Post-tensioning tendon force estimation of in-service prestressed concrete structure using cylindrical lamb waves

Abstract

This paper proposes an ultrasonic-based force estimation technique for the post-tensioning (PT) tendon of an in-service prestressed concrete structure. First, three macro fiber composite transducers are installed on the surface of an in-service PT tendon subjected to an unknown tensile force for the generation and measurement of cylindrical Lamb waves. Then, the velocities of the longitudinal and shear waves are calculated using the measured cylindrical Lamb waves at two ultrasonic input frequencies. Subsequently, the Poisson’s ratio, elastic modulus, and strain induced by the PT tendon force are estimated from the longitudinal and shear wave velocities. Finally, the PT tendon force is obtained from the estimated mechanical property values. The experimental validation was performed using a mono-strand PT tendon under various temperature conditions, and the developed technique estimated the unknown PT tendon force in the range of 10–100 kN with a 6.22 kN maximum error and a 2.99 kN root-mean-square error. The uniqueness of this paper includes (1) no requirement for temperature compensation for field deployment, (2) PT tendon force estimation without calibration of the PT tendon force and cylindrical Lamb wave relationship, and (3) easy sensor installation and no interference with the PT construction process.

Keywords

Post-tensioning tendon in-service monitoring prestressed concrete structure force estimation cylindrical Lamb wave

Introduction

Post-tensioning (PT) construction is widely used to assemble prefabricated and modularized concrete segments using bundles of prestressing strands, which are also called PT tendons. Figure 1 shows an overview of constructing a prestressed concrete (PSC) structure using PT tendons. The PT tendons are inserted into the concrete segments through holes, and a tensile force is applied to the PT tendons to achieve an integrated structural behavior of the concrete segments. The applied PT tendon force decreases over time owing to concrete shrinkage, creep, steel relaxation, and corrosion after the completion of PT construction.¹ Excessive force reduction in PT tendon can lead to performance deterioration, concrete cracking, and catastrophic failure of PSC structures. Failures of PSC structures due to PT tendon force reduction have been previously reported: Lake View Drive Bridge, USA (2005); Jeong Leung Cheon Overpass Bridge, Republic of Korea (2016); and Morandi Bridge, Italy (2018).^2–4 Therefore, the PT tendon force estimation for in-service PSC structures is essential for maintaining structural integrity and preventing unforeseen structural failures.

Figure 1.

Overview of PSC structure construction using PT tendons.

Several nondestructive techniques have been developed for estimating the PT tendon force. Zhang et al.⁵ estimated the PT tendon force by measuring the inductance variation in a magnetoelastic inductance (MI) sensor based on the Villari effect. To estimate the unknown PT tendon force, the induction variation of the sensor should first be measured at known force levels to establish a relationship between the PT tendon force and the sensor inductance. Furthermore, the application of MI sensor to in-service PSC structures is limited because the MI sensor must be embedded in concrete segments during PT construction. Lan et al.⁶ inserted a fiber Bragg grating (FBG) optical fiber into a PT tendon and measured the reflected wavelength variation caused by the PT tendon force change. The PT tendon force was estimated using the linear relationship between the reflected wavelength and the PT tendon force. Because the FBG optical fiber must also be inserted during the PT construction, it is not applicable to in-service PSC structures. In addition, inserting an FBG optical fiber into a PT tendon is challenging. Kim et al.⁷ installed an eddy current sensor on the surface of a wedge holding a PT tendon at the anchor head and correlated the PT tendon force with the measured eddy current response. The eddy current response should be measured at a known PT tendon force and only the relative force reduction from the known PT tendon force can be estimated. X-ray diffraction can be used to estimate the PT tendon force using lattice pattern measurements of the PT tendon based on the Bragg’s law.⁸ To establish the relationship between the lattice pattern and PT tendon force, calibration tests on PT tendons should be performed in a well-controlled laboratory setting. Furthermore, X-ray diffraction cannot be used for PT tendons embedded in concrete because X-rays cannot penetrate deep into concrete segments. Abdullah et al.⁹ installed a strain gauge in an anchor head and experimentally examined the relationship between the relative strain and PT tendon force. This technique requires a calibration test to calculate the friction coefficient between the anchor head and wedge. In addition, this technique can detect only PT tendon breakage rather than PT tendon force estimation. In one study, an accelerometer was used to measure the natural frequency of the PSC girder, and the PT tendon force was estimated from the measured natural frequency.¹⁰ This technique is attractive because the accelerometer can be installed on the PSC girder, rather than directly on the PT tendon. However, this technique also requires calibration tests in a well-controlled setting to estimate the relationship between the natural frequency and the PT tendon force.

Recently, ultrasonic techniques have been proposed to estimate the PT tendon force using the wave velocity, nonlinearity, and energy entropy spectra of measured ultrasonic waves.^11–13 For example, an ultrasonic technique based on the acoustoelastic theory has been proposed to estimate the PT tendon force.¹¹ Nucera and Lanza di Scalea¹² estimated the PT tendon force using the nonlinearity of ultrasonic waves. Qian et al.¹³ experimentally derived the relationship between the energy entropy spectrum and PT tendon force and used this relationship to estimate the PT tendon force. However, conventional ultrasonic techniques require an initial calibration test to establish the relationship between PT tendon force and ultrasonic wave features. In addition, because the ultrasonic wave features are affected by the temperature as well as PT tendon force, temperature compensation of the ultrasonic waves is necessary for field applications.

To address the limitations of the conventional techniques, this paper develops an ultrasonic technique using cylindrical Lamb waves. Three macro fiber composite (MFC) transducers are installed on the PT tendon to generate and measure cylindrical Lamb waves when the PT tendon is subjected to an unknown tensile force, as shown in Figure 2. The velocities of the longitudinal and shear waves are computed using the measured cylindrical Lamb waves. Subsequently, the unknown PT tendon force is estimated from the velocities of the longitudinal and shear waves based on the acoustoelastic effect. The uniqueness of this paper includes (1) no requirement for temperature compensation for field deployment, (2) PT tendon force estimation without calibration of the PT tendon force and cylindrical Lamb wave relationship, and (3) easy sensor installation and no interference with the PT construction process.

Figure 2.

Schematic of cylindrical Lamb wave generation and measurement.

This paper is organized as follows: Section “Development of PT tendon force estimation technique for in-service PSC structure” presents the cylindrical Lamb wave propagation in the PT tendon and the proposed PT tendon force estimation technique. Section “Experimental validation” describes the experimental procedure and reports the test results. Section “Conclusion” concludes the paper with a summary and suggestions for future research.

Development of PT tendon force estimation technique for in-service PSC structure

Acoustoelastic effect of longitudinal and shear waves

Longitudinal and shear waves are ultrasonic bulk waves, in which their particles vibrate parallel and perpendicular to the wave propagation direction, respectively. These two waves can estimate the strain induced by an applied force based on the acoustoelastic effect, where the wave velocities are linearly correlated with the strain induced by the force. The velocities of the longitudinal and shear waves at zero force are defined as¹⁴

c_{L}^{0} = \sqrt{\frac{E^{0} (1 - ν)}{(1 + ν) (1 - 2 ν) ρ^{0}}} and c_{S}^{0} = \sqrt{\frac{E^{0}}{2 (1 + ν) ρ^{0}}}

(1)

where $c_{L}^{0}$ and $c_{S}^{0}$ denote the velocities of the longitudinal and shear waves at zero force, respectively; $ν$ is the Poisson’s ratio; and $ρ^{0}$ and $E^{0}$ are the density and elastic modulus and of the structure at zero force, respectively. Because $ρ^{0}$ is a material property of the PT tendon, it can be easily determined from the design standards.¹⁵ When $ρ^{0}$ is known, $ν$ and $E^{0}$ can be estimated via Equation (1) using $c_{L}^{0}$ and $c_{S}^{0}$ .

ν = \frac{{(c_{L}^{0} / c_{S}^{0})}^{2} - 2}{2 {(c_{L}^{0} / c_{S}^{0})}^{2} - 2}

(2)

and

E^{0} = \frac{c_{L}^{0^{2}} (1 + ν) (1 - 2 ν) ρ^{0}}{(1 - ν)}

(3)

When a tensile force is applied to the structure in the $x$ -direction, strain is induced, and the density and elastic modulus of the structure change. Consequently, the velocities of the longitudinal and shear waves changed proportionally with the strain. Previous studies have reported that the longitudinal wave is more sensitive to strain changes than the shear wave.¹⁶ Therefore, in this paper, the strain is estimated using the longitudinal wave. The relationship between the strain along the $x$ -direction and longitudinal wave velocity is given by the following equation¹⁶:

c_{L}^{F} = c_{L}^{0} [1 - ε_{x} (ν + 10)]

(4)

where $c_{L}^{F}$ is the velocity of the longitudinal wave when tensile force, $F$ , is applied to the structure in the $x$ -direction, and $ε_{x}$ is the strain induced by the tensile force along the $x$ -direction. To generate longitudinal and shear waves, the ultrasonic input frequency has to be high so that the wavelengths of these waves are smaller than the thickness of the structure.¹⁶ However, the higher the ultrasonic input frequency, the larger the attenuation and the shorter the wave propagation distance. Furthermore, the propagation of longitudinal and shear waves in the PT tendon are practically impossible because the PT tendon is composed of smaller spirally wound strands.¹⁷ Instead, cylindrical Lamb waves in a lower frequency range with larger wavelengths are generated within the PT tendon, and the velocities of the longitudinal and shear waves are numerically calculated from the cylindrical Lamb waves. A detailed procedure for calculating the velocities of the longitudinal and shear waves is provided in the next section.

Characteristics of cylindrical Lamb wave propagation in PT tendon

Cylindrical Lamb wave is an ultrasonic-guided wave propagating in cylindrical structures such as PT tendons, cylinder, and strand. Unlike longitudinal and shear waves, a cylindrical Lamb wave is generated even at a low ultrasonic input frequency and propagates long distances with less attenuation in the PT tendon.¹⁸ Figure 3 provides an overview of cylindrical Lamb wave propagation in the PT tendon. Three MFC transducers are installed on the PT tendon that is subjected to an unknown tensile force ( $F)$ . The MFC transducer is a flexible ultrasonic transducer that can be installed on curved surfaces such as the PT tendon. MFC 1 is used for cylindrical Lamb wave generation and MFCs 2 and 3 are used to measure the cylindrical Lamb waves propagating along the PT tendon. The locations of MFCs 2 and 3 are selected such that the cylindrical Lamb wave along MFC 1 to MFC 3 is affected by $F$ , whereas the waves along MFC 1 to MFC 2 are not affected by $F$ .

Figure 3.

Overview of cylindrical Lamb wave propagation.

The cylindrical Lamb waves propagating in the PT tendon exhibit dispersive characteristics, where the wave velocities depend on the wave mode, ultrasonic input frequency (f), and the velocities of the longitudinal and shear waves. The cylindrical Lamb waves are divided into longitudinal, torsional, and flexural modes based on wave propagation and particle movement motions. It has been shown that, among the three cylindrical Lamb wave modes, the longitudinal mode is the most sensitive to tensile force variation.¹⁸ Therefore, the PT tendon force estimation is performed using the longitudinal mode. In this paper, the longitudinal mode is named as the cylindrical Lamb wave to prevent confusion between the longitudinal wave of the bulk waves and the longitudinal mode of the cylindrical Lamb waves. The dispersive characteristics of the cylindrical Lamb wave can be described by the Pochhammer–Chree frequency equation¹⁹:

\begin{matrix} \frac{2 α}{a} (β^{2} + {k^{F}}^{2}) J_{1} (α a) J_{1} (β a) - {(β^{2} - {k^{F}}^{2})}^{2} J_{0} (α a) J_{1} (β a) \\ - 4 {k^{F}}^{2} α β J_{1} (α a) J_{0} (β a) = 0 \end{matrix}

(5)

where $a$ is the radius of the PT tendon and can be easily obtained from the design standard of PT tendon¹⁵; $k^{F}$ is the wavenumber of the propagated cylindrical Lamb wave when the $F$ is applied to the PT tendon; and $J_{0}$ and $J_{1}$ are the Bessel functions of the first kind of zero and one orders, respectively. $α$ and $β$ are defined as

α = \sqrt{\frac{w^{2}}{c_{L}^{F^{2}}} - {k^{F}}^{2}} and β = \sqrt{\frac{w^{2}}{c_{S}^{F^{2}}} - {k^{F}}^{2}}

(6)

where $w$ is the angular frequency of the propagated cylindrical Lamb wave and can be written as $2 π f$ . $c_{L}^{F}$ and $c_{S}^{F}$ denote the velocities of the longitudinal and shear waves, respectively, when the tensile force applied to the PT tendon is F. Equation (5) is a function of $c_{L}^{F}$ , $c_{S}^{F}$ , $f$ , and $k^{F}$ . When $c_{L}^{F}$ , $c_{S}^{F}$ , and $f$ values are given, $k^{F}$ is obtained by solving Equation (5) numerically. When the $k^{F}$ value is computed, the velocity of the cylindrical Lamb wave ( $c^{F})$ corresponding to a specific $f$ value is computed as follows¹⁹:

c^{F} = \frac{d w}{d k^{F}} ≅ \frac{2 π Δ f}{Δ k^{F}}

(7)

In this paper, $c^{F}$ is measured, and $c_{L}^{F}$ and $c_{S}^{F}$ are estimated by following the previous steps in reverse order. First, $c^{F}$ is measured at two distinctive input frequencies, $f_{2}$ and $f_{3}$ . The measured $c^{F}$ is then expressed as follows:

c_{f 2}^{F} = \frac{2 π Δ f}{k_{f 2}^{F} - k_{f 1}^{F}} and c_{f 3}^{F} = \frac{2 π Δ f}{k_{f 3}^{F} - k_{f 2}^{F}}

(8)

where $k_{fi}^{F}$ is the $k^{F}$ value corresponding to frequency, $f_{i}$ ( $i = 1, 2, 3$ ), $Δ f$ is the frequency difference between $f_{2}$ and $f_{3}$ , and $c_{f 2}^{F}$ and $c_{f 3}^{F}$ are the $c^{F}$ value measured at $f_{2}$ and $f_{3}$ , respectively. Then, five unknown values, $c_{L}^{F}$ , $c_{S}^{F}$ , and $k_{fi}^{F}$ are obtained by solving the following three Pochhammer–Chree frequency equations and two velocity equations (Equation (8)) simultaneously:

\begin{matrix} \frac{2 α_{f_{i}}}{a} (β_{f_{i}}^{2} + k_{fi}^{F^{2}}) J_{1} (α_{f_{i}} a) J_{1} (β_{f_{i}} a) - {(β_{f_{i}}^{2} - k_{fi}^{F^{2}})}^{2} J_{0} (α_{f_{i}} a) J_{1} (β_{f_{i}} a) \\ - 4 k_{fi}^{F^{2}} α_{f_{i}} β_{f_{i}} J_{1} (α_{f_{i}} a) J_{0} (β_{f_{i}} a) = 0 \end{matrix}

(9)

where $α_{f_{i}}$ and $β_{f_{i}}$ are the $α$ and $β$ values corresponding to the frequency $f_{i}$ .

PT tendon force estimation procedure

Figure 4 presents a flowchart of the developed technique for estimating the PT tendon force, which does not require calibration or temperature compensation.

Figure 4.

Flowchart of developed technique for estimating the PT tendon force.

First, the MFC transducers generate a cylindrical Lamb wave at two distinctive ultrasonic input frequencies ( $f_{2}$ and $f_{3})$ in the PT tendon and measure $c_{f 2}^{F}$ and $c_{f 3}^{F}$ . Second, the velocities of the longitudinal and shear waves are calculated using the $c_{f 2}^{F}$ and $c_{f 3}^{F}$ . Here, the density and elastic modulus of the PT tendon in the PSC structure are altered owing to the applied PT tendon force, whereas the PT tendon force outside the concrete is zero. Therefore, the $c_{f 2}^{F}$ and $c_{f 3}^{F}$ values measured in MFC 2 are constant regardless of the PT tendon force. The $c_{f 2}^{F}$ and $c_{f 3}^{F}$ values measured in MFC 3 are dependent on the PT tendon force. Therefore, $c_{L}^{0}$ and $c_{S}^{0}$ are calculated using the measured cylindrical Lamb wave at MFC 2, whereas $c_{L}^{F}$ and $c_{S}^{F}$ are calculated using the measured cylindrical Lamb wave at MFC 3. Subsequently, the $ν$ and $E^{0}$ values are estimated from Equations (2) and (3) using $c_{L}^{0}$ , $c_{S}^{0}$ , and $ρ^{0}$ , respectively. In this paper, $ν$ is assumed to be constant even when the tensile force is applied to the PT tendon. The validity of this assumption is examined in Section “Poisson’s ratio measurement under various tensile force levels.” Next, the strain ( $ε_{x})$ induced by the PT tendon force along the $x$ -direction is estimated as follows¹⁶:

ε_{x} = \frac{c_{L}^{0} - c_{L}^{F}}{c_{L}^{0} (ν + 10)}

(10)

Finally, the PT tendon force is obtained from the estimated mechanical property values:

F = A^{0} E^{0} ε_{x}

(11)

where $A^{0}$ is the cross-sectional area of the PT tendon at zero force, which can easily be determined from the dimensions of the PT tendon.¹⁵

Conventional ultrasonic techniques for PT tendon force estimation require calibration tests to establish the relationship between ultrasonic features and PT tendons. Furthermore, calibration tests should be performed under controlled temperature conditions because the ultrasonic features are affected by the temperature and PT tendon force. However, the developed technique can estimate the PT tendon force using only cylindrical Lamb waves measured at the current unknown tensile force without any calibration or temperature compensation.

Experimental validation

Experimental setup

The performance of the developed PT tendon force estimation technique was tested using the experimental setup shown in Figure 5. A custom-designed universal testing machine (UTM) consisting of a 2400 mm × 220 mm × 220 mm steel frame, a load cell, and a hydraulic actuator with a load capacity of 250 kN was manufactured, as shown in Figure 5(a). The rated measurement range of the load cell was 0–300 kN with a force resolution of 0.1 kN. A 3.3 m mono-strand PT tendon with $ρ^{0}$ of 8.09 g/cm³, $A^{0}$ of 138.7 mm², $a$ of 7.6 mm, and an allowable strength of 2160 MPa was inserted into the custom-designed UTM for the experimental validation. Three MFC transducers (M0714-P2 type) were attached to the PT tendon specimen to generate and measure the cylindrical Lamb wave, as shown in Figures 5(b) and (c). The maximum operating temperature of used MFC transducers is 85°C. The various tensile force levels were applied to the PT tendon specimen using a hydraulic actuator. The reference PT tendon force (ground truth) was measured using a load cell.

Figure 5.

Experimental setup: (a) custom-designed UTM with a PT tendon specimen, (b) schematic of MFC transducer installation, and (c) MFC transducers installed on the PT tendon specimen.

A National Instruments (NI) PXI system consisting of a control unit (NI PXIe-8840), a 14-bit arbitrary waveform generator (AWG, NI PXI-5421), and a 10-bit two-channel digitizer (DIG, NI PXIe-5160) was used for data acquisition. The AWG generated a three-cycle tone-burst ultrasonic input signal, and the DIG obtained the corresponding response of the cylindrical Lamb wave. The duration of each response was 0.07 μs, and each response was sampled at 2.5 GHz (time resolution of 0.4 ns). Figure 6 shows a theoretical dispersion curve of the cylindrical Lamb waves in the tested PT tendon.^15,20 The ultrasonic input signal was applied at six different frequencies (150–175 kHz in 5-kHz increments). The input frequency range is selected so that the longitudinal mode velocity changes drastically and the longitudinal mode does not overlap with the other modes, as shown in Figure 6. To improve the signal-to-noise ratio, the response was measured 500 times at each ultrasonic input frequency and averaged in the time domain. The representative responses of the cylindrical Lamb waves measured at 150 kHz are shown in Figure 7. Both cylindrical Lamb waves measured at MFC 2 and MFC 3 are clearly shown in Figure 7(a) and (b).

Figure 6.

Dispersion curve for cylindrical Lamb wave in PT tendon.

Figure 7.

Representative responses measured at: (a) MFC 2 and (b) MFC 3.

Velocities measurement of cylindrical Lamb wave under tensile forces of 0 and 100 kN

The velocities of cylindrical Lamb waves were measured under tensile forces of 0 and 100 kN using the following equation:

c_{fi}^{F} = \frac{d}{t}

(12)

where $d$ and $t$ are the cylindrical Lamb wave propagation distance and arrival time, respectively. The $t$ value was measured based on the peak point method.²¹ When a cylindrical Lamb wave passes through an anchor, attenuation occurs but the velocity changes little. Because the developed technique estimates the PT tendon force solely based on the cylindrical wave velocity, the anchor has little effect on the tensile force estimation.¹¹ The variation in the velocities with respect to the tensile force over the ultrasonic input frequency range of 150–175 kHz is presented in Figure 8. When a tensile force of 100 kN was applied to the PT tendon, the velocity measured at MFC 2 was almost constant compared with that measured at 0 kN. However, the velocity measured at MFC 3 decreased, as mentioned in Section “PT tendon force estimation procedure.” Therefore, $c_{L}^{0}$ and $c_{S}^{0}$ were calculated using the velocity measured at MFC 2, whereas $c_{L}^{F}$ and $c_{S}^{F}$ were calculated using the velocity measured at MFC 3.

Figure 8.

Variation in the velocities measured at (a) MFC 2 and (b) MFC 3 when the tensile force in the precast concrete (metal frame in this case) increased from 0 and 100 kN.

Poisson’s ratio measurement under various tensile force levels

The assumption that the $ν$ value is constant even when the tensile force is applied to the PT tendon is examined in this section. Two strain gauges (resistance-type, FLA-1-11-1LJC) were installed on the PT tendon specimen to measure the strain in the tensile force direction and the direction transverse to the tensile force, as shown in Figure 9. The strains were measured using a strain input module (NI-9237) with a 100 kHz sampling rate when the tensile force was increased. A tensile force varying from 0 to 100 kN in 10-kN increments was applied to the PT tendon specimen. The $ν$ value was calculated as follows:

ν = - \frac{strain in the transverse direction}{strain in the tensile force direction}

(13)

Figure 9.

Experimental setup to measure Poisson’s ratio under various tensile force levels.

Figure 10 shows the $ν$ values measured at various tensile force levels. The average value of the $ν$ values measured under 10 tensile force levels was 0.2519. The maximum error between the measured and averaged $ν$ value was 0.0014, and the difference between the minimum and maximum measured $ν$ value was 0.0022. Furthermore, no clear relationship was observed between the measured $ν$ value and the applied tensile force. Therefore, the assumption of a constant $ν$ value under varying tensile forces is acceptable.

Figure 10.

Measurement of Poisson’s ratio under various tensile force levels using strain gauges.

Poisson’s ratio estimation

The $ν$ value of the PT tendon was estimated using the cylindrical Lamb wave measured at MFC 2 under 10 different tensile force levels that ranged from 10 to 100 kN in 10-kN increments. The $ν$ value at each tensile force level was estimated as follows:

ν = \frac{1}{5} \sum \frac{{(\frac{c_{L}^{0 j, j + 5}}{c_{S}^{0 j, j + 5}})}^{2} - 2}{2 {(\frac{c_{L}^{0 j, j + 5}}{c_{S}^{0 j, j + 5}})}^{2} - 2} (j = 150, 155, 160, 165, 170 kHz)

(14)

where $c_{L}^{0 j, j + 5}$ and $c_{S}^{0 j, j + 5}$ are the velocities of the longitudinal and shear waves at zero force estimated from the cylindrical Lamb wave measured at ultrasonic input frequencies $j$ and $j + 5 kHz$ , respectively. From the measured velocities of the cylindrical Lamb wave, $c_{L}^{0 j, j + 5}$ and $c_{S}^{0 j, j + 5}$ were obtained by numerically solving Equations (8) and (9) using the modified Newton–Raphson method.²² The estimated $ν$ values are shown in Figure 11. The average $ν$ value measured using the two strain gauges in Section “Poisson’s ratio measurement under various tensile force levels” was used as the reference value. The developed technique successfully estimated the $ν$ value of the PT tendon regardless of the tensile force with a maximum error of 0.0173.

Figure 11.

Poisson’s ratio estimation using Equation (2) of the developed technique.

Elastic modulus estimation at zero force

$E^{0}$ was estimated at each tensile force level using the cylindrical Lamb wave measured at MFC 2, as described in Section “Poisson’s ratio estimation.” To obtain the reference $E^{0}$ value, the stress–strain curve was obtained using the strain gauge and load cell built into the custom-designed UTM, as shown in Figure 12(a). Through linear regression of the stress and strain curve, after imposing the constraint that $y$ -intercept should be zero, the reference $E^{0}$ value was 209.9 GPa. The corresponding $E^{0}$ values were estimated using the following equation:

E^{0} = \frac{1}{5} \sum \frac{c_{L}^{0 j, j + 5} (1 + ν) (1 - 2 ν) ρ^{0}}{(1 - ν)}

(15)

Here, $ν$ at the current tensile force level was calculated using Equation (14). The $c_{L}^{0 j, j + 5}$ values were obtained as described in Section “Poisson’s ratio estimation.” The results of estimating the $E^{0}$ value under various tensile force levels are shown in Figure 12(b). The maximum error in the $E^{0}$ value estimation was 3.9 GPa. These estimated results were in good agreement with the reference $E^{0}$ value. As the strain gauge was not perfectly aligned with the direction of the tensile force, the measured strain value was smaller than the actual strain induced by the tensile force. Consequently, the estimated $E^{0}$ values were lower than the reference $E^{0}$ value.

Figure 12.

Elastic modulus at zero force estimation: (a) reference value measurement from stress–strain curve and (b) estimation using the developed technique.

PT tendon force estimation

The PT tendon force was estimated using cylindrical Lamb waves from 10 different tensile force levels (10–100 kN in 10-kN increments) under various temperature conditions. The laboratory temperature was controlled using the air conditioner and based on daily ambient temperature variations (16.25–29.33°C), and the temperature of the PT tendon was measured using a thermocouple, as shown in Figure 13.

Figure 13.

Temperature measurement of PT tendon using a thermocouple.

The PT tendon force was estimated using a cylindrical Lamb wave measured only at each tensile force level without any calibration or temperature compensation. At each tensile force level, the combinations of the ultrasonic input frequencies were identical to those described in Section “Poisson’s ratio estimation.” The reference PT tendon force (ground truth) was measured using a load cell of a custom-designed UTM. For quantitative performance estimation, the maximum error and root-mean-square error (RMSE) values were calculated as follows:

RMSE = \sqrt{\frac{1}{5} \sum {(F - \hat{F^{j, j + 5}})}^{2}}

(16)

where $F$ and $\hat{F^{j, j + 5}}$ are the reference PT tendon force measured by the load cell and the PT tendon force estimated using the developed technique using $c_{fj}^{F}$ and $c_{fj + 5}^{F}$ , respectively.

Figure 14 shows the estimation results of the PT tendon force for ten different force levels at various temperatures. On average, the maximum error and RMSE values were 4.42 and 2.87 kN, respectively. The blue dashed line represents the average value of the PT tendon force estimated for the five different input frequency combinations. The maximum error between the average PT tendon force and reference PT tendon force was 2.95 kN, as shown in Figure 14(e). The experimental results show that the developed technique accurately estimated the PT tendon force under various temperature conditions without calibrating the relationship between the PT tendon force and velocity of the cylindrical Lamb wave in advance.

Figure 14.

Estimation of PT tendon force under various temperatures from (a) 10 kN to (j) 100 kN.

Conclusion

This paper presents the development of a technique to estimate PT tendon force using a cylindrical Lamb wave to prevent the failure of in-service PSC structures. First, the cylindrical Lamb wave propagating to the PT tendon is measured and the velocity is calculated. Thereafter, the velocities of the longitudinal and shear waves are calculated by numerically solving nonlinear equations using the velocity of the cylindrical Lamb wave. Subsequently, the Poisson’s ratio, elastic modulus at zero force, and strain induced by unknown force are estimated from the velocities of the longitudinal and shear waves. Finally, the PT tendon force is obtained from the estimated mechanical properties without temperature compensation and calibration test for establishing the relationship between the PT tendon force and the features of the cylindrical Lamb wave. The performance of the developed PT tendon force estimation technique was experimentally investigated using blind test data obtained at ten different tensile force levels under various temperature conditions. The results of blind tests show that the average maximum error and the RMSE values were 4.42 and 2.87 kN, respectively. The uniqueness of the developed technique includes (1) no requirement for temperature compensation for field deployment, (2) PT tendon force estimation without calibrating the PT tendon force and cylindrical Lamb wave relationship, and (3) easy sensor installation and no interference with the PT construction process.

Although the developed technique sheds light for tensile force estimation of in-service PSC structures, the readers should be fully aware of the following limitations before actual field applications. First, the performance of the developed technique was verified only for ungrouted PT tendons. Therefore, further investigation is warranted for grouted PT tendons. Second, a more reliable PT tendon force estimation technique is required to compensate for the variation of the cylindrical Lamb wave velocity caused by the corroded segment of PT tendon. Future research is warranted to address these problems and improve the suitability of the developed technique for actual field applications.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIP) (No. 2017R1A5A1014883).

ORCID iD

Hoon Sohn

Data availability statement

The datasets generated during and/or analyzed during the current study are not publicly available due to privacy concerns and institutional policy but are available from the corresponding author on reasonable request.

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