this experiment uses the internationally emerging transfer function-based Polylscf, which is able to obtain a very clear steady-state graph, and the recognition accuracy of each modal parameter is very high. The method mainly includes three parts: frequency band selection, steady-state graph calculation, and frequency response curve fitting.(1)
Frequency band selection: the vertical cursor on the left side is typically placed at the far left when the band is selected. When moving the horizontal cursor to determine the number of nodes, in most cases the number of nodes should be greater than 3. After determining the frequency range and the number of nodes, the steady-state graph calculation can be performed (as shown in Figure11).(2)
Steady-state graph calculation: in the calculation of the steady-state graph, the letterS indicates that all three modal parameters have reached a steady state, while V indicates that only the frequency and modal participation factors have been stabilized. We then select the pole corresponding to the frequency point of S to perform the vibration mode calculation. For the normalization method, the maximum value of the mode shape is selected to be normalized, after which the calculation of the modal parameters can be completed (as shown in Figure 12).(3)
Frequency response curve fitting: after the modal parameter calculation has been completed, the effect of fitting the frequency response curve of each measuring point can then be viewed, and the fitting degree shown in Figure13 is 0.94.Figure 11
Frequency band selection.Figure 12
Steady-state graph calculation.Figure 13
Frequency response curve fitting.


### 3.1.4. Mode Shape Determination

After the modal parameters have been calculated, the modal shapes of each stage can be viewed in the mode animation display interface. As the test column of this paper is equivalent to the cantilever beam, the accuracy of the mode shape in the test can then be verified through comparison with the theoretical mode shape of the cantilever beam (as shown in Figure14). If the theoretical mode shape has not been met, then the test is repeated.Figure 14
(a) First mode shape; (b) second mode shape; (c) theoretical mode shape of the cantilever beam.
(a)
(b)
(c)


### 3.1.5. Modal Verification

The verification results of each modal parameter file, mainly MAC, MSF, and MOV, are viewed. This modal verification is as follows.Figure15 shows a schematic diagram for the distribution of the modal assurance criterion (MAC) and modal scale factor (MSF) array. As can be seen in the diagram, the majority of the modal analysis results were found to be ideal, with low modal influence and good decoupling observed. Among the three modal validation parameters detailed in Table 5, the MAC, MSF, and MOV (modal complexity) of the first modal frequency were all 1. These results indicated that the first modal frequency obtained from the test was the true physical modal frequency of the pier.Figure 15
(a) Modal assurance criterion (MAC); (b) modal scale factor (MSF).
(a)
(b)Table 5
Parameters relating to the modal verifications.
Serial number
Frequency
MAC
MSF
MOV
First order
Second order
First order
Second order
First order
27.34
1.00
0.29
1.00
1.18
1.00
Second order
56.66
0.29
1.00
0.85
1.00
0.25


## 3.1.1. Model Building

An analytical model corresponding to the distribution of the test pier and acceleration sensor was established (as shown in Figure9).Figure 9
Analysis model.


## 3.1.2. Data Import

When importing data, the measurement method selects the “measurement method,” and the method selects single-point excitation. The frequency response function is used to analyze the data (as shown in Figure10).Figure 10
Force measurement data import.


## 3.1.3. Parameter Identification

The parameter identification method applied in this experiment uses the internationally emerging transfer function-based Polylscf, which is able to obtain a very clear steady-state graph, and the recognition accuracy of each modal parameter is very high. The method mainly includes three parts: frequency band selection, steady-state graph calculation, and frequency response curve fitting.(1)
Frequency band selection: the vertical cursor on the left side is typically placed at the far left when the band is selected. When moving the horizontal cursor to determine the number of nodes, in most cases the number of nodes should be greater than 3. After determining the frequency range and the number of nodes, the steady-state graph calculation can be performed (as shown in Figure11).(2)
Steady-state graph calculation: in the calculation of the steady-state graph, the letterS indicates that all three modal parameters have reached a steady state, while V indicates that only the frequency and modal participation factors have been stabilized. We then select the pole corresponding to the frequency point of S to perform the vibration mode calculation. For the normalization method, the maximum value of the mode shape is selected to be normalized, after which the calculation of the modal parameters can be completed (as shown in Figure 12).(3)
Frequency response curve fitting: after the modal parameter calculation has been completed, the effect of fitting the frequency response curve of each measuring point can then be viewed, and the fitting degree shown in Figure13 is 0.94.Figure 11
Frequency band selection.Figure 12
Steady-state graph calculation.Figure 13
Frequency response curve fitting.


## 3.1.4. Mode Shape Determination

After the modal parameters have been calculated, the modal shapes of each stage can be viewed in the mode animation display interface. As the test column of this paper is equivalent to the cantilever beam, the accuracy of the mode shape in the test can then be verified through comparison with the theoretical mode shape of the cantilever beam (as shown in Figure14). If the theoretical mode shape has not been met, then the test is repeated.Figure 14
(a) First mode shape; (b) second mode shape; (c) theoretical mode shape of the cantilever beam.
(a)
(b)
(c)


## 3.1.5. Modal Verification

The verification results of each modal parameter file, mainly MAC, MSF, and MOV, are viewed. This modal verification is as follows.Figure15 shows a schematic diagram for the distribution of the modal assurance criterion (MAC) and modal scale factor (MSF) array. As can be seen in the diagram, the majority of the modal analysis results were found to be ideal, with low modal influence and good decoupling observed. Among the three modal validation parameters detailed in Table 5, the MAC, MSF, and MOV (modal complexity) of the first modal frequency were all 1. These results indicated that the first modal frequency obtained from the test was the true physical modal frequency of the pier.Figure 15
(a) Modal assurance criterion (MAC); (b) modal scale factor (MSF).
(a)
(b)Table 5
Parameters relating to the modal verifications.
Serial number
Frequency
MAC
MSF
MOV
First order
Second order
First order
Second order
First order
27.34
1.00
0.29
1.00
1.18
1.00
Second order
56.66
0.29
1.00
0.85
1.00
0.25


## 3.2. Experimental Modal Frequency to Identify Pier Damage

The first-order modal frequencies of five test pier models before and after impact are obtained by the abovementioned modal frequency acquisition and analysis method. The modal frequency is a function of pier stiffness. Therefore, when the pier is damaged, the modal frequency will change correspondingly. In this study, by analyzing the changes in the modal frequencies, the positions and degrees of the damages could be determined. It is known that the pier stiffness will change after damages occur. In this study, the change rates of component stiffness were adopted to represent the pier damages, and the damage factorD1 of pier damage was defined for the first time in this research study as follows:(2)D1=ΔKK=1−ωb2ω02,where ω0 is the frequency (Hz) before damage and ωb is the frequency (Hz) after damage.The experimentally measured data and the calculatedD1 are shown in the following table:It can be seen from Table6 that, with the increase of impact times, the first-order modal frequency gradually decreases, the damage factor gradually increases, and the change trend of modal frequency and damage factor remains consistent. This is due to the fact that the pier is gradually damaged with the increase of impact times, and the rigidity decreases. Therefore, the damage factor characterized by the modal frequency can be used as the basis for evaluating the damage of the pier.Table 6
Identification of the pier damages using the modal frequencies of the test columns.
Specimen no.
Impact time
Impact velocity (m/s)
Impact energy (J)
First-order frequency (Hz)
Damage factorD1
Before impactω0
After impactωb
Z16-1
1
1.181
836.8566
27.337
23.538
0.258626
2
1.376
1136.026
23.538
22.78
0.305606
3
1.589
1514.953
22.78
21.194
0.398931
4
1.717
1768.853
21.194
19.7
0.480685
Z20-1
1
1.047
657.7254
23.116
21.242
0.207939
2
1.354
1099.99
21.242
21.017
0.224629
3
1.603
1541.765
21.017
20.985
0.226989
4
1.806
1956.982
20.985
19.544
0.329506
Z16-2
1
1.378
1139.33
24.365
20.453
0.219390
2
1.336
1070.938
20.453
20.062
0.295337
3
1.316
1039.114
20.062
19.271
0.322022
4
1.316
1039.114
19.271
18.565
0.374430
Z16-3
1
1.496
1342.81
22.817
19.362
0.279916
2
1.496
1342.81
19.362
19.118
0.297950
3
1.471
1298.305
19.118
18.395
0.350046
4
1.549
1439.641
18.395
17.47
0.413769
Z20-2
1
1.587
1511.141
27.383
20.472
0.441069
2
1.587
1511.141
20.472
19.926
0.470485
3
1.587
1511.141
19.926
19.332
0.501585
4
1.587
1511.141
19.332
19.09
0.513985


## 3.3. Comparative Analysis of the Pier Crack and Modal Frequency Damage

After the cumulative impact, the final damage degree of the high damage area (the bottom area of the pier and the vicinity of the impact point) of each pier is shown in Table7.Table 7
Cracks in high damage area of the pier.
Specimen no.
Left side of impact point
Bottom area of impact point
Right side of impact point
Bottom back of impact point
Z16-1
Z20-1
Z16-2
Z16-3
Z20-2The left and the right side areas in the table are the areas below the side of the impact point. It can be seen that the horizontal bending cracks appear in the front bottom region of each member and that the crack pattern and failure mode are the same. These cracks are similar in nature to typical curved cracks. The number, length, and maximum width of the main cracks in each area are shown in Table8.Table 8
Comparison of the numbers, lengths, and widths of the main cracks.
Specimen no.
Left side area
Bottom area of impact point
Right side area
Number
Length (mm)
Width (mm)
Number
Length (mm)
Width (mm)
Number
Length (mm)
Width (mm)
Z16-1
0
0
0
6
500
2.25
0
0
0
Z20-1
4
590
0.39
2
455
0.39
3
470
0.32
Z16-2
0
0
0
1
560
0.72
0
0
0
Z16-3
0
0
0
2
600
2.16
0
0
0
Z20-2
0
0
0
4
720
3.54
0
0
0It can be seen from Table8 that the development of pier crack width is consistent with the change of damage factor proposed by the pier modal frequency, and thus, it is feasible to identify the impact damage of reinforced concrete pier by modal frequency, and the modal frequency of the pier can be used to evaluate the damage degree.


## 4. Feasibility Analysis of Identification of Pier Damage by Ultrasonic Wave Velocity Verification Mode Frequency

### 4.1. Theoretical Analysis of the Relationship between the Pier Modal Frequency and the Ultrasonic Wave Velocity

For the identification of the impact damages of the reinforced concrete piers, this study mainly used the modal frequencies to identify and evaluate the test results, followed by a traditional ultrasonic detection method to verify the findings. In the ship-bridge collision simulations, the key variation following the pier damages was the elastic modulusE. It is known that if the stiffness K satisfies the first-order function relationship with the elastic modulus E as follows [32]:(3)K=fE.Then it can be known from the structural dynamics that(4)K=mω2,where m is the mass of the pier specimen (kg) and ω is the natural vibration frequency (Hz) of the pier specimen.During the process of using the ultrasonic instruments to measure the damages, the function relationship between the wave velocity and elastic modulus in solid medium conformed to the following formula:(5)E=ρV2,where ρ is the density of the concrete material (kg/m3) and V is the ultrasonic wave velocity.Then, in accordance with the relational expression shown above, the natural frequencyω of the bridge pier satisfied the first-order function relationship with the ultrasonic wave velocity V:(6)V=fω.Therefore, the relationship between the damage coefficients of the pier damages which had been identified by the modal frequencies and the damage coefficients of pier damages identified by ultrasonic wave velocities is the first-order function relationship which was found to be consistent. In other words, the modal frequencies and ultrasonic waves were consistent in this study’s evaluation of the pier damages.


### 4.2. Damage Analysis of Experimental Ultrasonic Wave Velocity

The ultrasonic detection of impact damages has a high sensitivity [33]. The damage criteria of concrete mainly include sound velocity, main frequency, and wave amplitude. Currently, many researchers have defined the damage variables of material through the changes in the wave velocities and have defined the damage factor D2 as follows [34,35]:(7)D2=1−Vb2V02,where V0 is the wave velocity before damage (m/s) and Vb is the wave velocity after damage (m/s).The following table identifies the pier damages identified by the ultrasonic velocities of the test columns.With the increase in the number of impacts, the damage factor of the pier gradually increases, which is a result of the change of the sound transmission medium. It can be seen from Table9 that after each impact, the value of wave velocity gradually decreases and the change remains relatively stable, and thus, it can be used to judge the damage condition of the concrete after impact.Table 9
Identification of the pier damages using the ultrasonic velocities of the test columns.
Specimen
Impact time
Impact velocity (m/s)
Impact energy (J)
Average acoustic velocity (m/s)
Damage factorD2
Before impactV0
After impactVb
Z16-1
1
1.181
836.8566
4.796
4.577
0.089241
2
1.376
1136.026
4.577
4.565
0.094010
3
1.589
1514.953
4.565
4.471
0.130938
4
1.717
1768.853
4.471
4.315
0.190525
Z20-1
1
1.047
657.7254
4.554
4.419
0.058410
2
1.354
1099.99
4.419
4.355
0.085486
3
1.603
1541.765
4.355
4.315
0.102208
4
1.806
1956.982
4.315
4.104
0.187864
Z16-2
1
1.378
1139.33
4.369
4.181
0.084209
2
1.336
1070.938
4.181
4.08
0.127920
3
1.316
1039.114
4.08
3.886
0.208882
4
1.316
1039.114
3.886
3.677
0.291690
Z16-3
1
1.496
1342.81
4.597
3.901
0.279883
2
1.496
1342.81
3.901
3.888
0.284675
3
1.471
1298.305
3.888
3.681
0.358816
4
1.549
1439.641
3.681
3.542
0.406326
Z20-2
1
1.587
1511.141
4.449
3.802
0.269703
2
1.587
1511.141
3.802
3.648
0.327666
3
1.587
1511.141
3.648
3.596
0.346697
4
1.587
1511.141
3.596
3.504
0.379698


### 4.3. Comparative Analysis of Ultrasonic Damage and Modal Frequency Damage

#### 4.3.1. Comparative Analysis

Figure16 shows this study’s comparisons of the fitting curves for the modal frequency damage factors and ultrasonic velocity damage factors of each test column under the conditions of equal impact velocity or accumulated impact energy. It can be seen from the figure that the damage factors of the modal frequencies and those of the ultrasonic velocities displayed a first-order relationship with the impact velocities. The two specimens had similar curve slopes, which indicated that the development and extensions of the pier damages were consistent under the two evaluation criteria and that the definition of the damage factors proposed in this study was feasible. Therefore, it was feasible to identify the pier impact damages using the modal frequencies.Figure 16
Comparative fitting curves for the modal frequency damage factors and ultrasonic velocity damage factors. (a) Comparative fitting curve of Z16-1, (b) comparative fitting curve of Z20-1, (c) comparative fitting curve of Z16-2, (d) comparative fitting curve of Z16-3, and (e) comparative fitting curve of Z20-2.
(a)
(b)
(c)
(d)
(e)


#### 4.3.2.D1andD2 Fitting Curves

Figure17 details the fitting curves of the correlation between the modal frequency damage factors and the ultrasonic velocity damage factors. Table 10 shows the analysis report for the fitting parameters of the fitting curves. As can be seen in Figure 17, the modal frequency damage factors and the ultrasonic velocity damage factors showed a linear relationship. In other words, a first-order function relationship was observed, which confirmed that the modal frequencies and ultrasonic waves were consistent in characterizing the damages of the piers. It can be seen from Table 8 that the fitting goodness R2 of each fitting curve was greater than 0.9. Also, the errors of the fitting values were within the allowable range, which indicated that the fitting degree was better in this test. The empirical formulas of the modal frequency damage factors and ultrasonic velocity damage factors could be roughly obtained as follows:(8)D2=μD1+λ,where μ∈0.45869,1.65719,λ∈−0.40649,−0.00437.Figure 17
Fitting curves of the modal frequency damage factors and the ultrasonic velocity damage factors. (a) Fitting curve of Z16-1, (b) fitting curve of Z20-1, (c) fitting curve of Z16-2, (d) fitting curve of Z16-3, and (e) fitting curve of Z20-2.
(a)
(b)
(c)
(d)
(e)Table 10
Analysis report for the fitting parameters of the fitting curves.
Specimen number
Intercept
Slope
Statistics Adj.R-square
Value
Standard error
Value
Standard error
Z16-1
−0.03939
0.03046
0.45869
0.0821
0.910
Z20-1
−0.13641
0.03311
0.99044
0.13143
0.910
Z16-2
−0.40649
0.01626
1.65719
0.04566
0.998
Z16-3
−0.00437
0.03806
1.00409
0.11212
0.964
Z20-2
−0.33039
0.10927
1.37269
0.22641
0.923


## 4.1. Theoretical Analysis of the Relationship between the Pier Modal Frequency and the Ultrasonic Wave Velocity

For the identification of the impact damages of the reinforced concrete piers, this study mainly used the modal frequencies to identify and evaluate the test results, followed by a traditional ultrasonic detection method to verify the findings. In the ship-bridge collision simulations, the key variation following the pier damages was the elastic modulusE. It is known that if the stiffness K satisfies the first-order function relationship with the elastic modulus E as follows [32]:(3)K=fE.Then it can be known from the structural dynamics that(4)K=mω2,where m is the mass of the pier specimen (kg) and ω is the natural vibration frequency (Hz) of the pier specimen.During the process of using the ultrasonic instruments to measure the damages, the function relationship between the wave velocity and elastic modulus in solid medium conformed to the following formula:(5)E=ρV2,where ρ is the density of the concrete material (kg/m3) and V is the ultrasonic wave velocity.Then, in accordance with the relational expression shown above, the natural frequencyω of the bridge pier satisfied the first-order function relationship with the ultrasonic wave velocity V:(6)V=fω.Therefore, the relationship between the damage coefficients of the pier damages which had been identified by the modal frequencies and the damage coefficients of pier damages identified by ultrasonic wave velocities is the first-order function relationship which was found to be consistent. In other words, the modal frequencies and ultrasonic waves were consistent in this study’s evaluation of the pier damages.


## 4.2. Damage Analysis of Experimental Ultrasonic Wave Velocity

The ultrasonic detection of impact damages has a high sensitivity [33]. The damage criteria of concrete mainly include sound velocity, main frequency, and wave amplitude. Currently, many researchers have defined the damage variables of material through the changes in the wave velocities and have defined the damage factor D2 as follows [34,35]:(7)D2=1−Vb2V02,where V0 is the wave velocity before damage (m/s) and Vb is the wave velocity after damage (m/s).The following table identifies the pier damages identified by the ultrasonic velocities of the test columns.With the increase in the number of impacts, the damage factor of the pier gradually increases, which is a result of the change of the sound transmission medium. It can be seen from Table9 that after each impact, the value of wave velocity gradually decreases and the change remains relatively stable, and thus, it can be used to judge the damage condition of the concrete after impact.Table 9
Identification of the pier damages using the ultrasonic velocities of the test columns.
Specimen
Impact time
Impact velocity (m/s)
Impact energy (J)
Average acoustic velocity (m/s)
Damage factorD2
Before impactV0
After impactVb
Z16-1
1
1.181
836.8566
4.796
4.577
0.089241
2
1.376
1136.026
4.577
4.565
0.094010
3
1.589
1514.953
4.565
4.471
0.130938
4
1.717
1768.853
4.471
4.315
0.190525
Z20-1
1
1.047
657.7254
4.554
4.419
0.058410
2
1.354
1099.99
4.419
4.355
0.085486
3
1.603
1541.765
4.355
4.315
0.102208
4
1.806
1956.982
4.315
4.104
0.187864
Z16-2
1
1.378
1139.33
4.369
4.181
0.084209
2
1.336
1070.938
4.181
4.08
0.127920
3
1.316
1039.114
4.08
3.886
0.208882
4
1.316
1039.114
3.886
3.677
0.291690
Z16-3
1
1.496
1342.81
4.597
3.901
0.279883
2
1.496
1342.81
3.901
3.888
0.284675
3
1.471
1298.305
3.888
3.681
0.358816
4
1.549
1439.641
3.681
3.542
0.406326
Z20-2
1
1.587
1511.141
4.449
3.802
0.269703
2
1.587
1511.141
3.802
3.648
0.327666
3
1.587
1511.141
3.648
3.596
0.346697
4
1.587
1511.141
3.596
3.504
0.379698


## 4.3. Comparative Analysis of Ultrasonic Damage and Modal Frequency Damage

### 4.3.1. Comparative Analysis

Figure16 shows this study’s comparisons of the fitting curves for the modal frequency damage factors and ultrasonic velocity damage factors of each test column under the conditions of equal impact velocity or accumulated impact energy. It can be seen from the figure that the damage factors of the modal frequencies and those of the ultrasonic velocities displayed a first-order relationship with the impact velocities. The two specimens had similar curve slopes, which indicated that the development and extensions of the pier damages were consistent under the two evaluation criteria and that the definition of the damage factors proposed in this study was feasible. Therefore, it was feasible to identify the pier impact damages using the modal frequencies.Figure 16
Comparative fitting curves for the modal frequency damage factors and ultrasonic velocity damage factors. (a) Comparative fitting curve of Z16-1, (b) comparative fitting curve of Z20-1, (c) comparative fitting curve of Z16-2, (d) comparative fitting curve of Z16-3, and (e) comparative fitting curve of Z20-2.
(a)
(b)
(c)
(d)
(e)


### 4.3.2.D1andD2 Fitting Curves

Figure17 details the fitting curves of the correlation between the modal frequency damage factors and the ultrasonic velocity damage factors. Table 10 shows the analysis report for the fitting parameters of the fitting curves. As can be seen in Figure 17, the modal frequency damage factors and the ultrasonic velocity damage factors showed a linear relationship. In other words, a first-order function relationship was observed, which confirmed that the modal frequencies and ultrasonic waves were consistent in characterizing the damages of the piers. It can be seen from Table 8 that the fitting goodness R2 of each fitting curve was greater than 0.9. Also, the errors of the fitting values were within the allowable range, which indicated that the fitting degree was better in this test. The empirical formulas of the modal frequency damage factors and ultrasonic velocity damage factors could be roughly obtained as follows:(8)D2=μD1+λ,where μ∈0.45869,1.65719,λ∈−0.40649,−0.00437.Figure 17
Fitting curves of the modal frequency damage factors and the ultrasonic velocity damage factors. (a) Fitting curve of Z16-1, (b) fitting curve of Z20-1, (c) fitting curve of Z16-2, (d) fitting curve of Z16-3, and (e) fitting curve of Z20-2.
(a)
(b)
(c)
(d)
(e)Table 10
Analysis report for the fitting parameters of the fitting curves.
Specimen number
Intercept
Slope
Statistics Adj.R-square
Value
Standard error
Value
Standard error
Z16-1
−0.03939
0.03046
0.45869
0.0821
0.910
Z20-1
−0.13641
0.03311
0.99044
0.13143
0.910
Z16-2
−0.40649
0.01626
1.65719
0.04566
0.998
Z16-3
−0.00437
0.03806
1.00409
0.11212
0.964
Z20-2
−0.33039
0.10927
1.37269
0.22641
0.923


## 4.3.1. Comparative Analysis

Figure16 shows this study’s comparisons of the fitting curves for the modal frequency damage factors and ultrasonic velocity damage factors of each test column under the conditions of equal impact velocity or accumulated impact energy. It can be seen from the figure that the damage factors of the modal frequencies and those of the ultrasonic velocities displayed a first-order relationship with the impact velocities. The two specimens had similar curve slopes, which indicated that the development and extensions of the pier damages were consistent under the two evaluation criteria and that the definition of the damage factors proposed in this study was feasible. Therefore, it was feasible to identify the pier impact damages using the modal frequencies.Figure 16
Comparative fitting curves for the modal frequency damage factors and ultrasonic velocity damage factors. (a) Comparative fitting curve of Z16-1, (b) comparative fitting curve of Z20-1, (c) comparative fitting curve of Z16-2, (d) comparative fitting curve of Z16-3, and (e) comparative fitting curve of Z20-2.
(a)
(b)
(c)
(d)
(e)


## 4.3.2.D1andD2 Fitting Curves

Figure17 details the fitting curves of the correlation between the modal frequency damage factors and the ultrasonic velocity damage factors. Table 10 shows the analysis report for the fitting parameters of the fitting curves. As can be seen in Figure 17, the modal frequency damage factors and the ultrasonic velocity damage factors showed a linear relationship. In other words, a first-order function relationship was observed, which confirmed that the modal frequencies and ultrasonic waves were consistent in characterizing the damages of the piers. It can be seen from Table 8 that the fitting goodness R2 of each fitting curve was greater than 0.9. Also, the errors of the fitting values were within the allowable range, which indicated that the fitting degree was better in this test. The empirical formulas of the modal frequency damage factors and ultrasonic velocity damage factors could be roughly obtained as follows:(8)D2=μD1+λ,where μ∈0.45869,1.65719,λ∈−0.40649,−0.00437.Figure 17
Fitting curves of the modal frequency damage factors and the ultrasonic velocity damage factors. (a) Fitting curve of Z16-1, (b) fitting curve of Z20-1, (c) fitting curve of Z16-2, (d) fitting curve of Z16-3, and (e) fitting curve of Z20-2.
(a)
(b)
(c)
(d)
(e)Table 10
Analysis report for the fitting parameters of the fitting curves.
Specimen number
Intercept
Slope
Statistics Adj.R-square
Value
Standard error
Value
Standard error
Z16-1
−0.03939
0.03046
0.45869
0.0821
0.910
Z20-1
−0.13641
0.03311
0.99044
0.13143
0.910
Z16-2
−0.40649
0.01626
1.65719
0.04566
0.998
Z16-3
−0.00437
0.03806
1.00409
0.11212
0.964
Z20-2
−0.33039
0.10927
1.37269
0.22641
0.923


## 5. Conclusions

In order to explore the feasibility of the identification of pier impact damages using modal frequencies, horizontal impact tests were carried out on five reinforced pier specimens in this study. The relationships between the modal frequencies and ultrasonic velocities in characterizing and evaluating the pier impact damage were examined. The main conclusions which were reached in this study were as follows:(1)
The modal frequencies and ultrasonic waves had displayed consistency in the characterization and evaluations of the pier damages(2)
This study proposed that it was feasible to define the damage factors for identifying pier impact damages using a modal frequency method(3)
An empirical formula for the damage factors of the modal frequencies and ultrasonic velocities was obtained as follows:D2=μD1+λ, where με0.45869,1.65719,λε−0.40649,−0.00437(4)
Based on the abovementioned conclusions, it can be considered that it is feasible to identify and evaluate the impact damages of bridge piers using modal frequencies


## 6. Discussion

This research study concluded that the first-order function relationship between the damage factors of modal frequencies and the damage factors of the ultrasonic velocities could be written as follows:D2=μD1+λ, where με0.45869,1.65719andλε−0.40649,−0.00437. The scopes of the parameters μ and λ were given according to the results of this study’s experimental tests. It was impossible to give the determined values of μ and λ due to the limited test conditions. A large number of repeated tests would be required in order to obtain the accurate values. Therefore, in accordance with the deductions of the author, μ would approach 1 and λ would approach 0 after a large number of repeated tests were completed.



---

*Source: 6365354-2020-01-22.xml*