Experimental Study on Residual Bending Strength of Corroded Reinforced Concrete Beam Based on Micromagnetic Sensor

Abstract

This paper presents a nondestructive test method to evaluate the residual bending strength of corroded reinforced concrete beam by analyzing the self-magnetic flux leakage (SMFL) signals. The automatic scanning device was equipped with a micromagnetic sensor and sensor-based experimental details were introduced. Next, the theoretical formula of the normal component H_S(z) of the SMFL signal that originated from the corroded region was derived based on the magnetic dipole model and the experimental results were discussed. The results indicate that the experimental data of H_S(z) are consistent with the theoretical calculations, both location and extent of the steel bars corrosion can be qualitatively determined by using H_S(z). The gradient K of H_S(z) is approximately linearly related to the loss rate, S, of the bending strength, which can be used to evaluate the residual bending strength of the corroded reinforced concrete beam. This work lays the foundation for evaluating the residual bending strength of corroded reinforced concrete beams using the SMFL signal; the micromagnetic sensor is further applied to the civil engineering.

1. Introduction

Reinforced concrete structures are widely used in civil engineering due to their high bearing capacity, low cost, and easy construction. However, long-term exposure to the external aggressive environment, means that the reinforcement steel bars of the reinforced concrete structure will be corroded inevitably. The steel corrosion will lead to reduction of the steel bar cross-section, failure of the bond between the steel bar and the concrete, and deterioration of the concrete, which will ultimately weaken the bearing capacity and durability of the reinforced concrete structure [1,2,3]. Therefore, the measurement of corroded steel bar is crucial to determine the bearing capacity and durability of the reinforced concrete structures.

In the past, traditional electrochemical methods were used to measure corrosion of steel bars, but there are many limitations with these methods. In recent years, smart sensor-based methods that are useful to measure different defects (corrosion, cracks, etc.) of ferromagnetic materials (steel bars, etc.) and other materials have become increasingly popular, such as fiber optic coil winding [4], fiber Bragg grating [5,6], laser scanning technology [7], line scanning thermography (LST) [8,9], inductive thermography [10,11], optical excitation thermography [12], and other thermographic methods [13] such as acoustic emission and ultrasound testing using traditional and novel transducers [14,15]. Especially, the electromagnetic nondestructive test (NDT) including magnetic flux leakage (MFL) [16], metal magnetic memory (MMM) [17], magnetic acoustic emission (MAE) [18], magnetic Barkhausen noise (MBN) [19], and self-magnetic flux leakage (SMFL) [20] are widely used to measure corrosion and defects of ferromagnetic materials.

Based on the magnetic mechanical effect in the weak magnetic field, factors such as stress and defects within the ferromagnetic material will cause SMFL on its surface by orienting the magnetic domains, dislocating the lattices, and distorting the magnetic field lines [20,21]. Steel bars are a typical kind of ferromagnetic material; the corrosion of steel bars will reduce the bearing capacity and change the SMFL signals of the reinforced concrete structure at the same time. Based on this unified change, the relationship between the bearing capacity and the SMFL signals of corroded reinforced concrete structure can be established. Compared with MFL, SMFL has the advantages of rapid, effective, and simpler operation without any external excitation magnetic field required [22]. Liu [23,24,25] established a full electronic potential magneto-mechanical model of the ferromagnetic material and explored the relationship between magnetic memory signal and the stress concentration defects. Zhang [26,27] studied the relationship between the SMFL signals and the corrosion extent of corroded reinforced concrete. Zhang [28] investigated the fundamental relationship between corrosion rate and magnetic induction surrounding steel reinforcement. N. Polydorides [29] realized the magnetic induction tomography scanning of corroded reinforced concrete columns. Sun [30] had quantitatively studied the magnetic anomalies of reinforcement rods in bored in situ concrete piles for the first time and summarized their magnetic anomaly character. A. Orbe [31] proposed a magnetic scanning methodology to infer, nondestructively, the spatial dispersion of mechanical properties throughout the steel fiber reinforced concrete (SFRC) structure. H.-J Krause [32] developed a four-channel SQUID system based on magnetometers for detection of tendon ruptures in prestressed members of bridges. B.T. Fernandes [33] described a method of extracting positional information from images of steel bars embedded in concrete using a set of image preprocessing algorithms combined with a modified Hough transform. Chen [34] studied the corrosion of steel bars in reinforced concrete columns based on MFL. However, experimental study of the relationship between the bending strength and SMFL signals of corroded reinforced concrete beam is rarely reported.

In this paper, by detecting and analyzing the bending strength and SMFL signals of the corroded reinforced concrete beam, not only the location and extent of corrosion can be determined, but also the relationship between the bending strength and the SMFL signals. It will provide a rapid, effective, and simpler operation NDT method for the bending strength of corroded reinforced concrete beams.

2. Experimental Details and Theoretical Background

2.1. Experimental Details Based on Micromagnetic Sensor

To carry out the experiment, 10 identical reinforced concrete tested beams, numbered 1–10, were prepared. In order to explore the relationship between the bending strength and the SMFL signals of the corroded reinforced concrete tested beams more intuitively, central sections (corrosion region) of the tested beams were only reinforced with two tensile steel bars. Ordinary Portland cement, coarse aggregate (stone), with a maximum size of 22 mm, water, and sand were used to mix concrete, the detailed material parameters and dimensional drawings of the tested beam are shown in

Table 1. Material parameters of tested beam.

Item	Label	Material	Cement	Water	Sand	Stone	Standard Strength (Mpa)	Design Strength (Mpa)
Concrete	C30	Unit dosage (kg/m3)	461	175	512	1252	fck = 20.1	fcd = 14.3
		weight ratio	1	0.38	1.11	2.72	ftk = 2.01	ftd = 1.43
Item		Reinforcement					Yield Strength (Mpa)	Design Strength (Mpa)
Steel bars		Tensile bars: 2Φ14; Stirrup: Φ8@100					fyk = 335	fyd = 300

and Figure 1. An electrochemical method [35] was used to corrode steel bars of the tested beams, and the amount of corroded iron could be calculated according to the Faraday’s 1st Law, as expressed in Equation (1):

$$\Delta m=\frac{M}{nF}\Delta Q=\frac{M}{nF}I\Delta t$$

(1)

where Δm is the amount of corroded iron, M is the molar mass of iron (Fe), n is the valence state of Fe²⁺, F is the Faraday constant, ΔQ is the quantity of electric charge, I is the direct current flow through steel bars, and Δt is the corrosion time. The corrosion region is located at the midpoint of the tested beam with a width of approximately 15 cm, wrapped with a towel impregnated with a 5% sodium chloride solution and kept moist by the capillary action. The carbon rod placed in the solution is connected to the negative pole of the current source, and the steel bars are connected to the positive pole, forming a closed circuit for corrosion. Corrosion parameters of all tested beams are presented in

Table 2. Corrosion parameters of all tested beams.

Parameter	No.
	1#	2#	3#	4#	5#	6#	7#	8#	9#	10#
Corrosion current/A	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05
Corrosion time/h	0	12	24	36	48	72	96	120	156	504
Δm/g	0	13.1	26.3	39.4	52.5	78.8	96.9	121.2	157.5	206.0

, the layout of the corroding device is shown in Figure 2.

Each tested beam was corroded in a periodic manner until the corrosion was completed. After each period of corrosion was completed, the SMFL signals of the tested beam were measured and collected. The self-designed 3-dimensional (3D) device for SMFL signals acquisition based on the 3D mechanical displacement system and the high-precision micromagnetic sensor is shown in Figure 3. HMR2300 magnetometer (Honeywell International, Morristown, NJ, USA) was used as the micromagnetic sensor with a resolution to less than 70 μGs. The device was connected to a computer and can measure the 3D spatial magnetic signal with a controllable scanning speed and path, and then output a data file that contains X, Y, and Z coordinates and its corresponding magnetic signals components H_P(y), H_P(x), H_P(z). The top surface midline of the tested beam is defined as the scanning path for SMFL signals measurement using the 3D scanning device, which is shown in Figure 4. In addition, Figure 4 also shows the different scanning lift-off heights (LFH) from 5 mm to 810 mm.

All the tested beams were subjected to the “4-point” bending test after the corrosion and SMFL signals acquisition were all completed. The layout of the “4-point” bending test is shown in Figure 5.

2.2. Theoretical Background Based on Magnetic Dipole Model

The equivalent magnetic dipole is a physical model commonly used in the theoretical study of the mechanism of SMFL signal of the ferromagnet [22,27,36]. According to the equivalent magnetic charge theory, the exterior magnetic field including SMFL would be considered to originated from the magnetic charge: ρ = −∇M. M is the magnetization satisfying M = (μ_r − 1) H_mL, where μ_r is the relative magnetic permeability and the Weiss field H_mL is the effective field producing self-magnetization in the ferromgnet. Figure 6 shows the calculated diagram of the corroded steel bars of the tested beam based on the magnetic dipole model. The corrosion notch of the steel bar is assumed to a rectangular pit with dimensions of 2b × h and the distribution of charge density ±ρ_ms at the edges of the corroded region is considered as uniform for simplicity.

Based on the calculated diagram shown in Figure 6, the surface SMFL signal at point P(x, y, z) due to these concentrated magnetic charges can be expressed as in Equation (2), and its normal component dH_p(z) can be expressed as in Equation (3):

$$\overrightarrow{d{H}_P}={\displaystyle \sum _{i=1}^4\overrightarrow{d{H}_{Pi}}}={\displaystyle \sum _{i=1}^4\frac{\pm {\rho }_{ms}\cdot dh}{2\pi {\mu }_0{r}_i^2}\overrightarrow{r_i}}$$

(2)

$$\{\begin{array}{l}d{H}_{P1}(z)=\frac{(-{\rho }_{ms})\cdot (z+h)dh}{2\pi {\mu }_0[{(x+a)}^2+{(y-b)}^2+{(z+h)}^2]}\\ d{H}_{P2}(z)=\frac{(+{\rho }_{ms})\cdot (z+h)dh}{2\pi {\mu }_0[{(x+a)}^2+{(y+b)}^2+{(z+h)}^2]}\\ d{H}_{P3}(z)=\frac{(-{\rho }_{ms})\cdot (z+h)dh}{2\pi {\mu }_0[{(x-a)}^2+{(y-b)}^2+{(z+h)}^2]}\\ d{H}_{P4}(z)=\frac{(+{\rho }_{ms})\cdot (z+h)dh}{2\pi {\mu }_0[{(x-a)}^2+{(y+b)}^2+{(z+h)}^2]}\end{array}$$

(3)

where r_i is the space vector from the magnetic charge element to the space point P(x, y, z); μ₀ = 1.0 is the vacuum magnetic permeability; a is half of the spacing of the two parallel steel bars, and z is the lift-off height (LFH). Then, the normal component H_p(z), expressed in Equation (4), can be obtained by the integral of Equation (3). The theoretical calculation results of Equation (4) are shown in Figure 7.

It can be seen from Figure 7 that as the corrosion extent increases (h increases), the amplitude of H_p(z) increases continuously. While the H_p(z) amplitude decreases with increasing z when the corrosion amount Δm is constant. The H_p(z) curves has a zero-crossing intersection at the midpoint of the corroded region, and the peak-valley spacing equals the corrosion width 2b.

$$\begin{array}{ll}{H}_P(z)& ={\displaystyle \sum _{i=1}^4{\displaystyle {\int }_{-h}^{0}d{H}_{Pi}(z)}}\\ & =\frac{{\rho }_{ms}}{4\pi {\mu }_0}(\ln \frac{{(x+a)}^2+{(y-b)}^2+{(z-h)}^2}{{(x+a)}^2+{(y-b)}^2+z^2}+\ln \frac{{(x+a)}^2+{(y+b)}^2+z^2}{{(x+a)}^2+{(y+b)}^2+{(z-h)}^2}\\ & +\ln \frac{{(x-a)}^2+{(y-b)}^2+{(z-h)}^2}{{(x-a)}^2+{(y-b)}^2+z^2}+\ln \frac{{(x-a)}^2+{(y+b)}^2+z^2}{{(x-a)}^2+{(y+b)}^2+{(z-h)}^2})\\ & =\mathrm{A}\frac{{\rho }_{ms}}{4\pi {\mu }_0}\end{array}$$

(4)

3. Results and Discussion

3.1. Experimental Measurement Results of SMFL Signal

Figure 8 shows the normal component H_P(z) curves of the nine tested beam SMFL signals with different corrosion amounts Δm, where Y = 400~550 mm corresponds to the corrosion region. The big spikes of amplitude outside of the corrosion region in Figure 8 are due to the effect of stirrups. As the corrosion amount Δm increased, the amplitude of the H_P(z) curves in the corroded region increased, and then developed obvious peak-to-valley values and intersection; the smaller the LFH is, the larger the amplitude of H_P(z) in the corroded region of the same Δm. The location and extent of the steel bars corrosion can be qualitatively determined by H_P(z), apparently. All the experimental phenomena are consistent with the results of theoretical analysis, apparently.

In fact, the H_P(z) shown in Figure 8 is a superposition of H_B, H_S, and H_E. H_B is the demagnetizing field of the steel bars, H_S is the SMFL field originated from the corrosion region, and H_E is the environmental magnetic field. H_B and H_E can be considered constant because the steel bars’ magnetization condition and the environment had not changed during the whole experimental process, which can be confirmed by the fact the H_P(z) curves shown in Figure 8 did not significantly change within the uncorroded region. Therefore, the changes of H_P(z) within the corroded region are mainly caused by the normal component H_S(z) of the SMFL field that originated from the corrosion.

For further analysis, H_P(z) curves of each tested beam with the smallest LFH = 5 mm but different Δm were extracted from original H_P(z) curves shown in Figure 8. The H_P(z) curve with Δm = 0 g is defined as the background magnetic field (H_B + H_E), which is subtracted from each of the H_P(z) curves to obtain the H_S(z) curves originated from the corrosion region. Figure 9 shows the H_S(z) curves of 2–10 tested beams (a small number of meaningless curves that do not affect the analysis results were deleted) and the ΔL (peak-valley spacing, unit: cm) corresponding to each H_S(z) curve. As can be seen from Figure 9, the H_S(z) curves are consistent with the theoretical analysis results, especially tested beam number four.

3.2. Analysis of SMFL Signal

In order to analyze the relationship between bending strength and SMFL signals of the corroded tested beams, the gradient K is defined. K is a direct and effective criterion for further description of the variation of SMFL signals, which is given by Equation (5):

$$K=\left|\frac{\Delta H_S(z)}{\Delta L}\right|$$

(5)

where ΔH_S(z) is the difference in the values of peak and valley H_S(z) of the corroded area and ΔL is the peak-valley spacing of H_S(z) curves, which are shown in Figure 9.

Table 3. The calculation parameters of gradient K.

Parameter	Δm/g
	0	13.1	26.3	39.4	52.5	78.8	96.9	121.2	157.5	206.2
ΔHS(z)Ea/mGs ΔLEa/cm R/% h/cm A ρms/mGs ΔHS(z)T/mGs K1/mGs·m−1 K2/mGs·m−1	0 - 100 0 0 - 0 0 0	44.6 9.1 96.4 0.050 0.037 7574 40.4 269.5 490.1	84.1 11.5 92.80 0.101 0.075 7046 81.0 540.0 731.3	120.4 13.2 89.2 0.151 0.112 6754 121.7 811.6 912.1	159.5 15.0 85.6 0.202 0.150 6681 162.6 1084 1063	230.0 17.2 78.4 0.302 0.225 6423 244.8 1632 1337	313.0 19.6 73.4 0.372 0.278 7074 302.1 2014 1597	350.6 22.5 66.8 0.465 0.348 6330 377.9 2520 1558	501.8 24.7 56.8 0.605 0.454 6945 493.2 3288 2032	623.4 29.6 43.5 0.791 0.595 6583 646.5 4310 2106

shows the calculation parameters of gradient K of all tested beams. ΔH_S(z)_Ea refers to the average value of the experimental ΔH_S(z) with the same corrosion amount Δm; ΔL_Ea is that of the experimental ΔL with the same corrosion amount Δm. R is the calculation cross-section remaining percentage of the corroded steel bars based on the uniform corrosion assumption, and the corrosion width 2b is assumed to be 15 cm when R is calculated. The conversion model corrosion depth h, integral value A, and magnetic charge density ρ_ms are the theoretical calculation parameters of Equation (4). Depth h is calculated according to R, and the calculation parameters of A are x = 0 cm, y = 7.5 cm, z = 3 cm, a = 2.5 cm, and b = 7.5 cm. Each ρ_ms is calculated according to its corresponding experimental ΔH_S(z)_Ea. ΔH_S(z)_T is the theoretical value of ΔH_S(z). K₁ and K₂ refer to the theoretical gradient and the experimental gradient, respectively. The calculation formulas for all parameters are summarized in Equation (6):

$$\{\begin{array}{l}R=100\frac{15\times \pi \times {0.7}^2\times 7.9\times 2-\Delta m}{15\times \pi \times {0.7}^2\times 7.9\times 2}=100-\frac{\Delta m}{3.648}\\ h=1.4(1-0.01R)\\ {\rho }_{ms}=\frac{1}{2}\Delta H_S{(z)}_{Ea}\frac{4\pi {\mu }_0}{\mathrm{A}}=\frac{2\pi {\mu }_0\Delta H_S{(z)}_{Ea}}{\mathrm{A}}\\ \Delta H_S{(z)}_T=2\frac{\overline{{\rho }_{ms}}\cdot \mathrm{A}}{4\pi {\mu }_0}=\frac{\overline{{\rho }_{ms}}\cdot \mathrm{A}}{2\pi {\mu }_0}\\ K_1=\frac{\Delta H_S{(z)}_T}{2b}=\frac{\overline{{\rho }_{ms}}\cdot \mathrm{A}}{0.3\pi {\mu }_0}\\ K_2=\frac{\Delta H_S{(z)}_{Ea}}{\Delta L_{Ea}}\end{array}$$

(6)

Figure 10 shows the relationship diagram between K and Δm according to Table 3. K₁ is linearly related to Δm, the linear fitting equation is K₁ = 20.8Δm with an R-squared value of 0.999. K₂ is exponentially related to Δm, the exponential fitting equation is K₂ = 2204 (1 − 0.985^Δm) with an R-squared value of 0.946. The difference between K₁ and K₂ is mainly caused by the fact that the calculation of K₁ is based on the constant assumed corrosion width, but the calculation of K₂ is based on the continuous increasing actual corrosion width measured in the experiment.

3.3. Analysis of Bending Strength

Table 4. The bending strength M of all tested beams.

Parameter	No.
	1#	2#	3#	4#	5#	6#	7#	8#	9#	10#
Δm/g M1/(KN·m) S1/% F/kN M2/(KN·m) S2/%	0 13.2 0 48.7 15.1 0	13.1 12.8 3.0 42.6 13.2 12.6	26.3 12.3 6.8 40.0 12.4 17.9	39.4 12.0 9.1 39.4 12.2 19.2	52.5 11.6 12.1 38.4 11.9 21.2	78.8 10.8 18.2 35.8 11.1 26.5	96.9 10.2 22.7 32.3 10.0 33.8	121.2 9.5 28.0 27.7 8.6 40.0	157.5 8.2 37.9 24.5 7.6 49.7	206.0 6.5 50.8 23.2 7.2 52.3

shows the bending strength M and its corresponding loss rate S of all tested beams, where the theoretical bending strength M₁ and the experimental bending strength M₂ are calculated according to the reinforced concrete structure design principle expressed in Equation (7):

$$\{\begin{array}{l}{\alpha }_1{f}_{c}bx=f_y{\mathrm{A}}_s\\ M_1={\alpha }_1{f}_{c}bx(h_0-\frac{x}{2})\\ {\mathrm{A}}_s=0.01\mathrm{A}R\\ M_2=\frac{1}{2}Fd\end{array}$$

(7)

where α₁ = 1.0 is the simplified calculation factor, f_c = 14.3 MPa is the axial compressive design strength of the concrete, b = 100 mm is the section width of tested beams, x is the height of the concrete pressured zone, f_y = 300 MPa is the tensile strength of the steel bar, for structure design, A_s is the total cross-sectional area of corroded steel bars, h₀ = 175 mm is the effective section height, A = 308 mm² is the total cross-sectional area of uncorroded steel bars, F is the loading force measured in the experiment, and d = 0.62 m is the arm length of the force couple.

Figure 11 shows the relationship diagram between M and Δm according to Table 4. It can be seen that both M₁ and M₂ decrease approximately linearly with the increase of Δm, especially M₁. The values of M₂ are not much different from that of M₁, which indicates that the results of the experiment are consistent with that of the theoretical calculation. As for the loss rate S, S₁ is positive linearly related to Δm, but S₂ is more inclined to increase exponentially with the increase of Δm. The difference, mainly caused by the continuous increasing actual corrosion width in the experiment, leads to the cross-section of the steel bar being reduced more and more slowly with the same increment of Δm. Then, M₂ decreased more and more slowly and S₂ increased more and more slowly with the increase of Δm, which led to S₂ increasing more and more slowly.

3.4. Analysis of Relationship Between SMFL Signal and Bending Strength

According to the above analysis, both K and S increase monotonically with the increase of Δm. For evaluating the relationship between SMFL signals and the bending strength of the corroded reinforced concrete beams, scatters and the linear fitting line of S-K are shown in Figure 12. It can be seen that with the increase of S₁ and S₂, both K₁ and K₂ increase approximately linearly. The fitting function of the theoretical data is K₁ = 86.7S₁, with an R-squared value of 0.998, and that of the experimental data is K₂ = 42.4S₂, with an R-square value of 0.964.

Compared with the theoretical data, the experimental data is more discrete and its gradient grows more slowly. In addition to being affected by the corrosion width, this difference may originate from the fact that some factors are ignored in the idealized theoretical analysis. Such as the magnitude and the distribution of magnetic charge, meaning that the shape of the corrosion notch will be affected by nonuniform corrosion or corrosion expansion stress.

However, in general, the variation and distribution laws of S-K obtained from the experiment are consistent with that of the theoretical analysis, which shows that the gradient K is a reliable and effective indicator for evaluating the residual bending strength of corroded reinforced concrete beams.

4. Conclusions

In this work, the experiment details of 10 corroded reinforced concrete tested beams based on micromagnetic sensors are introduced, followed by the derivation of the formulas of H_P(z) (H_S(z)) based on the magnetic dipole model. Finally, both the experimental data and theoretical data were analyzed. The following conclusions can be drawn:

(1) The experimental H_S(z) curves of all tested beams are consistent with the theoretical calculation results of the magnetic dipole model, both the location and extent of the steel bars’ corrosion can be qualitatively determined using H_S(z);

(2) The bending strength loss rate S of all the tested beams are approximately linearly related to the gradient K of H_S(z), thus the residual bending strength of the corroded reinforced concrete beams can be evaluated based on the gradient K of H_S(z).

This paper lays the foundation for the experimental study of the relationship between the bending strength and the SMFL signals of corroded reinforced concrete beams. A simpler, low-cost, and more efficient new method for evaluating the residual bending strength of corroded reinforced concrete beams based on the micromagnetic sensor is also proposed.