110Equation Chapter 0 Section 1Absolute Stress Measurement of Structural Steel Members with Ultrasonic Shear Wave Spectral Analysis Method

Zuohua Li1, Jingbo He1, Jun Teng1, Qin Huang1, Ying Wang2

Abstract Absolute stress in structural steel members is an important parameter for the design,

construction, and servicing of steel structures. However, it is difficult to measure via

traditional approaches to structural health monitoring. The ultrasonic time-of-flight

(TOF) method has been widely studied for monitoring absolute stress by measuring the

change of ultrasonic propagation time induced by stress. The TOF of the two separated

shear-wave modes induced by birefringence, which is particular to shear waves, is also

affected by stress to different degrees. Their synthesis signal amplitude spectrum exhibits

a minimum that varies with stress, which makes it a potential approach to evaluating

uniaxial stress using the shear-wave amplitude spectrum. In this study, the effect of steel-

member stress on the shear-wave amplitude spectrum from the interference of two shear

waves produced by birefringence is investigated, and a method of uniaxial absolute stress

measurement using shear-wave spectral analysis is proposed. Specifically, a theoretical

expression is derived for the shear-wave pulse-echo amplitude spectrum, leading to a

formula for evaluating uniaxial absolute stress. Three steel-member specimens are 1Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, People’s Republic of China2Department of Civil and Environmental Engineering, University of Surrey, Guildford GU2 7XH, UK

Corresponding author: Qin Huang, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, People’s Republic of ChinaE-mail: tmxhq@126.com

employed to investigate the influence of uniaxial stress on the shear-wave pulse-echo

amplitude spectrum. The testing results indicate that the amplitude spectrum changes

with stress, and that the inverse of the first characteristic frequency in the amplitude

spectrum and its corresponding stress exhibit a near-perfect linear relationship. On this

basis, the uniaxial absolute stress of steel members loaded by a test machine is measured

by the proposed method. Parametric studies are further performed on three groups of steel

members made of 65# steel and Q235 steel to investigate the factors that influence the

testing results. The results show that the proposed method can measure and monitor steel-

members uniaxial absolute stress on the laboratory scale and has potential to be used in

practical engineering with specific calibration.

KeywordsUniaxial absolute stress monitoring, structural steel members, shear-wave birefringence,

amplitude spectrum, acoustoelastic effect

Introduction

Absolute Stress and Stress Measurement Methods

Steel is widely used in civil engineering structures for its high strength and toughness.

Many large structural-bearing members, such as huge steel columns, giant steel beams,

and large joints, are made of steel. The use of steel improves structural load resistance

and prolongs structural service life. The safe and proper service of the overall structures

is directly affected by the key steel members. When steel structures have been serviced

for a long time or have suffered natural disasters, their spatial stress state may be

redistributed; in particular, stress concentration may occur in key components. This may

lead to abnormal functioning and even failures of aging structures. Absolute stress in

structural steel members plays an important role in evaluating the safety of existing

structures. Evaluating the absolute stress of structural steel members is of great

significance in judging structural safety performance, predicting structural service life,

and determining aging structural reinforcement and reconstruction.1

Structural health monitoring technologies can be used to monitor the structural state by

applying different types of stress/strain sensor to key members.2–4 In recent years, crack

identification5 of local components, damage detection, and vibration control of structures6

have been widely studied. However, studies that investigate the application of measuring

the absolute stress of structural steel members remain rare in the literature. The stress

measured in traditional way is the relative value from the start time of monitoring to the

current state, rather than the absolute stress.7 If the stress/strain sensors are applied at the

beginning of structure construction, the absolute stress can be measured. However, the

absolute stress distribution cannot be measured over long-term servicing of the structures

because the stress monitoring system and sensors have a certain lifetime. After replacing

the stress monitoring system and sensors, the absolute stress of structures still cannot be

monitored. Generally speaking, the absolute stress of steel members is difficult to be

measured using traditional structural health-monitoring approaches.

Stress measurement methods1 such as sectioning, hole drilling, X-ray diffraction, neutron

diffraction, the magnetoelastic method, and ultrasound can be used to measure absolute

stress. The sectioning8 and hole-drilling9 methods are both destructive to members and

thus are not recommended for use on existing structures. X-ray diffraction,10 neutron

diffraction,11 and the magnetoelastic method12 can only be used on the laboratory scale

and are generally not suitable for on-site applications because the required equipment is

complex. However, ultrasound is not limited by the types of material under study and can

be used with portable instruments to measure absolute stress. The existing literature13

shows that ultrasound can be used to measure the internal stress of steel members and is

regarded as a promising approach for the non-destructive evaluation of absolute stress.

Ultrasonic Stress Measurement Methods

The ultrasonic method is based on the acoustoelastic effect.14 In the elastic range, the

velocity of ultrasound varies linearly with material stress when a wave propagates in the

material. However, because the change in ultrasonic velocity due to stress is too small to

be measured, the TOF method15 is proposed by transforming the relationship between

stress and velocity to one between stress and TOF along a fixed acoustic path. Thus, the

material stress can be evaluated by measuring the ultrasonic TOF. According to the

different ultrasonic wave patterns, ultrasonic stress evaluation based on TOF

measurement can be subdivided into the following methods.

The first method involves the use of longitudinal waves. The estimation of clamping

force is regarded as the main issue in the maintenance of high-tension bolts. Because the

ultrasonic wave velocity changes minimally in the range of actual stress in the bolt, the

determination of stress requires precise and accurate measurement of the wave velocity.

Jhang et al.16 proposed a phase-detection method to measure the TOF of longitudinal

waves precisely, and the experimental results showed that longitudinal-wave TOF

decreases linearly with stress. Karabutov et al.17 used an optoacoustic transducer to excite

and detect longitudinal ultrasonic pulses, leading to TOF measurements with an accuracy

of around 0.5 ns. This technique was used to measure plane residual stress in welds, and

the results agreed well with those of conventional testing. Vangi18 employed a pulse-echo

technique to evaluate concentrated stress gradients. Combining with automated ultrasonic

inspection techniques, an approach to evaluate the structural integrity of components or

structures was presented19. Overall, the longitudinal-wave method has been widely

studied and used.

The second method involves the use of shear waves. Schramm20 measured the velocity of

two orthogonally polarized ultrasonic shear waves, which allowed the calculation of rim

stress in railroad wheels; the results were consistent with those obtained by destructive

methods. Clark and Moulder21 measured the velocity difference of orthogonally polarized

shear-waves, which can be related to the difference in principal stresses; the

experimentally determined specimen stresses were in good agreement with theoretical

values. Allen and Sayers22 measured residual stress in textured steel by measuring the

time delay between two orthogonally polarized shear waves used in the birefringence

technique; the method was further demonstrated by measuring near the tip of a crack in a

compact tension specimen. The use of the shear-wave method to measure stress has

unique advantages. By varying the shear-wave velocity simultaneously in two different

polarization directions, the two-dimensional stress distribution in the propagation

direction of the shear wave can be obtained.23

The third method employs a critically refracted longitudinal (Lcr) wave to measure stress.

The Lcr wave is a bulk longitudinal wave that travels underneath the surface of the

material, with the penetration depth being a function of excitation frequency.24 The Lcr

technique does not require opposite parallel surfaces and so does not impose any strict

geometric limitations on the test specimens. Of all ultrasonic waves, Lcr waves exhibit

the largest sensitivity to stress and are least affected by material texture.25 All these

features make Lcr waves the best candidate for use in stress evaluation. Bray and

Junghans24 and dos Santos and Bray25 used Lcr waves to detect the residual stresses of an

aluminum welded seam and a steel welded seam to study the relaxation of welding

residual stress. Chaki et al.26 investigated characterization of the Lcr beam profile both

numerically and experimentally to optimize the choice of incident angle to obtain

sufficient energy in the experimental signal. Javadi and colleagues23,27 used Lcr waves to

measure the distribution of welding residual stress in an austenitic stainless-steel member

and verified the experimental results by hole drilling. Combining experimental results

with those of numerical simulations using ANSYS, the distribution of welding residual

stress was obtained at different depths, and a stress-distribution diagram was drawn. Li et

al.13 used Lcr waves to measure the absolute stress of existing structural steel members,

and the testing results agreed well with those measured using a strain gauge. Vangi28

presented an ultrasonic critical-angle reflectometry technique to determine the velocity of

the longitudinal wave by measuring the amplitude of the refracted signal in relation to the

angle of incidence. This technique can be applied to the measurement of stress states and

is not based on the TOF data, which offers the possibility of a significant increase in

spatial resolution. In general, applications of the Lcr wave method have been widely

researched in recent years. In addition to the aforementioned stress measurement

methods, Lamé waves,29 Rayleigh waves,30 and surface waves31 are also used to evaluate

stress.

Generally speaking, the majority of studies on ultrasonic TOF measurement are still in

their infancy, with a limited number of industrial applications. There are three possible

reasons for this. First, ultrasonic propagation characteristics are influenced by factors

such as grain size,32 material texture,33 and coupling conditions,34 all of which influence

the change in ultrasonic wave velocity or TOF more so than does absolute stress. In

particular, the propagation times of ultrasonic waves in the probe and coupling layer

cannot be measured accurately,13,23,27 leading to inaccurate values of detected stress.

Second, the TOF method requires high-precision measurement of the travelling times of

the ultrasonic waves. Based on previous studies, the ultrasonic wave velocity is not very

sensitive to stress. For example, a 100-MPa stress change corresponds to a velocity

change of less than 1%.16 Thus, the results of measuring absolute stress depend directly

on the TOF measurement accuracy. To solve this problem, the sing-around method35 was

proposed to reduce the TOF measurement error by extending the ultrasonic propagation

distance. However, the energy of ultrasonic waves is seriously attenuated when they

travel a long distance.36 Third, some parameters require calibration in field experiments,

and the fitted results are inaccurate. In fact, the fitted parameters contain information

such as material density, elastic modulus, and elastic constant,13,23,27 all of which are

greatly influenced by the component material. Still, there are a few industrial applications

of the ultrasonic method. For example, Vangi37 designed a methodology to remotely

monitor thermal stresses on continuous welded rails. The methodology has been used to

monitor about 3 km of railway track.

The ultrasonic TOF method has some intrinsic defects, leading some scholars to propose

the use of ultrasonic shear waves to test stress. This method is based on the birefringence

of shear waves.38 The presence of stress in a solid material leads to acoustic anisotropy.

When an ultrasonic shear wave encounters a stressed solid, it separates into two modes

that propagate simultaneously with different velocities and produce interference effects.

The shear-wave echo contains both fast and slow components of the two shear-wave

modes, and its amplitude spectrum exhibits a periodic minimum value that sensitive to

stress and therefore could be used to indicate the stress in the material. Blinka and

Sachse39 investigated the interference between two decomposed shear waves in stressed

aluminum, and found that the echo power spectrum exhibited periodic stress-induced

minima. In addition, their experimental results showed that material stress and the

frequency corresponding to the minimum value were linearly related. That research laid

the foundation for stress measurement using spectral analysis. One advantage of this

method is that it is less affected by environmental noise. However, research into the use

of shear-wave birefringence to measure absolute stress has yet to yield a systematic

method for doing so. In fact, it is inaccurate to claim a perfectly linear relationship

between stress and frequency in the power spectrum. In general, stress measurement

using shear-wave birefringence requires further investigation.

Goals and Objectives of This Study

Accepting the above difficulties, the aim of this paper is a new non-destructive evaluation

approach to determine the uniaxial absolute stress of structural steel members using

spectrum analysis of ultrasonic shear waves. In accordance with ultrasonic shear-wave

birefringence and acoustoelasticity theory, a formula is derived for uniaxial stress

measurement. An important parameter, the interference factor, is emphatically analyzed.

Expressions for shear wave frequency and polarization angle are solved theoretically

when the shear-wave echo amplitude spectrum exhibits a minimum value. On this basis,

three steel members are tested using the proposed method and the results are validated.

The influencing factors, namely the material and the thickness of the steel members, are

investigated by combining theoretical and experimental studies on three other groups of

specimens made of 65# steel and Q235 steel.

Theory

Theoretical Derivation of Shear-wave Pulse-echo Amplitude Spectrum Based on Birefringence Effect

In a free state, the velocity of ultrasonic shear waves in an acoustically isotropic metallic

material is independent of the directions of propagation and polarization. When a metallic

material is subjected to stress, the stress causes acoustic anisotropy of the material. When

a beam of shear waves travels from one medium to another, it separates into two shear-

wave modes at the interface. Shear waves whose polarization directions are parallel and

perpendicular to the stress direction have different velocities. This phenomenon is known

as birefringence, and is particular to shear waves. In this paper, we first derive a

theoretical expression for the amplitude spectrum of a shear-wave pulse echo in a

stressed steel member.

Figure 1 shows the propagation of shear waves generated and received in a parallelepiped

specimen. The contact point between the shear-wave probe and the steel member is the

origin o. The propagation direction of the shear wave is in the positive direction of the

horizontal axis x3. The angle between the polarization of the shear wave and the

horizontal axis x1 is θ. The length of the steel member along the horizontal axis x3 is l,

and the direction of the steel member’s force is in the direction of the vertical axis x2. For

the sake of illustration, compressive stress of the steel members is positive and tensile

stress is negative.

σ

σl

x1

x2

x3

Transceiver shearwave probe

θo

Figure 1. Sketch of shear-wave propagation in steel member

According to Fourier analysis, a shear-wave pulse can be regarded as the superposition of

sine or cosine waves with different frequencies. The equation of motion for the shear

wave can be written as

. (1)

Here, u0 is the amplitude of vibration source, Ai is the amplitude of the ith vibrational

component, wi is the angular frequency of the ith vibrational component, φi is the initial

phase of the ith vibrational component, and t is the vibration time of the vibration source.

Because steel is birefringent, shear waves containing fast and slow components parallel

to the x1 and x2 axes propagate through the specimen in the x3-direction. After being

reflected from the specimen’s rear side, the shear wave travels back to the source. The

equations of motion for two components propagating in the positive x3 direction are given

by38

, (2)

,(3)

where vij is the velocity of shear waves traveling in direction i with particle vibration in

direction j. Synthesis of the two wave components parallel to the polarization direction θ

received by the transducer (x3=0) is

. (4)

Let

, (5)

, (6)

where Δt is the arrival time difference of the two components. Equation (4) can then be

written as

. (7)

We define U(f ) and G(f ) to be the Fourier transforms of ur(t) and g(t), respectively.

Transforming Eq. (7), a relationship is obtained between the shear-wave echo power

spectrum and frequency:

. (8)

According to Euler’s formula, Eq. (8) can be written as

. (9)

Taking the modulus of Eq. (9), the following expression is obtained:

.(10)

Here, |U(f )| is the shear-wave echo amplitude spectrum when the steel member is under

stress. It contains both fast and slow shear-wave components; |G(f )| is the reference

spectrum, which is the amplitude spectrum of an echo when the steel member is under

zero stress. The remaining part in Eq. (10) can be defined as the interference factor, the

expression of which can be written as

. (11)

The interference factor holds information on the interference between the fast and slow

shear-wave components, and is a function of polarization angle and frequency. The

functional image is shown in Fig. 2 when the arrival time difference of the two shear-

wave components is 0.1 μs. It can be seen that the interference factor varies periodically

with polarization angle and frequency. The value range of the interference factor is [0, 1].

Figure 2. Functional image of the interference factor

Equation (10) can be further simplified as

. (12)

Equation (12) is the theoretical expression of the shear-wave echo amplitude spectrum; it

is the product of the reference spectrum and the interference factor. After adjusting the

reference spectrum by the interference factor, the amplitude spectrum can be obtained. In

fact, when the steel member is under zero stress, the velocities of the two shear-wave

components are equal and there is no interference. Then, the interference factor is 1, and

the echo amplitude spectrum is the same as the reference amplitude spectrum. When the

steel member is under stress, the interference factor changes periodically with

polarization angle and frequency.

Next, we deduce theoretical expressions for the shear-wave polarization angle and

frequency when the interference factor is a minimum. To obtain this minimum value, we

use the following:

. (13)

The solution of Eq. (13) is

, (14)

where f * is defined as the characteristic frequency when the interference factor is a

minimum. The characteristic frequencies corresponding to N1=1, 2, 3, … are recorded as

f1*, f2

*, f3*, …, respectively, which are referred to as the first characteristic frequency

(FCF), the second characteristic frequency, the third characteristic frequency, ...,

respectively.

By substituting Eq. (6) into Eq. (14), a new formula is obtained:

. (15)

Equation (15) illustrates the fact that the characteristic frequency is influenced by the

velocities of the two shear-wave components. Because the shear-wave velocities are

influenced by the stress in the steel member, the characteristic frequency is similarly

affected. Thus, the relationship between characteristic frequency and stress is established

in this way.

By substituting Eq. (13) into Eq. (11), the latter becomes

, (16)

whereupon the shear-wave polarization angle can be solved as

. (17)

Figure 3 shows the relationship between the extreme value of the interference factor and

the polarization angle. The maximum value of the interference factor is constant at 1. The

minimum value changes periodically with polarization angle every 90°. When the

polarization angle is an odd multiple of 45°, the minimum value of the interference factor

is 0. The analysis results are consistent with those shown in Fig. 2.

Figure 3. Relationship between polarization angle and extreme value of interference factor

Theoretical Derivation of Uniaxial Absolute Stress Measurement Based on Acoustoelasticity Effect

In this study, the material of the structural steel member is assumed to be isotropic and

homogeneous. Furthermore, it is assumed to be in its elastic range. According to

acoustoelastic theory, the shear-wave velocity in the steel member is related to the

particle vibration and stress direction.23 The specific expressions are as follows:

, (18)

. (19)

Here, v31 is the velocity of the shear wave when its propagation and particle-vibration

directions are perpendicular to the stress direction; v32 is the velocity of the shear wave

when its propagation and particle-vibration directions are perpendicular and parallel,

respectively, to the stress direction; λ and μ are the second-order elastic constants (Lamé

constants); l, m, and n are the third-order elastic constants (Murnaghan constants); and ρ0

is the material density.

When the steel member is under zero-stress condition, the velocity of the shear-wave is:

, (20)

where v0 is the shear-wave velocity when the steel member is under zero stress.

It can be deduced from Eqs. (18) and (19) that

. (21)

By substituting Eq. (20) into Eq. (21), it becomes:

. (22)

In fact, the ultrasonic wave velocity is not sensitive to stress conditions of the steel

members40. Based on previous experimental studies41, when the strain of a steel sample is

about 1‰ (the stress is about 210 MPa), the relative difference of velocity of two types of

shear waves are about 1.15‰ and -0.2‰. The term in Eq. (22) is thus equal to

1.0005, which is approximately equal to 1. Referring to the derivation process by Allen

and Sayers42, Eq. (22) is simplified as the following form

. (23)

Considering that the initial state of the steel material may not be isotropic, a factor, α, is

included to reflect the initial acoustic anisotropy of the steel member. Then the following

formula can be obtained:

. (24)

By substituting Eq. (15) into Eq. (24), the latter becomes

, (25)

where l0 is the initial thickness of steel member; ν is Poisson’s ratio; and E is elastic

modulus.

Let

, (26)

, (27)

whereupon Eq. (25) can be simplified as follows:

. (28)

In Eq. (26), parameter κ’ is a function of stress. For structural steel members, the stress

range of the commonly used steel materials is from 0 MPa to 500 MPa43,44. When a steel

member is under the stress of 500 MPa, with ν=0.26 and E=205 GPa, then based on Eq.

(26), the maximum influence of stress on the variation of parameter κ’ is only 0.6‰.

Further, the maximum transverse deformation is about 0.634‰ of its initial thickness,

which has little influence on the parameter κ’, and little effect on the interference of the

two separated shear-wave modes.

Therefore, for the convenience of parametric calibration, we neglect the influence of the

transverse deformation and assume l is equal to l0. Then Eq. (26) can be simplified as

. (29)

where t0 is the TOF, and equal to l0/v0.

To determine the optimum N1 value in Eqs. (15) and (28), the following tests are carried

out. A shear-wave transceiver probe is attached on the surface of the steel member of

dimensions 50.00 mm × 29.71 mm × 19.27 mm. The polarization angle of the shear wave

is 45°. The first echo signals are collected when the steel-member stresses are zero and

400 MPa. The amplitude spectra are normalized by the spectrum of the shear-wave

transducer whose central frequency, f0, is about 5 MHz. Specifically, the peak amplitude

for this spectrum for the unloaded sample is defined as unity. Their amplitude spectra are

shown in Fig. 4.

Figure 4. The normalized amplitude spectra for steel-member stresses of zero and 400 MPa when polarization angle is 45°

Fig. 4 shows the stress-induced change in the energy distribution of the shear-wave echo

amplitude spectrum; the change is most obvious at shear-wave frequencies around

5.7 MHz, which is the FCF. It should be noted that when the shear wave polarization

angle is different from an odd multiple of 45°, the frequency corresponding to the

minimum value of interference factor is also the FCF, as shown in Fig. 2. But its

amplitude spectrum does not change as obviously as that when the shear-wave

polarization angle is 45°. Because the amplitude-spectrum energy is centered on the low-

frequency band (0–10 MHz) and the FCF is in the range of 0–10 MHz (i.e., the spectrum

energy concentration region), the change associated with it is most easily observed. To

facilitate the FCF data collection, we set N1 = 1. By integrating Eq. (29), Eq. (28) can be

further simplified to the following form:

, (30)

which is the proposed formula for measuring uniaxial absolute stress in the steel member,

and indicates that uniaxial absolute stress is a function of the FCF. By calibrating the

linear relationship between uniaxial stress and the inverse of FCF, the parameters κ and γ

can be obtained.

Experimental Studies

Absolute Stress Measurement System

Based on the above theory, a measurement system is designed and developed in this

study. A schematic diagram and a photograph of the developed system are shown in

Figs. 5 and 6, respectively. As can be seen, there are four components in the system,

specifically ① an ultrasonic generator (5072PR; Olympus, USA); ② a shear-wave

transceiver probe (Olympus, USA; central frequency: 5 MHz); ③ an oscilloscope

(MDO3024; Tektronix, USA); the sampling rate is 100 MSa/s in all experiments; and ④

a universal testing machine (DNS300; Changchun Research Institute for Mechanical

Science Co., Ltd., China). In addition, a GW-type-III high-temperature ultrasound

coupler is used to effectively couple the probe to the surface of the steel member.

The main objective of the measurement system is to accurately collect the shear-wave

echo signals. To achieve this, the following process is proposed. The ultrasonic generator

transmits pulse signals and then shunts them into the probe to generate a pure shear wave.

The shear wave propagates in the steel member and is then reflected from its rear side.

After being collected by the same probe, the echo signal is transmitted to the ultrasonic

generator and displayed on the oscilloscope. The first pulse echo signal of the collected

data is collected, and Fourier analysis is used to study the signal’s amplitude spectrum.

Echo signal output

Transmitting signal

Receiving signal LoadingSteel member

12

3

Figure 5. Measurement system: schematic diagram

1

2

3

4

4

Figure 6. Measurement system: photograph

Test Sample

In this study, experiments were carried out on Q235 steel and 65# steel, which are used

not only in steel structures but also in mechanical welding and parts machining. They are

the main materials of structural load-bearing components and mechanical processing. The

dimensions and materials of the specimens are summarized in Table 1. To reduce the

lateral restraint of the machine bearing pad, lubricant is smeared on the end surface of the

specimen before loading. All tests were carried out at room temperature.

Determination of Shear Wave Polarization Angle

Equation (10) shows that the shear-wave polarization angle exerts a direct influence on

the echo amplitude spectrum. Because Figs. 2 and 3 show that the interference factor

varies periodically with polarization angle, we study the variation of the echo amplitude

spectrum only for changes in the polarization angle in the range of [0, 45°].

Table 1. Material and dimensions of test samples

Name Material Dimensions

Sample A 65# steel 60.0 mm×30.0 mm×20.0 mm

Sample B 65# steel 50.0 mm×30.0 mm×20.0 mm

Sample C 65# steel 55.0 mm×30.0 mm×18.0 mm

Sample D 65# steel 60.0 mm×30.0 mm×16.0 mm

Sample E 65# steel 60.0 mm×30.0 mm×18.6 mm

Sample F Q235 steel 50.0 mm×30.0 mm×20.0 mm

Sample G Q235 steel 55.0 mm×30.0 mm×18.0 mm

Sample H Q235 steel 60.0 mm×30.0 mm×16.0 mm

Sample I 65# steel 45.0 mm×30.0 mm×12.8 mm

Sample II 65# steel 45.0 mm×30.0 mm×13.6 mm

Sample III 65# steel 45.0 mm×30.0 mm×14.4 mm

Sample IV 65# steel 45.0 mm×30.0 mm×15.2 mm

Sample V 65# steel 45.0 mm×30.0 mm×16.0 mm

Sample VI 65# steel 45.0 mm×30.0 mm×16.8 mm

Sample VII 65# steel 45.0 mm×30.0 mm×17.6 mm

Sample VIII 65# steel 45.0 mm×30.0 mm×18.4 mm

Sample IX 65# steel 45.0 mm×30.0 mm×19.2 mm

Sample A is chosen as the research object. The shear-wave probe is coupled to the

surface of sample A and the polarization angle is set to 0°. The universal testing machine

is used to apply loads on sample A to generate the stresses of zero, 100 MPa, 200 MPa,

300 MPa, and 400 MPa. The first echo signals under each stress condition are recorded.

We adjust the polarization angle to 15°, 30°, and 45°, and repeat the above process in

each case. By using a Fourier transform, we obtain the amplitude spectra under different

stress conditions and polarization angles, as shown in Fig. 7.

a) θ = 0° b) θ = 15°

c) θ = 30° d) θ = 45°

Figure 7. Amplitude spectra under different polarization angles

It can be seen that the amplitude spectra are most obviously affected by stress when the

shear-wave polarization angle is 45°, which verifies the correctness of Eq. (17), Fig. 2

and Fig.3. If the shear-wave polarization angles are different from 45°, the FCF in the

amplitude spectrum can also be collected, but the change of amplitude spectrum is not as

obvious as that of the polarization angle is 45°, which may lead to greater errors.

Therefore, the polarization angle is determined to be 45°, which is beneficial for

capturing the FCF accurately. It should be noted that in general it is not so immediate to

know the appropriate polarization angle to adopt. Eq. (16) shows that when shear-wave

polarization is an odd multiple of 45°, the minimum amplitude-spectrum value is 0. We

can change the polarization angle to observe in which direction the minimum value in

amplitude spectrum is 0. And this direction is the appropriate polarization angle. In this

study, all the steel members are subject to axial stress and the shear-wave polarization

angle is 45°.

Calibration of Parameters

Equation (30) indicates that the parameters κ and γ must be calibrated before measuring

the absolute stress in the steel member. Equations (27) and (29) illustrate that κ and γ are

related to the shear-wave propagation path and the steel-member material. Therefore,

parameter calibration must be done on specimens of the same thickness and material as

those of the tested object. In this study, samples B, C, and D were used to calibrate

parameters κ and γ. The experimental setup is shown in Fig. 6.

Samples B, C, and D were loaded in compression from zero stress, and each load was

held for 10 min. Because the amplitude-spectrum values corresponding to the FCF are

most obviously affected by stress when the shear-wave polarization angle is an odd

multiple of 45°, the shear-wave polarization angle is set as 45° during the calibration of

parameters. Then, the shear-wave echo signals and the corresponding stress data were

recorded. The first pulse-echo time-domain signals were converted into the amplitude

spectrum using a Fourier transform. As the amplitude spectra of sample B, C and D

influenced by stress is nearly the same, the amplitude spectra of sample B is provided as

an example, as shown in Fig. 8. The FCFs were extracted from the amplitude spectra, and

the FCF and the corresponding stress value under each load state were obtained. Using

the least-squares method, linear relationships between the stress and the inverse of FCF

were fitted, and the parameters κ and γ were calibrated as shown in Fig. 9.

Figure 8. Illustration of amplitude spectra of sample B influenced by stress

Figure 9. Fitting lines between stress and the inverse of FCF for samples B, C, and D

Validation of the Proposed Method

Based on Eq. (30) and the calibrated parameters κ and γ, the absolute stress of the

structural steel member can be calculated via measurement of the FCF. Actually, it is

difficult to measure steel-member absolute stress using traditional methods, which causes

difficulties in verifying the proposed method. Therefore, the specimens loaded by the

testing machine were taken as the measurement objects. Under laboratory conditions, a

set of random loads was applied to samples B, C, and D using the universal testing

machine. The shear-wave polarization angle was set to 45°. The echo signal under each

loading condition was recorded, and the applied stress was measured using the proposed

method. The measurement results are listed in Table 2.

To verify the stress measured by the proposed method, the loading indicators of the

universal testing machine were recorded to calculate the true stress values of the

specimens. The true stress values of samples B, C, and D are listed in Table 2.

Table 2. Comparison between two methods for samples B, C, and D

Sample B Sample C Sample D

Proposed method

Testing machine value

DifferenceProposed method

Testing machine value

DifferencePropose

d method

Testing machine value

Difference

(MPa) (MPa) (%) (MPa) (MPa) % (MPa) (MPa) %

29.53 28.42 3.91 25.71 25.07 2.55 26.62 25.92 2.70

59.18 57.60 2.74 48.41 47.05 2.89 50.38 49.46 1.86

85.44 81.67 4.62 64.92 62.89 3.23 73.96 72.46 2.07

105.72 108.76 2.80 90.47 92.39 2.08 98.20 96.28 1.99

131.18 127.60 2.81 113.39 109.69 3.37 123.39 124.01 0.50

158.84 153.58 3.42 138.64 141.85 2.26 144.05 139.48 3.28

172.51 175.25 1.56 159.06 153.37 3.71 168.45 164.56 2.36

199.45 192.37 3.68 183.17 176.37 3.86 183.49 179.92 1.98

222.74 230.99 3.57 206.55 213.06 3.05 204.43 197.82 3.34

249.87 242.52 3.03 233.96 227.69 2.75 228.88 221.83 3.18

271.03 263.23 2.96 258.84 254.35 1.77 253.29 256.83 1.38

297.13 288.16 3.11 283.32 274.31 3.28 279.49 272.36 2.62

318.35 312.84 1.76 304.58 306.83 0.73 304.48 299.21 1.76

344.15 334.89 2.77 327.68 319.03 2.71 327.54 319.91 2.39

367.52 356.38 3.13 352.49 346.21 1.81 351.74 344.61 2.07

393.21 380.31 3.39 371.50 362.44 2.50 376.61 366.88 2.65

Investigation of Influencing Factors

Equations (27) and (29) indicate that parameters κ and γ are influenced by the material,

and that parameter κ is also related to steel-member thickness. To further investigate the

effects of steel-member material and thickness on the results, extensive experimental

studies were conducted and are reported in this section.

Materials

The influence of probe position on the parametric fitting results was studied. Samples E

and VIII were used for this analysis; the corresponding fitting positions are shown in

Fig. 10. The values of κ and γ were fitted using the proposed method, as shown in

Fig. 11. The fitted parameter values of the different locations are listed in Table 3.

a) Sample E b) Sample VIII

Figure 10. Fitting positions on samples E and VIII

a) Sample E b) Sample VIII

Figure 11. Fitting lines between stress and the inverse of FCF for samples E and VIII at positions ** Expression is faulty **–** Expression is faulty **

Table 3. Fitted parameters values for samples E and VIII at different locations

Parameter κ(MPa·MHz) γ(MPa)

Samples E

Position ① 11793.86 1567.54

Position ② 11737.57 1559.14

Difference 0.48% 0.54%

Samples VIII

Position ③ 12325.13 1550.03

Position ④ 12302.16 1546.12

Difference 0.19% 0.25%

The influence on the parametric fitting results of steel members made of different

materials was further studied. Two groups of specimens were employed here. The first

group of specimens made of 65# steel contains samples B, C, and D, while the second

group of specimens made of Q235 steel contains samples F, G, and H. Their fitted linear

relationships are shown in Figs. 9 and 12, respectively. The fitted parameter values are

listed in Table 4.

Figure 12. Fitting lines between stress and the inverse of FCF for samples F, G, and H

Table 4. Fitted parameter values of two groups of specimens

Parameter κ(MPa·MHz) γ(MPa)

Samples B & F

Sample B 11054.36 1556.01

Sample F 7597.33 1157.64

Difference 37.07% 29.36%

Samples C & G

Sample C 13007.81 1550.65

Sample G 8194.17 1152.76

Difference 45.41% 29.44%

Samples D & H

Sample D 15155. 69 1548. 45

Sample H 9292. 68 1144.22

Difference 47.96% 30.02%

Steel Member Thickness

Equation (29) shows that parameter κ is inversely proportional to the shear-wave

propagation TOF, that is, parameter κ is inversely proportional to steel-member

thickness. Samples I–IX were employed to investigate the influence of steel-member

thickness on parameter κ. Parameter κ and the corresponding steel-member thickness

were measured using the proposed framework as shown in Fig. 13 and Table 5. The

fitting trend line between parameter κ and steel-member thickness is shown in Fig. 14.

Figure 13. Fitting lines between stress and the inverse of FCF for samples I–IX

Table 5. Fitted parameter values for samples I–IX

Nameκ

(MPa·MHz)

γ

(MPa)R2

Sample I 20090.31 1553.96 0.9996

Sample II 19669.68 1548.42 0.9995

Sample III 16969.39 1565.37 0.9994

Sample IV 15552.84 1556.00 0.9996

Sample V 15155.69 1548.45 0.9971

Sample VI 14402.77 1568.06 0.9996

Sample VII 13567.69 1561.73 0.9997

Sample VIII 12325.13 1550.03 0.9997

Sample IX 11309.08 1548.24 0.9997

Figure 14. Fitting trend line between parameter κ and steel-member thickness

Results and Discussion

Influence of Shear-wave Polarization Angle on the Amplitude Spectrum

Figure 7(a) indicates that the amplitude spectra are almost the same for a polarization

angle of 0°. Theoretically, when the shear-wave polarization angle is 0° in Eq. (11), the

interference factor is constant at 1. The echo amplitude spectrum and the reference

spectrum are then the same. However, the FCF appears in the amplitude spectra for

polarization angles of 15°, 30°, and 45°. As shown in Fig. 7(b)–(d), the FCF values

decrease with stress, a phenomenon that can be explained by Eq. (14). When the stress in

sample A increases, the TOF difference between the two shear-wave components also

increases, whereupon the FCF values decrease. It means that the proposed method can

measure absolute stress with different polarization angles (except 0°). By comparing the

amplitude spectra in Fig. 7(a)–(d), it can be further found that the amplitude spectra do

not change with polarization angle when sample is unstressed. It demonstrates that the

initial anisotropy exerts little influence on the measurement process and that stress causes

the change of the amplitude spectra.

Parameter Calibration and Absolute Stress Measurement Results

Figures 9, 11–13 show that the inverse of FCF and its corresponding stress in the steel

members exhibit an almost perfect linear relationship with a fitting error of less than 1%.

The fitting results verify the correctness of Eq. (30). The absolute-stress measurement

results in Table 2 show that the difference between the stresses measured by the two

methods is less than 5% for every single test. This demonstrates the reliability and

accuracy of the proposed method in the laboratory. Because the amplitude-spectrum

energy is centered on the low-frequency band, high-frequency noise exerts little influence

on the change of FCF.

To facilitate validation of the proposed method, the loaded specimens were taken as the

measurement objects. It should be pointed out that the stress distribution of a real

structural steel member is different from that of the loaded specimen. Structural steel

members are generally slender and have both ends fixed. In many cases, the steel

members are mainly subjected to axial force. Therefore, to verify the proposed method, it

is reasonable to some extent to use loaded specimens as the measured objects instead of

structural steel components.

Actually, structural steel members are not removable after installation, which makes

parametric calibration difficult. Since the calibration of parameters should ideally be

performed on the original on-site structural steel member while it cannot be removed, a

duplicate steel member with the same material properties and thickness to the on-site

structural steel member is needed for parametric calibration. In such scenario, the

parameters of the duplicate steel member and the on-site structural steel member need to

be assumed as the same. But there are limitations in this assumption. Most importantly,

there should be no damage on the on-site structural steel member and duplicate steel

member, because the effect of fatigue, corrosion, and other kinds of damage may change

the amplitude spectrum.

Influence of Materials on Parameter Calibration

Figure 11 and Table 3 show that the fitted linear functions are almost identical for the

same specimen, with a difference of less than 1%, which shows that the fitting-point

location exerts little influence on the fitting results. The small difference may be due to

the fitting process and can usually be ignored. Therefore, the probe position has little

influence on the parametric calibration results.

The fitted values of parameters κ and γ in Table 4 are quite different even though

specimens B, C, and D are the same sizes as specimens F, G, and H, respectively. The

difference is caused by the material, which illustrates that the specimen material exerts a

great influence on the two parameters. The essential reason for this is that the second- and

third-order elastic constants of 65# steel and Q235 steel are very different, which can

directly influence the calibration results. This matches with the conclusion from the

theoretical study as given by Eq. (27) and Eq. (29). Figures 9, 11, and 13 show that the

fitted values of parameter γ for specimens made of 65# steel are around 1555.21 MPa,

with a maximum error of 0.65%. This illustrates that specimens made of the same

material have little influence on the calibration of parameter γ. The little difference may

be caused by the initial birefringence of the material and can be ignored.

Influence of Steel Member Thickness on Parameter Calibration

Figure 14 illustrate that the fitted value of parameter κ decreases in inverse proportion to

the specimen thickness. Because specimen thickness is proportional to shear-wave TOF,

the fitted value of parameter κ is inversely proportional to shear-wave TOF. The change

trend of the fitted line in Fig. 15 verifies the correctness of Eq. (29). However, Fig. 14

also shows discrepancies between the fitted function and the experimental results, which

confirms the complexity of the problem. The discrepancies may come from fitting

parameter κ to the data, which will introduce certain fitting errors. The fitting of

parameter κ with initial errors can lead to greater errors. In fact, the quantitative

relationship between parameter κ and specimen thickness requires further study, which in

turn requires accurate measurement of the second- and third-order elastic constants of the

steel members.

ConclusionsThe main aim of this study is to evaluate the uniaxial absolute stress of steel members. A

shear-wave spectrum analysis method was employed to reach this goal. According to the

achieved results, the conclusions are summarized as follows.

1) Absolute stress in a steel member causes interference between the fast and slow shear-

wave components induced by birefringence, leading to a change in the shear-wave echo

amplitude spectrum. This establishes the basis for measuring the uniaxial absolute stress

of a steel member.

2) The proposed method focuses on the uniaxial stress evaluation of steel-member, rather

than biaxial stress. The relationship between the inverse of the FCF and its corresponding

uniaxial stress showed a near-perfect linear relationship. The uniaxial stresses of steel

members loaded by a test machine were measured. Based on the results, acceptable

agreement was observed between those from the proposed method and those from the

test-machine loading results. The present method measures the change of the FCF in the

amplitude spectrum, and does not consider the variation of shear-wave velocity or TOF

directly.

3) The probe position has little influence on the absolute-stress measurement results,

whereas the material properties and thickness of steel members have significant effects

on the fitted parameters. Therefore, parametric calibration is a necessary step for the

evaluation of steel-member uniaxial stress. If the to-be-tested steel member is in service,

parametric calibration should be performed on specimens of the same material and

thickness as the measured steel member.

4) The proposed method is non-destructive, and the equipment costs are low compared

with those of X-ray diffraction or the magnetoelastic method. The experimental

verification of this method is implemented on the laboratory scale. With specific

calibration, the method has potential to be used in practical engineering situations in

limited cases. Future research efforts can be placed on:

1) the influence of biaxial and complicated stress on the shear-wave pulse-echo

amplitude spectrum and the biaxial stress measurement method;

2) the combined effects of damage and stress on the shear-wave pulse-echo amplitude

spectrum;

3) the quantitative relationship between parameter κ and specimen thickness by

measuring the second- and third-order elastic constants of the steel members;

4) the influence of environmental factors, such as temperature, on measurement results.

Declaration of Conflicts of InterestThe author(s) declared no potential conflicts of interest with respect to the research,

authorship, and/or publication of this article.

FundingThe author(s) disclosed receipt of the following financial support for the research,

authorship, and/or publication of this article: This work was financially supported by

National Nature Science Foundation of China under Grant 51538003 and 51378007,

Shenzhen Technology Innovation Program under Grant JSGG20150330103937411 and

JCYJ20150625142543473.

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