# Study on the Main Factors of Frequency Variation of LC Resonant Prestress Detection Method

Authors: Benniu Zhang; Yuanyuan Zhao; Xingxing Li; Yanpeng Su; Qiang Yin; Jianhui Jiang
Journal: Advances in Materials Science and Engineering
 (2020)

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## Abstract

Stress loss of prestressed steel strands of existing bridges influences their bearing capacity, so it is of great significance to realize the stress detection. A steel strand that has an inductive property is designed into the resonant circuit, which can realize the stress measurement of the steel strands by testing the resonant frequency. This method is a promising approach for the stress detection of the steel strands. Previous research found that structural stress made the permeability of steel strands change due to the influence of magnetoelastic effect. In the process, the length of steel strands is also changed. Therefore, further research needs to be done to verify the main influence parameter affecting the resonant frequency of the circuit. Furthermore, it is very important to know how the stress affects the resonant frequency to realize the detection of the prestressed force of the steel strands. Therefore, in this paper, the relationship between stress and relative permeability and length is analyzed theoretically, and the theory of stress frequency of steel strands is modified and verified by experiments. The stress-frequency experiments of steel strands and aluminum strands with great difference in relative permeability are carried out. Experiments on stress frequencies of 7-Ф15.20 mm steel strands with different lengths are carried out. The influence of length and permeability parameters on resonance frequency is analyzed. The experimental results show that under the same conditions, the resonant frequencies of steel strands and aluminum strands are almost the same on LC electric circuits, and the resonant frequency decreases linearly with the increase of the natural length of the component and increases linearly with the increase of stress. Consequently, compared with the influence of length change on LC electromagnetic resonance frequency, the relative permeability of the stress change component can be ignored. The stress changes the resonant frequency mainly by changing the length of the strands.


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## Body

## 1. Introduction

Prestressed concrete structure is to use steel strand to exert pressure on the structure before it is subjected to external load, and the resulting prestressed state is used to reduce or offset the tensile stress caused by the external load. Because the prestressing concrete bridges take advantage of the compressive performance of concrete, while perfectly avoiding the disadvantage of poor crack resistance, it has been widely used in long-span bridges all over the world. However, for the shrinkage and creep of concrete, steel relaxation and other factors will lead to loss of prestressing force of steel strands. Prestress loss threatens the durability and serviceability of structures. Therefore, it is of great significance to detect the stress of steel strands in prestressed concrete bridges. At present, many nondestructive testing methods have been tried to apply to the prestress detection of steel strands, but no one has successfully applied to the stress detection of steel strands of prestressed concrete bridges.Fiber Bragg grating (FBG) method can monitor the stress history and spatial distribution of prestressing steel strands. Shin et al. [1] used four different sensors to carry out experimental research on stress monitoring of steel strands. Huynh et al. [2] used this sensor to study the stress loss in the prestressed concrete structure caused by temperature. Experiments show that the use of fiber Bragg grating embedded sensors can effectively evaluate the stress loss of steel strands. But, it needs to be embedded in the concrete structure in advance, which has limitations for the detection of the stress loss of the steel strands of the existing prestressed bridge; ultrasound-guided wave method [3, 4] is a common nondestructive testing method, which is widely used to detect crack lines on metal structures and corrosion defects on steel plates. Qian et al. [5] carried out experimental research on the relationship between the stress of steel strands and the guided wave and proved that there is a good correlation between the tension and the identification index. However, due to the loss of the ultrasonic guided wave when it passes through concrete, it can only be used in grouting bonded prestressing steel. Magnetoelastic method estimates stress by measuring the magnetic properties of supermagnetic materials. Duan et al. [6–11], based on magnetoelastic effect, has carried out an experimental study on stress monitoring of the high-strength steel wire, reinforced bar, and prestressed steel strands. Based on this method, Joh et al. [12] conducted a feasibility study on noncontact stress detection of steel strands. Kim et al. [13] proposed a tension monitoring system of a prestressed beam based on an embedded electromagnetic sensor. However, the magnetoelastic method needs to be demagnetized many times before stress measurement. It is difficult to ensure consistency between the state after demagnetization and the initial state. In addition, the magnetic flux leakage method [14] can be used to estimate the stress of external prestressing tendons, but it is difficult to apply to the stress detection of steel strands wrapped in concrete.The steel strand stress detection of the LC resonant circuit is to connect both ends of the steel strands into the circuit through the conductor. The change of the stress of the steel strands leads to the change of the inductance characteristic of the steel strands, which affects the resonance frequency of the circuit.It has been proved that the stress of strands has a good correlation with the electromagnetic resonance frequency [15, 16]. It is feasible to detect the prestress of strands based on the LC resonant circuit. In the early stage, Zhang et al. [17] carried out experimental research on 1.2 m steel strands. It was proved that the stress of steel strands has a good negative linear relationship with electromagnetic resonance frequency. It is feasible to detect the prestress of steel strands based on the LC resonant circuit. It was found that the stress of 10 m long steel strands is positively correlated with the resonance frequency in the later study. In order to prove this conflict, experimental study was carried out on the 1.2 m, 10 m, and 15 m steel strand and established two sets of theoretical analysis to study the length effect based on the LC resonant electric circuit. As a kind of ferromagnetic material, according to the magnetoelastic properties of the material, the change of stress will lead to the change of permeability [15, 16]. At the same time, considering the principle of material elasticity, we know that the length of steel strands will also change. However, it is still unknown which parameter of relative permeability and length is the main factor affecting the resonant frequency. In this paper, research on this problem is carried out in order to provide guidance for the later detection methods of LC resonant prestress.Based on the research, this paper theoretically analyzes the relationship between stress and relative permeability and length, and verifies it by experiments.In the experiment, the steel strands (ur = 1500) and aluminum strands (ur = 1) with great difference in relative permeability are selected for stress-frequency test. The resonant frequency with the same stress and length of 7-Ф9 mm steel strands/aluminum strands are measured by experiment, and the influence of permeability on the frequency of the LC resonant circuit is analyzed. In addition, the length frequency and stress-frequency experiments are carried out on the 1,860 steel strands of 7-Ф15.2 mm, which is commonly used in the project. The influence of the length of steel strands on the frequency of the LC resonant circuit is analyzed. According to the experimental results, the stress of the strands mainly affects the resonant frequency of the circuit by changing the length of the strands. Compared with the influence of the length, the influence of the change of permeability on the resonant frequency of the circuit can be neglected.


## 2. Theoretical Model

### 2.1. Introduction of Resonance Circuit

Based on the LC resonant circuit, an experimental study on stress detection of steel strands is carried out in this paper. The circuit model diagram is shown in Figure1, the initial inductance in the circuit is simplified to a fixed inductance, and steel strands are simplified as an inductance element. They are connected in series with the inherent inductance in the circuit and in parallel with the capacitor. In the resonance circuit, there is energy resonance between inductance and capacitance. That is to say, when discharging, the electric energy in the capacitor is converted into the magnetic energy in the inductor, and when charging, the magnetic energy in the inductor is converted into the electric energy in the capacitor, forming the electromagnetic resonance relationship.Figure 1
LC resonant circuit model diagram.


### 2.2. Derivation of Resonant Frequency

As shown in Figure1, the voltage VC at both ends of the capacitor is equal to the voltage VL at both ends of the total inductance due to the parallel connection of inductance and capacitance. Because in(1)VC=VL,the resonant circuit, the principle of energy resonance exists between inductance and capacitance; electric energy in capacitance and magnetic energy in inductance form electromagnetic resonance. External power supply continuously supplements the energy consumption in the circuit. Thus, the current flowing into the inductor is equal to the current flowing into the capacitor:(2)iC=−iL.From the constitutive relation of circuit elements, it can be seen that(3)VLt=LdiLdt,iCt=CdVCdt,iCt=CLd2iLdt2=CLd2iLtdt2,d2iLtdt2+1LCiLt=0.The current expression is as follows:(4)iLt=C1cos1LCx+C2sin1LCx.The current angular frequency is(5)w=w0=1LC.The resonant frequency of the circuit is(6)f=f0=12πLC.


### 2.3. Inductance Deduction

As a ferromagnetic material, considering the elastic principle of the material, the length of steel strands changes under the action of stress. At the same time, the change of length will also change its magnetic properties [15]. Through the analysis and study of the steel strand structure, we can see that because the strands are made of six outer wires twisted around the central wire, it is simplified to a spiral coil model [17]. At the same time, there is a high-speed conversion between electric energy and magnetic energy between the capacitance and the test piece, simplifying the strands into a high-frequency AC circuit. According to the skin effect of the current, the current distributes on the surface of six strands around the strands, and the magnetic field inside the strands is approximately 0. The magnetic field along the axis of a long wireless solenoid is uniform, and its values are μ0NI/l.(7)B=μ0NIl,Ψ=NBS=μ0N2SIl,L=ΨI=μ0N2Sl,N1=la,N=6N1.In these formulas,Ψ is the magnetic flux, S is the cross-section area of steel strands, N is the number of coil turns of steel strands, N1 is the number of coil turns of the single steel wire, L is the length of steel strands, u0 is the vacuum permeability, and a is the twist distance of steel strands.


### 2.4. Stress-Frequency Relationship

According to material mechanics, the elongation of steel strands under certain stress is as follows:(8)Δl=FlEA=σlE,L=L0+Ll+Lσ.Because the silver-coated enameled wires are connected at both ends before being connected to the strands, there is a self-contained frequency in the circuit. So, the inductance in the circuit consists of three parts, that is, inherent inductanceL0, inductance increment Ll caused by natural length of steel strands, and stress-induced inductance increment Lσ; L0 is obtained by connecting the two ends of the conductor in the experiment. The inductance of the inherent length of the strands can be obtained from the following formula:(9)Ll=u0N2Sl=36u0lSa2.The inductance increment caused by stress is(10)Lσ=−Δlμ0N2Sll+Δl=−36σ/Eμ0lSa21+σ/E.Therefore, the resonant frequency can be obtained as follows:(11)f=12πL0+36μ0lS/a2+−36σ/Eμ0lS/a21+σ/EC.


### 2.5. Simulation Results

By substitutingμ0=4π×10−7H/m, σ = 7.194 MPa, L0=0.000008H, S=139×10−6m2, a=0.212m, E=1.95×105MPa, and C=0.1μF into formula (11), the σ-f diagram can be obtained, as shown in Figure 2.(12)f=12π0.0000008+l×1.3984×10−8.Figure 2
l-f diagram.By substituting the parameterl=14m into formula (11), the l-f diagram can be obtained as shown in Figure 3.(13)f=12π9.958×10−7−1.958×10−7σ/195000+σ.Figure 3
σ-f diagram.As can be seen from formula (11), the frequency of the LC resonant circuit is related to the stress and natural length of strands. From Figure 3, it can be seen that the resonant frequency of the circuit is positively linearly related to the stress of the strands, and the resonant frequency increases with the increase of stress. As shown in Figure 2, the resonant frequency of the circuit is inversely proportional to the natural length of the strands, and the resonant frequency decreases with the increase of the natural length of the strands. It can be concluded that the stress of the strands in the circuit mainly changes the resonant frequency of the circuit by changing the length of the strands, and the change of relative permeability of steel strands can be ignored. In order to prove this theory, the following experiments are carried out in this paper.


## 2.1. Introduction of Resonance Circuit

Based on the LC resonant circuit, an experimental study on stress detection of steel strands is carried out in this paper. The circuit model diagram is shown in Figure1, the initial inductance in the circuit is simplified to a fixed inductance, and steel strands are simplified as an inductance element. They are connected in series with the inherent inductance in the circuit and in parallel with the capacitor. In the resonance circuit, there is energy resonance between inductance and capacitance. That is to say, when discharging, the electric energy in the capacitor is converted into the magnetic energy in the inductor, and when charging, the magnetic energy in the inductor is converted into the electric energy in the capacitor, forming the electromagnetic resonance relationship.Figure 1
LC resonant circuit model diagram.


## 2.2. Derivation of Resonant Frequency

As shown in Figure1, the voltage VC at both ends of the capacitor is equal to the voltage VL at both ends of the total inductance due to the parallel connection of inductance and capacitance. Because in(1)VC=VL,the resonant circuit, the principle of energy resonance exists between inductance and capacitance; electric energy in capacitance and magnetic energy in inductance form electromagnetic resonance. External power supply continuously supplements the energy consumption in the circuit. Thus, the current flowing into the inductor is equal to the current flowing into the capacitor:(2)iC=−iL.From the constitutive relation of circuit elements, it can be seen that(3)VLt=LdiLdt,iCt=CdVCdt,iCt=CLd2iLdt2=CLd2iLtdt2,d2iLtdt2+1LCiLt=0.The current expression is as follows:(4)iLt=C1cos1LCx+C2sin1LCx.The current angular frequency is(5)w=w0=1LC.The resonant frequency of the circuit is(6)f=f0=12πLC.


## 2.3. Inductance Deduction

As a ferromagnetic material, considering the elastic principle of the material, the length of steel strands changes under the action of stress. At the same time, the change of length will also change its magnetic properties [15]. Through the analysis and study of the steel strand structure, we can see that because the strands are made of six outer wires twisted around the central wire, it is simplified to a spiral coil model [17]. At the same time, there is a high-speed conversion between electric energy and magnetic energy between the capacitance and the test piece, simplifying the strands into a high-frequency AC circuit. According to the skin effect of the current, the current distributes on the surface of six strands around the strands, and the magnetic field inside the strands is approximately 0. The magnetic field along the axis of a long wireless solenoid is uniform, and its values are μ0NI/l.(7)B=μ0NIl,Ψ=NBS=μ0N2SIl,L=ΨI=μ0N2Sl,N1=la,N=6N1.In these formulas,Ψ is the magnetic flux, S is the cross-section area of steel strands, N is the number of coil turns of steel strands, N1 is the number of coil turns of the single steel wire, L is the length of steel strands, u0 is the vacuum permeability, and a is the twist distance of steel strands.


## 2.4. Stress-Frequency Relationship

According to material mechanics, the elongation of steel strands under certain stress is as follows:(8)Δl=FlEA=σlE,L=L0+Ll+Lσ.Because the silver-coated enameled wires are connected at both ends before being connected to the strands, there is a self-contained frequency in the circuit. So, the inductance in the circuit consists of three parts, that is, inherent inductanceL0, inductance increment Ll caused by natural length of steel strands, and stress-induced inductance increment Lσ; L0 is obtained by connecting the two ends of the conductor in the experiment. The inductance of the inherent length of the strands can be obtained from the following formula:(9)Ll=u0N2Sl=36u0lSa2.The inductance increment caused by stress is(10)Lσ=−Δlμ0N2Sll+Δl=−36σ/Eμ0lSa21+σ/E.Therefore, the resonant frequency can be obtained as follows:(11)f=12πL0+36μ0lS/a2+−36σ/Eμ0lS/a21+σ/EC.


## 2.5. Simulation Results

By substitutingμ0=4π×10−7H/m, σ = 7.194 MPa, L0=0.000008H, S=139×10−6m2, a=0.212m, E=1.95×105MPa, and C=0.1μF into formula (11), the σ-f diagram can be obtained, as shown in Figure 2.(12)f=12π0.0000008+l×1.3984×10−8.Figure 2
l-f diagram.By substituting the parameterl=14m into formula (11), the l-f diagram can be obtained as shown in Figure 3.(13)f=12π9.958×10−7−1.958×10−7σ/195000+σ.Figure 3
σ-f diagram.As can be seen from formula (11), the frequency of the LC resonant circuit is related to the stress and natural length of strands. From Figure 3, it can be seen that the resonant frequency of the circuit is positively linearly related to the stress of the strands, and the resonant frequency increases with the increase of stress. As shown in Figure 2, the resonant frequency of the circuit is inversely proportional to the natural length of the strands, and the resonant frequency decreases with the increase of the natural length of the strands. It can be concluded that the stress of the strands in the circuit mainly changes the resonant frequency of the circuit by changing the length of the strands, and the change of relative permeability of steel strands can be ignored. In order to prove this theory, the following experiments are carried out in this paper.


## 3. Experiments

### 3.1. Experimental Introduction

In order to verify the effect of the change of permeability caused by the stress of strands on the resonant frequency of the LC circuit, experiments were carried out on 7-Ф9 mm strands (ur = 1500) and aluminum strands (ur = 1) with large difference in relative permeability. The circuit frequency of steel strands and aluminum strand tests under the same conditions, respectively, are carried out to determine whether the change of the component permeability will affect the electromagnetic resonance frequency. By testing the circuit frequency of 7-Ф15.20 mm steel strands with different lengths and the same stress, the influence of strand length on electromagnetic resonance frequency of the access circuit is simulated. Finally, the stress-frequency test of 14 m long 7-Ф15.20 mm steel strands is carried out to verify the revised theory of measuring the stress of steel strands based on the LC resonant circuit.Schematic diagram of the experimental device for comparative test of steel strands and aluminum strands is shown in Figure4. The experimental device is divided into two parts: the force transfer device and the test circuit. Loading experiments of specimens with different lengths were carried out by passing the specimens through the reaction frame. Because the reaction frame is fixed on the ground and cannot be moved, it is impossible to obtain enough test data; a force transmission device is made as shown in Figure 4. Two 16 mm diameter screw rods are used to penetrate the reaction frame. Two 3 cm-thick steel plates are customized and connected with nuts. The screw jack (measuring range is 5 t) is installed between the outer steel plate and the reaction frame for additional force (Figure 5). The test circuit mainly connects the test piece into the circuit. One end of the test piece fixes the test piece with the steel plate by anchorage, and the other end fixes the test piece on the reaction frame by anchorage (Figure 6). In order to prevent the pressure sensor from falling into the anchor, a cushion plate is installed between the anchor and the pressure sensor. The specimen is connected to the circuit by the silver-plated copper core-enameled wire, and the circuit diagram is shown in Figure 7. Bluetooth is used to transmit the resonant frequency data of the circuit to the computer equipment. The field test drawings are shown in Figure 8.Figure 4
Schematic diagram of the experimental device.Figure 5
Jack apparatus.Figure 6
Diagram of anchorage, gaskets, and pressure sensor.Figure 7
Circuit device diagram.Figure 8
Field experimental diagram of aluminum/steel strands.In order to prevent the reaction frame from introducing the circuit, the PVC tube is used to wrap the test piece at the junction of the test piece and the reaction frame to prevent the contact between the test piece and the reaction frame. The test piece is wound with an insulating tape at the connection of the pressure sensor, steel plate, and cushion plate to prevent the outside from introducing the circuit. In the stress-frequency experiment of 14 m steel strands, the screw rod cannot bear enough tension, so the transmission device is removed, and the hydraulic jack is used to load the steel strands directly on the reaction frame. The schematic diagram of the experimental device is shown in Figure9.Figure 9
6 m/10 m/14 m experimental device of steel strands.


### 3.2. Experimental Process

In order to study the effect of permeability on stress detection of strands, the resonant frequency of the same tension force of 9 mm diameter strands/aluminum strands was measured. Because the maximum tensile stress of the aluminum strands is 7.8 kN, considering the deformation and safety factors, the tension force of 1 kN is exerted on the strands. For the sake of the authenticity of the experimental data, many groups of experiments were carried out, so experiments were carried out on steel strands/aluminum strands with different lengths, ranging from 600 cm to 1650 cm.Before the experiment, the experimental equipment is debugged, and the pressure sensor is calibrated. The sensitivity coefficient of the pressure sensor connecting the strain box is set to 0.3846. When the specimen is relaxed, the display of the strain box is set to 0. When the tension is 1 kN, the resonant frequency of the circuit is recorded, and each group of experiments is repeated three times. The same experiment was carried out on steel strands with diameter of 15.20 mm.Figure 10
l-f analysis diagram of steel strands/aluminum strands.Figure 11
l-f analysis diagram of steel strands.Figure 12
14 m steel strandσ-f analysis chart.In order to analyze the stress-frequency relationship, the stress-frequency experiment of 14 m-long 7-Ф15.20 mm strands was carried out, and the tension was controlled from 10 kN to 126 kN. The resonant frequencies of the circuits under different stresses were measured, and the experiments were repeated four times in each group. Detailed changes in parameters are shown in Table1.Table 1
Table of experimental parameters.
Parameter
Permeability
Length (m)
Stress (MPa)
Numerical value
7-Ф9 mm steel/aluminum strands
16.5, 16, …, 6
100, 200, …, 900
Note appended
u
r ≈ 1500, ur ≈ 1, see Figure 10
See Figure11
See Figure12


### 3.3. Experimental Data

In order to study whether the change of stress-relative permeability will affect the resonant frequency of the circuit, experiments were carried out on 7-Ф9 mm steel strands and aluminum strands with length of 6m–16 m. The experimental data are shown in Tables2 and 3, and the analysis results are shown in Figure 10.Table 2
Ф9 mm experimental data of aluminum strands.
l/m
f1 (kHz)
f2 (kHz)
f3 (kHz)
f4 (kHz)
SD
RE (%)
f (kHz)
6.0
104.612
104.625
104.606
104.608
0.0086
0.0182
104.613
6.5
103.273
103.277
103.262
103.281
0.0082
0.0184
103.273
7.0
101.836
101.847
101.864
101.856
0.0122
0.0278
101.851
7.5
101.166
101.179
101.191
101.196
0.0134
0.0296
101.183
8.0
100.815
100.784
100.787
100.794
0.0140
0.0308
100.795
10.0
98.421
98.363
98.465
98.397
0.0426
0.1030
98.411
10.5
97.621
97.666
97.643
97.633
0.0192
0.0464
97.641
11.0
97.009
97.014
97.018
96.994
0.0106
0.0247
97.009
11.5
96.350
96.326
96.339
96.365
0.0166
0.0405
96.345
12.0
95.704
95.707
95.719
95.705
0.0072
0.0164
95.709
12.5
94.965
94.938
94.934
94.951
0.0140
0.0326
94.947
14.0
94.186
94.199
94.171
94.184
0.0115
0.0297
94.185
14.5
93.541
93.566
93.542
93.547
0.0117
0.0267
93.549
15.0
93.219
93.228
93.210
93.222
0.0076
0.0197
93.220
15.5
92.676
92.680
92.665
92.676
0.0064
0.0158
92.674
16.0
92.480
92.448
92.492
92.465
0.0188
0.0469
92.471Table 3
Ф9 mm experimental data of steel strands.
l/m
f1 (kHz)
f2 (kHz)
f3 (kHz)
f4 (kHz)
SD
RE (%)
f (kHz)
6.0
103.813
103.819
103.815
103.805
0.0057
0.0128
103.813
6.5
101.991
102.009
102.035
102.013
0.0180
0.0428
102.012
7.0
102.379
102.378
102.385
102.374
0.0047
0.0111
102.379
7.5
101.660
101.663
101.662
101.653
0.0044
0.0098
101.660
8.0
100.993
101.011
101.023
101.026
0.0149
0.0327
101.013
10.0
99.016
99.044
99.043
99.042
0.0136
0.0283
99.036
10.5
98.125
98.121
98.126
98.132
0.0047
0.0115
98.126
11.0
97.460
97.471
97.473
97.474
0.0068
0.0150
97.470
11.5
95.727
95.715
95.726
95.739
0.0098
0.0251
95.727
12.0
94.963
94.976
94.973
94.981
0.0073
0.0183
94.973
12.5
94.376
94.371
94.378
94.379
0.0035
0.0081
94.376
14.0
93.786
93.790
93.789
93.792
0.0023
0.0060
93.789
14.5
93.520
93.528
93.539
93.534
0.0081
0.0203
93.530
15.0
92.771
92.786
92.774
92.794
0.0106
0.0244
92.781
15.5
92.658
92.655
92.653
92.674
0.0096
0.0227
92.660
16.0
92.415
92.422
92.450
92.434
0.0152
0.0375
92.430In order to study the influence of strand length on circuit resonance frequency, experiments of different lengths of 7-Ф15.20 mm strands commonly used in engineering are carried out, ranging from 6–16 m. The experimental data are shown in Table4, and the analysis drawings are shown in Figure 11.Table 4
Ф15.2 mm experimental data of strand length.
l/m
f1 (kHz)
f2 (kHz)
f3 (kHz)
f4 (kHz)
SD
RE (%)
f (kHz)
6.0
103.907
103.909
103.900
103.912
0.0051
0.0115
103.907
6.5
103.183
103.183
103.213
103.225
0.0214
0.0407
103.201
7.0
102.522
102.536
102.538
102.540
0.0082
0.0176
102.534
7.5
101.394
101.395
101.372
101.379
0.0113
0.0227
101.385
8.0
100.745
100.734
100.753
100.748
0.0080
0.0189
100.745
10.0
98.675
98.641
98.652
98.648
0.0147
0.0345
98.654
10.5
97.485
97.462
97.502
97.479
0.0165
0.0410
97.482
11.0
97.287
97.293
97.285
97.283
0.0043
0.0103
97.287
11.5
96.640
96.643
96.642
96.635
0.0036
0.0083
96.640
12.0
95.778
95.784
95.776
95.774
0.0043
0.0104
95.778
12.5
95.482
95.446
95.447
95.485
0.0214
0.0409
95.465
14.0
94.153
94.139
94.162
94.158
0.0100
0.0244
94.153
14.5
93.705
93.698
93.695
93.714
0.0084
0.0203
93.703
15.0
93.138
93.164
93.127
93.123
0.0185
0.0440
93.138
15.5
92.490
92.476
92.512
92.470
0.0187
0.0454
92.487
16.0
92.211
92.202
92.200
92.223
0.0105
0.0249
92.209In order to study the effect of stress on resonance frequency and verify the theoretical formula, theσ-f experiment of 14 m 7-Ф15.20 mm steel strands is carried out. The experimental data are shown in Table 5, and the analysis diagram is shown in Figure 12.Table 5
14 m Ф15.20 mm experimental data of steel strands.
σ (Mpa)
f1 (kHz)
f2 (kHz)
f3 (kHz)
f4 (kHz)
SD
RE (%)
f (kHz)
71.94
95.007
95.024
95.012
95.025
0.0089
0.0189
95.017
100.72
95.073
95.066
95.080
95.070
0.0059
0.0147
95.072
143.88
95.169
95.166
95.177
95.178
0.0059
0.0126
95.173
201.44
95.232
95.236
95.231
95.239
0.0037
0.0084
95.235
215.83
95.250
95.241
95.248
95.246
0.0039
0.0094
95.246
287.77
95.270
95.266
95.260
95.264
0.0042
0.0105
95.265
302.16
95.279
95.290
95.293
95.293
0.0067
0.0147
95.289
359.71
95.369
95.371
95.361
95.376
0.0062
0.0157
95.369
402.88
95.396
95.392
95.397
95.403
0.0045
0.0115
95.397
431.65
95.446
95.451
95.453
95.452
0.0031
0.0073
95.451
503.60
95.484
95.480
95.481
95.486
0.0028
0.0063
95.483
575.54
95.568
95.564
95.568
95.567
0.0019
0.0042
95.567
604.32
95.554
95.551
95.553
95.558
0.0029
0.0073
95.554
647.48
95.575
95.574
95.579
95.580
0.0029
0.0063
95.577
705.04
95.606
95.599
95.605
95.608
0.0039
0.0094
95.605
719.42
95.637
95.630
95.634
95.635
0.0029
0.0073
95.634
791.37
95.657
95.656
95.658
95.663
0.0031
0.0073
95.659
805.76
95.659
95.655
95.647
95.660
0.0059
0.0136
95.655
863.31
95.693
95.692
95.698
95.687
0.0045
0.0115
95.693
906.47
95.726
95.733
95.756
95.731
0.0133
0.0313
95.737


### 3.4. Data Analysis

(1)
When the tension force of Ф9 mm steel strands is measured at 1 kN, the error rate of data repetition is controlled at 0.038%. Under the same conditions, the data repetition error rate of Ф9 mm diameter aluminum strands is controlled at 0.047% and that of Ф15.20 mm diameter steel strands is controlled at 0.045%. The steel strands with 14 m length are tested inσ-f experiment, and data duplication error rate is only 0.010%.
(2)
As shown in Figure10, Ф9 mm steel strands and aluminum strands access the circuit, and the resonant frequency of steel strands/aluminum strands with the same stress and length tends to be the same. When the length is 6 m, the maximum frequency difference is 1 kHz, and the frequency difference of other length specimens is very small. Error data when 6 m are considered. At the same time, the frequency of strands/aluminum strands is consistent with the length, that is, it decreases with the increase of length. Because the permeability of steel strands (ur = 1500) and aluminum strands (ur = 1) is quite different, therefore, both theoretical and experimental data are available. If other conditions are consistent, the difference of relative permeability of strands itself has little effect on resonance frequency. It can be concluded that the effect of stress change-relative permeability on resonance frequency is smaller.
(3)
Figure11 shows that under the same stress, the resonance frequency of Ф15.20 mm strands changes with the length; the frequency decreases linearly with the increase of length, and it is consistent with that of Ф9 mm steel strands/aluminum strands varying with length. Comparing l-f analysis with Figures 2 and 11, we can see that, under the same stress, the resonance frequency of different length strands decreases with the increase of length, and the experimental results are consistent with the theoretical ones.
(4)
Figure12 shows that the resonance frequency of 14 m Ф15.20 mm steel strands increases with the increase of stress. There is a good positive linear correlation between stress and frequency. The data fitting equation is as follows: f=7.94606×10−4σ+95.0528, R2=0.96752. Comparing with Figure 3, it can be found that there is a consistent correlation between experimental data and theoretical data in stress and frequency. But, the slope of fitting formula is 10 times that of theoretical formula. Considering that, under the action of stress, while the length of strands changes, the compactness of seven steel wires in the strand structure changes. Therefore, the increase of the length produced by the stress cannot be simply equal to the increase of the length of the strands. Further research is needed in the next study.
(5)
The theoretical and experimental data show that compared with the influence of specimen length on relative permeability, the influence of relative permeability on resonance frequency can be ignored. The smaller the natural length of strands is, the more significant the resonance frequency is. Therefore, it can be deduced that the effect of stress-changing permeability on resonance frequency can be ignored. The stress mainly changes resonance frequency by changing the length of steel strands.


## 3.1. Experimental Introduction

In order to verify the effect of the change of permeability caused by the stress of strands on the resonant frequency of the LC circuit, experiments were carried out on 7-Ф9 mm strands (ur = 1500) and aluminum strands (ur = 1) with large difference in relative permeability. The circuit frequency of steel strands and aluminum strand tests under the same conditions, respectively, are carried out to determine whether the change of the component permeability will affect the electromagnetic resonance frequency. By testing the circuit frequency of 7-Ф15.20 mm steel strands with different lengths and the same stress, the influence of strand length on electromagnetic resonance frequency of the access circuit is simulated. Finally, the stress-frequency test of 14 m long 7-Ф15.20 mm steel strands is carried out to verify the revised theory of measuring the stress of steel strands based on the LC resonant circuit.Schematic diagram of the experimental device for comparative test of steel strands and aluminum strands is shown in Figure4. The experimental device is divided into two parts: the force transfer device and the test circuit. Loading experiments of specimens with different lengths were carried out by passing the specimens through the reaction frame. Because the reaction frame is fixed on the ground and cannot be moved, it is impossible to obtain enough test data; a force transmission device is made as shown in Figure 4. Two 16 mm diameter screw rods are used to penetrate the reaction frame. Two 3 cm-thick steel plates are customized and connected with nuts. The screw jack (measuring range is 5 t) is installed between the outer steel plate and the reaction frame for additional force (Figure 5). The test circuit mainly connects the test piece into the circuit. One end of the test piece fixes the test piece with the steel plate by anchorage, and the other end fixes the test piece on the reaction frame by anchorage (Figure 6). In order to prevent the pressure sensor from falling into the anchor, a cushion plate is installed between the anchor and the pressure sensor. The specimen is connected to the circuit by the silver-plated copper core-enameled wire, and the circuit diagram is shown in Figure 7. Bluetooth is used to transmit the resonant frequency data of the circuit to the computer equipment. The field test drawings are shown in Figure 8.Figure 4
Schematic diagram of the experimental device.Figure 5
Jack apparatus.Figure 6
Diagram of anchorage, gaskets, and pressure sensor.Figure 7
Circuit device diagram.Figure 8
Field experimental diagram of aluminum/steel strands.In order to prevent the reaction frame from introducing the circuit, the PVC tube is used to wrap the test piece at the junction of the test piece and the reaction frame to prevent the contact between the test piece and the reaction frame. The test piece is wound with an insulating tape at the connection of the pressure sensor, steel plate, and cushion plate to prevent the outside from introducing the circuit. In the stress-frequency experiment of 14 m steel strands, the screw rod cannot bear enough tension, so the transmission device is removed, and the hydraulic jack is used to load the steel strands directly on the reaction frame. The schematic diagram of the experimental device is shown in Figure9.Figure 9
6 m/10 m/14 m experimental device of steel strands.


## 3.2. Experimental Process

In order to study the effect of permeability on stress detection of strands, the resonant frequency of the same tension force of 9 mm diameter strands/aluminum strands was measured. Because the maximum tensile stress of the aluminum strands is 7.8 kN, considering the deformation and safety factors, the tension force of 1 kN is exerted on the strands. For the sake of the authenticity of the experimental data, many groups of experiments were carried out, so experiments were carried out on steel strands/aluminum strands with different lengths, ranging from 600 cm to 1650 cm.Before the experiment, the experimental equipment is debugged, and the pressure sensor is calibrated. The sensitivity coefficient of the pressure sensor connecting the strain box is set to 0.3846. When the specimen is relaxed, the display of the strain box is set to 0. When the tension is 1 kN, the resonant frequency of the circuit is recorded, and each group of experiments is repeated three times. The same experiment was carried out on steel strands with diameter of 15.20 mm.Figure 10
l-f analysis diagram of steel strands/aluminum strands.Figure 11
l-f analysis diagram of steel strands.Figure 12
14 m steel strandσ-f analysis chart.In order to analyze the stress-frequency relationship, the stress-frequency experiment of 14 m-long 7-Ф15.20 mm strands was carried out, and the tension was controlled from 10 kN to 126 kN. The resonant frequencies of the circuits under different stresses were measured, and the experiments were repeated four times in each group. Detailed changes in parameters are shown in Table1.Table 1
Table of experimental parameters.
Parameter
Permeability
Length (m)
Stress (MPa)
Numerical value
7-Ф9 mm steel/aluminum strands
16.5, 16, …, 6
100, 200, …, 900
Note appended
u
r ≈ 1500, ur ≈ 1, see Figure 10
See Figure11
See Figure12


## 3.3. Experimental Data

In order to study whether the change of stress-relative permeability will affect the resonant frequency of the circuit, experiments were carried out on 7-Ф9 mm steel strands and aluminum strands with length of 6m–16 m. The experimental data are shown in Tables2 and 3, and the analysis results are shown in Figure 10.Table 2
Ф9 mm experimental data of aluminum strands.
l/m
f1 (kHz)
f2 (kHz)
f3 (kHz)
f4 (kHz)
SD
RE (%)
f (kHz)
6.0
104.612
104.625
104.606
104.608
0.0086
0.0182
104.613
6.5
103.273
103.277
103.262
103.281
0.0082
0.0184
103.273
7.0
101.836
101.847
101.864
101.856
0.0122
0.0278
101.851
7.5
101.166
101.179
101.191
101.196
0.0134
0.0296
101.183
8.0
100.815
100.784
100.787
100.794
0.0140
0.0308
100.795
10.0
98.421
98.363
98.465
98.397
0.0426
0.1030
98.411
10.5
97.621
97.666
97.643
97.633
0.0192
0.0464
97.641
11.0
97.009
97.014
97.018
96.994
0.0106
0.0247
97.009
11.5
96.350
96.326
96.339
96.365
0.0166
0.0405
96.345
12.0
95.704
95.707
95.719
95.705
0.0072
0.0164
95.709
12.5
94.965
94.938
94.934
94.951
0.0140
0.0326
94.947
14.0
94.186
94.199
94.171
94.184
0.0115
0.0297
94.185
14.5
93.541
93.566
93.542
93.547
0.0117
0.0267
93.549
15.0
93.219
93.228
93.210
93.222
0.0076
0.0197
93.220
15.5
92.676
92.680
92.665
92.676
0.0064
0.0158
92.674
16.0
92.480
92.448
92.492
92.465
0.0188
0.0469
92.471Table 3
Ф9 mm experimental data of steel strands.
l/m
f1 (kHz)
f2 (kHz)
f3 (kHz)
f4 (kHz)
SD
RE (%)
f (kHz)
6.0
103.813
103.819
103.815
103.805
0.0057
0.0128
103.813
6.5
101.991
102.009
102.035
102.013
0.0180
0.0428
102.012
7.0
102.379
102.378
102.385
102.374
0.0047
0.0111
102.379
7.5
101.660
101.663
101.662
101.653
0.0044
0.0098
101.660
8.0
100.993
101.011
101.023
101.026
0.0149
0.0327
101.013
10.0
99.016
99.044
99.043
99.042
0.0136
0.0283
99.036
10.5
98.125
98.121
98.126
98.132
0.0047
0.0115
98.126
11.0
97.460
97.471
97.473
97.474
0.0068
0.0150
97.470
11.5
95.727
95.715
95.726
95.739
0.0098
0.0251
95.727
12.0
94.963
94.976
94.973
94.981
0.0073
0.0183
94.973
12.5
94.376
94.371
94.378
94.379
0.0035
0.0081
94.376
14.0
93.786
93.790
93.789
93.792
0.0023
0.0060
93.789
14.5
93.520
93.528
93.539
93.534
0.0081
0.0203
93.530
15.0
92.771
92.786
92.774
92.794
0.0106
0.0244
92.781
15.5
92.658
92.655
92.653
92.674
0.0096
0.0227
92.660
16.0
92.415
92.422
92.450
92.434
0.0152
0.0375
92.430In order to study the influence of strand length on circuit resonance frequency, experiments of different lengths of 7-Ф15.20 mm strands commonly used in engineering are carried out, ranging from 6–16 m. The experimental data are shown in Table4, and the analysis drawings are shown in Figure 11.Table 4
Ф15.2 mm experimental data of strand length.
l/m
f1 (kHz)
f2 (kHz)
f3 (kHz)
f4 (kHz)
SD
RE (%)
f (kHz)
6.0
103.907
103.909
103.900
103.912
0.0051
0.0115
103.907
6.5
103.183
103.183
103.213
103.225
0.0214
0.0407
103.201
7.0
102.522
102.536
102.538
102.540
0.0082
0.0176
102.534
7.5
101.394
101.395
101.372
101.379
0.0113
0.0227
101.385
8.0
100.745
100.734
100.753
100.748
0.0080
0.0189
100.745
10.0
98.675
98.641
98.652
98.648
0.0147
0.0345
98.654
10.5
97.485
97.462
97.502
97.479
0.0165
0.0410
97.482
11.0
97.287
97.293
97.285
97.283
0.0043
0.0103
97.287
11.5
96.640
96.643
96.642
96.635
0.0036
0.0083
96.640
12.0
95.778
95.784
95.776
95.774
0.0043
0.0104
95.778
12.5
95.482
95.446
95.447
95.485
0.0214
0.0409
95.465
14.0
94.153
94.139
94.162
94.158
0.0100
0.0244
94.153
14.5
93.705
93.698
93.695
93.714
0.0084
0.0203
93.703
15.0
93.138
93.164
93.127
93.123
0.0185
0.0440
93.138
15.5
92.490
92.476
92.512
92.470
0.0187
0.0454
92.487
16.0
92.211
92.202
92.200
92.223
0.0105
0.0249
92.209In order to study the effect of stress on resonance frequency and verify the theoretical formula, theσ-f experiment of 14 m 7-Ф15.20 mm steel strands is carried out. The experimental data are shown in Table 5, and the analysis diagram is shown in Figure 12.Table 5
14 m Ф15.20 mm experimental data of steel strands.
σ (Mpa)
f1 (kHz)
f2 (kHz)
f3 (kHz)
f4 (kHz)
SD
RE (%)
f (kHz)
71.94
95.007
95.024
95.012
95.025
0.0089
0.0189
95.017
100.72
95.073
95.066
95.080
95.070
0.0059
0.0147
95.072
143.88
95.169
95.166
95.177
95.178
0.0059
0.0126
95.173
201.44
95.232
95.236
95.231
95.239
0.0037
0.0084
95.235
215.83
95.250
95.241
95.248
95.246
0.0039
0.0094
95.246
287.77
95.270
95.266
95.260
95.264
0.0042
0.0105
95.265
302.16
95.279
95.290
95.293
95.293
0.0067
0.0147
95.289
359.71
95.369
95.371
95.361
95.376
0.0062
0.0157
95.369
402.88
95.396
95.392
95.397
95.403
0.0045
0.0115
95.397
431.65
95.446
95.451
95.453
95.452
0.0031
0.0073
95.451
503.60
95.484
95.480
95.481
95.486
0.0028
0.0063
95.483
575.54
95.568
95.564
95.568
95.567
0.0019
0.0042
95.567
604.32
95.554
95.551
95.553
95.558
0.0029
0.0073
95.554
647.48
95.575
95.574
95.579
95.580
0.0029
0.0063
95.577
705.04
95.606
95.599
95.605
95.608
0.0039
0.0094
95.605
719.42
95.637
95.630
95.634
95.635
0.0029
0.0073
95.634
791.37
95.657
95.656
95.658
95.663
0.0031
0.0073
95.659
805.76
95.659
95.655
95.647
95.660
0.0059
0.0136
95.655
863.31
95.693
95.692
95.698
95.687
0.0045
0.0115
95.693
906.47
95.726
95.733
95.756
95.731
0.0133
0.0313
95.737


## 3.4. Data Analysis

(1)
When the tension force of Ф9 mm steel strands is measured at 1 kN, the error rate of data repetition is controlled at 0.038%. Under the same conditions, the data repetition error rate of Ф9 mm diameter aluminum strands is controlled at 0.047% and that of Ф15.20 mm diameter steel strands is controlled at 0.045%. The steel strands with 14 m length are tested inσ-f experiment, and data duplication error rate is only 0.010%.
(2)
As shown in Figure10, Ф9 mm steel strands and aluminum strands access the circuit, and the resonant frequency of steel strands/aluminum strands with the same stress and length tends to be the same. When the length is 6 m, the maximum frequency difference is 1 kHz, and the frequency difference of other length specimens is very small. Error data when 6 m are considered. At the same time, the frequency of strands/aluminum strands is consistent with the length, that is, it decreases with the increase of length. Because the permeability of steel strands (ur = 1500) and aluminum strands (ur = 1) is quite different, therefore, both theoretical and experimental data are available. If other conditions are consistent, the difference of relative permeability of strands itself has little effect on resonance frequency. It can be concluded that the effect of stress change-relative permeability on resonance frequency is smaller.
(3)
Figure11 shows that under the same stress, the resonance frequency of Ф15.20 mm strands changes with the length; the frequency decreases linearly with the increase of length, and it is consistent with that of Ф9 mm steel strands/aluminum strands varying with length. Comparing l-f analysis with Figures 2 and 11, we can see that, under the same stress, the resonance frequency of different length strands decreases with the increase of length, and the experimental results are consistent with the theoretical ones.
(4)
Figure12 shows that the resonance frequency of 14 m Ф15.20 mm steel strands increases with the increase of stress. There is a good positive linear correlation between stress and frequency. The data fitting equation is as follows: f=7.94606×10−4σ+95.0528, R2=0.96752. Comparing with Figure 3, it can be found that there is a consistent correlation between experimental data and theoretical data in stress and frequency. But, the slope of fitting formula is 10 times that of theoretical formula. Considering that, under the action of stress, while the length of strands changes, the compactness of seven steel wires in the strand structure changes. Therefore, the increase of the length produced by the stress cannot be simply equal to the increase of the length of the strands. Further research is needed in the next study.
(5)
The theoretical and experimental data show that compared with the influence of specimen length on relative permeability, the influence of relative permeability on resonance frequency can be ignored. The smaller the natural length of strands is, the more significant the resonance frequency is. Therefore, it can be deduced that the effect of stress-changing permeability on resonance frequency can be ignored. The stress mainly changes resonance frequency by changing the length of steel strands.


## 4. Conclusions

The stress loss of the prestressed steel strands of the bridges in service makes the deflection of the structure to decrease and the main beam to crack, which seriously threaten the bearing capacity of the structure. It is of great significance to realize the detection of steel strand prestress. LC resonance connects the steel strands as an inductance through the conductor into the circuit and realizes the stress detection of the prestressed steel strands by testing the resonance frequency of the circuit. The previous research team has proved that the resonance frequency of the circuit has a good linear correlation with the stress of the steel strands, and this method has great potential for the stress detection of the prestressed steel strands. As a kind of magnetic material, the magnetic properties of the steel strands will change when it deforms under the stress. Therefore, it is very important to know which parameters affect the resonant frequency of the stress through changing. In this paper, the influence of permeability and length parameters on the resonant frequency of the circuit is studied in order to provide guidance for the future prestressing detection method.In this paper, the relationship between stress and relative permeability and length is analyzed theoretically. The theory of stress detection of prestressed steel strands based on the LC resonant circuit is modified and verified by experiments. The inductance in the circuit is simplified into three parts: the inherent inductance generated by the circuit, the inductance caused by different lengths of steel strands, and the inductance increment caused by stress change.Through theoretical analysis, it is found that the length is the main influence factor of resonance frequency. The theory is verified by establishing experiments of different lengths of steel strands and aluminum strands; according to the experiment, compared with the effect of length, the change of resonance frequency caused by the change of stress on permeability can be ignored. The stress mainly changes the resonance through the change of length; the resonance frequency has a negative linear correlation with the natural length, and the frequency decreases linearly with the increase of the natural length, and the experimental data are consistent with the theoretical law; the frequency of a certain length of steel strands is proportional to the stress, and the stress is linearly related to the frequency. The law of experimental data is consistent with the theoretical derivation.In the next research, we need to further study the influence of the length change of the stress on the resonant frequency. In addition, the research on the stress frequency of the prestressed concrete structure with bond is carried out to make further efforts to realize the stress detection of the steel strands of the prestressed concrete bridges in service.



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*Source: 9526843-2020-01-14.xml*