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Techniques based on ultrasonic guided waves (UGWs) play important roles in the structural health monitoring (SHM) of large-scale civil infrastructures. In this paper, dispersion equations of longitudinal wave propagation in reinforced concrete member are investigated for the purpose of monitoring steels embedded in concrete. For a steel bar embedded in concrete, not the velocity but the attenuation dispersion curves will be affected by the concrete. The effects of steel-to-concrete shear modulus ratio, density ratio, and Poisson’s ratio on propagation characteristics of guided wave in steel bar embedded in concrete were studied by the analysis of the real and imaginary parts of the wave number. The attenuation characteristics of guided waves of steel bar in different conditions including different bar concrete constraint and different diameter of steel bar are also analyzed. Studies of the influence of concrete on propagation characteristics of guided wave in steel bars embedded in concrete will increase the accuracy in judging the structure integrity and promote the level of defect detection for the steel bars embedded in concrete.

In recent years, techniques based on ultrasonic guided waves (UGWs) have gained popularities and played important roles in the structural health monitoring (SHM) of large-scale civil infrastructures as guided waves have some important advantages, such as the capability of testing over long range with a greater sensitivity, the ability to test multilayered structures, and relatively cheapness due to simplicity and sensor cost [

Among the current available corrosion monitoring techniques in reinforced concrete, the technique based on ultrasonic guided wave has gained more and more popularity in the recent years due to its advantages for monitoring corrosion related damage in reinforcing bars, so it has gained popularities in the recent years [

All the above test results indicated that the received waveform is less attenuated with the increase in debonding for both low and high frequencies. However, the lower frequencies showed more sensitivity to the change in bond. There was no significant change in the waveform arrival time reported. The location of debonding is not discernible through pulse transmission as reported by Evin et al. [

These studies were mostly carried out with simulated debonding. In these cases, it is in fact the free steel bar within the concrete. There is difference in the propagation characteristics of guided waves between the free steel bars and the steel bar surrounded by concrete. However, the research on the effects of concrete on propagation characteristics of guided wave in steel bar embedded in concrete is less reported, because there is a difficulty of limitation of monitoring range of guided wave in reinforced concrete [

In an infinite isotropic solid medium, only two types of independent wave propagation exist, that is, compression and shear waves. Both waves propagate with constant velocities and are nondispersive. When geometry constraints are introduced and the dimensions are close to the wavelength, the wave becomes dispersive and is called a guided wave. Longitudinal, torsional, and flexural waves propagate in isotropic cylinders. The characteristic equation for solid isotropic cylinders was originally independently derived for the special case of longitudinal propagation in the late 19th century [

In civil infrastructures, steel bars are usually embedded in concrete, so the existence of concrete is of strong interference with the integrity evaluation of steel bars, which lead to the difficult problem in the detection of steel bars using guided wave based techniques [

For a steel bar embedded in concrete, the guided waves propagate not only along the steel bar but also spread outward, which leads to the wave energy that diffuses from steel bar into the concrete. Furthermore, the change of wave impedance of concrete can also cause the reflection of the waves. Thus, the wave equations of guided waves in steel bar-concrete system can be established based on the 3D elastic wave theory.

The simplified model of a steel bar embedded in concrete is shown in Figure

Simplified model of steel bar embedded in concrete.

Displacement of guided waves propagating in steel bar can be expressed by Bessel function [

It is assumed that bonding condition at the interface between steel bar and concrete is good and the displacement and stress at the interface are continuous, so

Then, substituting displacement and stress into (

If there is a nonzero solution to (

There are axisymmetric and nonaxisymentric modes of waves included in the propagation of guided waves in the steel bar [

When subdeterminant

When subdeterminant

Further, the following definitions are introduced [

The following variables are introduced:

Other dimensionless quantities are given by

Then, the following equations can be derived from the theory of elasticity [

On the substitution of these variables into (

If the parameters of steel bar and concrete are known, only two unknown variables (dimensionless frequency and dimensionless wave number) are involved in (

Related material parameters of steel bar and concrete.

Parameters | Steel bar | Ordinary concrete | High strength concrete |
---|---|---|---|

Poisson’s ratio | 0.2865 | 0.27 | 0.20 |

Density (kg/m^{3}) |
7932 | 2200 | 2400 |

Elastic modulus (MPa) | 210000 | 22000 | 38000 |

Shear modulus (MPa) | 81600 | 8600 | 15800 |

Velocity of longitudinal |
5960 | 3540 | 4190 |

Velocity of transverse |
3260 | 1980 | 2570 |

Density ratio |
3.6 | 3.3 | |

Shear modulus ratio |
9.5 | 5.2 |

In order to solve the relationship between dimensionless frequency and dimensionless wave numbers in the dispersion equation (

Find the cutoff frequency of each order mode from a sweep frequency with a specified range, which is the initial root of the equation and also the initial point of the disperse curve of each order mode. Generally, there will be the same number of order modes as that of initial points which have been found within a specified range of frequency, and the corresponding number of disperse curves can be obtained.

The routine starts from the initial point, in the given step and direction, and converges to the next point which is the second root of the equation.

Take the second root as the initial point and repeat step 2 and so forth. The routine will end until the specified frequency or upper limit wave number is reached.

Connect all the found points and a disperse curve will be achieved after.

Repeat the above process from another initial point and another disperse curve will be achieved in this way. The complete solution to the equation will be gotten until all the disperse curves are finished.

Based on the solutions, the disperse curves between dimensionless frequency and dimensionless wave number of the round steel bar embedded in concrete with different material parameters will be gained by solving (

Disperse curves of the first three modes of guided waves in steel bar in the air.

Disperse curves of the first three modes of guided waves in steel bar embedded in ordinary concrete.

Disperse curves of the first three modes of guided waves in steel bar embedded in high strength concrete.

From Figures

It can also be drawn from Figures

As shown in Figures

From the disperse curves in Figure

The impact of concrete on the propagation characteristics of guided wave in steel bar can be evaluated by changing the material properties of concrete, which include the factors such as steel-to-concrete shear modulus ratio, density ratio, and ratio of Poisson’s ratio. Furthermore, from previous studies by the authors [

When the steel bar is embedded in ordinary concrete, the shear modulus ratio is about 10, which will increase when the surrounding concrete becomes softer. Until it is up to a certain extent, the boundary of steel bar can be regarded as a free boundary. By changing the ratio, the relationship between dimensionless wave number and dimensionless frequency can be gained as shown in Figure

Influence on the real and imaginary parts of

From Figure

When the density of concrete is changed and other parameters remain constant, the disperse relation will surely be affected. While steel bar is embedded in high strength concrete, the steel-to-concrete density ratio is 3.6. Then, decreasing gradually the density of concrete, the steel-to-concrete density ratio will be increased and the change of the disperse curve between dimensionless wave number and dimensionless frequency will be gotten as shown in Figure

Influence on the real and imaginary parts of

From Figure

As Poisson’s ratio of steel bar and concrete can be changed and large changes in the Poisson’s ratio of concrete from low grade to high grade take place, the effect on the disperse curve by Poisson’s ratio can be gotten by assuming Poisson’s ratio of steel bar as a constant of 0.2865 and changing that of concrete, as shown in Figure

Influence on the real and imaginary parts of

From Figure

Through the previous definition and analysis, it usually regarding the disperse curve between the imaginary part of dimensionless wave number and dimensionless frequency as the attenuation curve. In fact, the forward propagation of guided waves in steel bar is a process in which energy attenuates gradually due to the effects of constrains of concrete, impact excitation energy, material damping of steel, and defects on steel bar. If there are much residues or defects in steel bar or its characteristics is close to the surrounding concrete (close density or shear modulus), the waves will vanish due to attenuation after a short distance of propagation and then it will be very difficult to detect the defect of steel bar using the traditional echo method. Based on the research of the attenuation curve, the energy dissipation of guided waves propagating along the steel bar and the effects on the attenuation by material properties, propagation distance, and frequency can be known more clearly, which will be a benefit for the defect detection in the reinforced concrete.

When the steel bar is in the air (free steel bar), the solved imaginary wave number is very small and even close to zero because there is great difference in the physical properties between steel bar and the air. Therefore, its propagation mode is approximated free or no attenuation mode.

For the steel bar embedded in concrete, the imaginary wave number depends on the shear modulus ratio and density ratio between steel bar and concrete. The energy loss of wave can be quantified by the displacement

Thus, the attenuation of guided wave along the steel bar can be measured using the amplitude variation of the signal along the

In (

The disperse curves for steel bar with the diameter of 22 mm embedded in ordinary concrete are shown in Figure

Disperse curves of guided waves in steel bar embedded in ordinary concrete.

Phase velocity disperse curves

Energy velocity disperse curves

Attenuation disperse curves

By comparing the energy velocity curve to phase velocity curve in Figure

From the phase velocity disperse curves in Figure

By (

Attenuation coefficient and the signal amplitude ratio of mode

Propagating distance (m) | 0 | 1 | 1.5 | 2 | 2.5 | 3 | 5 |
---|---|---|---|---|---|---|---|

Attenuation (dB) | 0 | −18 | −27 | −36 | −45 | –54 | −90 |

Amplitude ratio | 1.000 | 0.1259 | 0.0450 | 0.0159 | 0.0056 | 0.0020 | 0.00003 |

As can be seen from Table

Figure

Mode shape of

The disperse curves for steel bar embedded in high strength concrete with the diameter of 22 mm are shown in Figure

Disperse curves of guided waves in steel bar embedded in high strength concrete.

Phase velocity disperse curves

Energy velocity disperse curves

Attenuation disperse curves

The phase velocity disperse curves in Figure

Attenuation coefficient and the signal amplitude ratio of mode

Propagating distance (m) | 0 | 1 | 1.5 | 2 | 2.5 | 3 | 5 |
---|---|---|---|---|---|---|---|

Attenuation (dB) | 0 | −36.8 | −55.2 | −73.6 | −92 | −110.4 | −184 |

Amplitude ratio | 1.000 | 0.0150 | 0.0017 | 0.0002 | 0.00002 | 3.0 × 10^{−6} |
6.2 × 10^{−10} |

Figures

Mode shape of

Mode shape of

From the analysis, it can be drawn that the higher the strength concrete, the more the energy leakage into concrete, the faster the attenuation of guided waves, and the smaller the amplitude ratio of signals at the same position. Guided waves at low frequency mode can propagate very far along the steel bar in the air for the little attenuation, so a few meters of steel bar can be monitored by guided waves based technique [

For the given reinforced concrete model, the disperse curves of attenuation of longitudinal mode

Attenuation disperse curves of mode

As can be seen from Figure

In reinforced concrete structures, the propagation characteristics of guided waves in steel bar are influenced greatly by its surrounded concrete. In this paper, the impact of concrete on the propagation characteristics of guided wave in steel bar is studied by changing the material properties of concrete. It is found that the higher the excitation frequency, the more obvious the dispersion phenomenon. However, not the velocity but attenuation dispersion curves will be affected by the concrete. The shear modulus and density of concrete have no effect on the real parts of wave number; that is, the propagating velocity of guided waves in steel bar depends only on the material properties of steel bar and has nothing to do with the surrounding concrete. The imaginary part of wave number depends on the steel-bar-to-concrete shear modulus ratio and density ratio. When the concrete has lower shear modulus or density, the spread of the wave attenuation in steel bar is slow and the transmission distance is further.

For a given steel-concrete model, the attenuation extent in signal propagation can be obtained by the attenuation and frequency dispersion curve. It is found that, for the same steel bar with larger diameter embedded in the same concrete, the energy leaks less into concrete, the spread of wave energy attenuation is slower, and the spread distance is shorter. Studying the influence on propagation characteristics of guided wave in steel bars by concrete will increase the accuracy in judging the structure integrity and promote the level of defect/corrosion detection for the steel bars embedded in concrete.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research is supported by Key Science and Technology Project from Fujian Province, China (no. 2013Y0079) and the Research Fund SLDRCE10-MB-01 from the State Key Laboratory for Disaster Reduction in Civil Engineering at Tongji University, China.