The defect-length effect in corrosion detection with magnetic method for bridge cables

Abstract

Quantitative evaluation of the steel corrosion in cables is significant for the safe operation of cable-supported bridges. The magnetic flux (MF) examination shows great potential to detect the corrosion defect, or loss of metallic cross-sectional area (LMA). An LMA defect in steel cables can be measured accurately when it is longer than a certain length. However, for defects in early stage, where the length of corrosion area is short, the MF examination may produce unacceptable error. In this study, the effect of defect length on the MF examination for corrosion detection of bridge cables is investigated through theoretical analysis and model experiments. An original analytical model to quantify the influence of defect length is proposed based on the equivalent magnetic circuit method. Then, MF examination experiments are performed on a series of cable models with different defect lengths and locations to verify the analytical model. Further, parameter study is conducted based on the proposed analytical model to clarify the mechanism of the defect-length effect. The results show that the area loss caused by short corrosion damage will be underestimated if the defect-length effect is neglected, and this effect can be quantified efficiently with the proposed analytical model.

Keywords

bridge cable / corrosion detection / defect length / MF examination / quantitative evaluation

Introduction

The recent decades have witnessed a rapid development and construction of cable-supported bridges around the world. However, as one of the main load-carrying members in these bridges, the stay cable often suffers rust or even breakage at the early stage of the designed service life. The non-destructive testing (NDT) and evaluation of bridge cables and suspenders is of great significance for the maintenance and safe operation of cable-supported bridges. Investigation of the NDT techniques shows that magnetic testing method has a good potential to detect the internal defects in the free length of bridge cables [¹,²]. Magnetic detecting instruments were developed to adjust the cable configuration [³,⁴], and field test in bridges were also reported [⁵]. Application in the Innoshima Bridge in Japan indicates that the result of the magnetic flux (MF) method is similar to the corrosion actually observed on the opened suspender [⁶].

In magnetic testing, the defects in steel wires are divided into two groups: the loss of metallic cross-sectional area (LMA), which refers to the uniform corrosion, and the local flaws (LF), such as corrosion pits and wire breakage. These two types of defects are inspected with two different techniques respectively: the LMA is inspected with the MF examination because the defect will trigger magnetic flux variation (MFV) in the magnetic circuit, while the LF is inspected with the magnetic flux leakage (MFL) examination [⁷] since obvious MF leakage can be measured around the defect. Typical signal from the MFL examination and the MF examination is shown in Fig. 1(a) and Fig. 1(b). Because of the complementation of these two techniques, they are usually integrated into a dual-function instrument [⁸,⁹] as shown schematically in Fig. 1(c).

In the evaluation of the uniform corrosion, the MF variation caused by the defect is proportional to the percent LMA, i.e. the percentage of LMA [³]. However, if the rust section is not long enough, the signal variation caused by the defect will decrease with the defect length. Thus, the percentage of LMA will probably be underestimated [¹⁰,¹¹]. The minimum length of defects that can be detected accurately is called the scan length, the average length or the quantitative resolution [¹⁰].

To improve the quantitative resolution, Hamelin and Kitzinger [¹²] designed a high-pass filter to improve the profile inverse of the uniform corrosion; Weischedel [¹³] proposed a signal enhancement method that increases the quantitative resolution of a clamshell auxiliary coil device to the same level of the annular coil device. Although these efforts help to provide better measurement, the quantitative resolution still closely relates to the length of the device. According to a survey into the worldwide instruments for the NDT of steel ropes [¹⁴], the quantitative resolution, or the critical length, might be as large as 750mm in some devices. For convenience, the effect of the defect length on the MF variation is named the defect-length effect. To diminish the defect-length effect and improve the evaluating accuracy of a specific device, a linear correction of the MFV based on the defect length was applied in steel rope test [¹⁵]. However, in the previous study of the authors [¹¹], the defect-length effect was found to be nonlinear in bridge cables. To improve the evaluation accuracy of short corrosion defects, deeper inspection into the defect-length effect and better correction method of the defect length are in need.

In this work, the mechanism of the defect-length effect is deduced theoretically, and a defect-length factor is proposed to quantify this effect. Then, an innovative theoretical model is established to quantitatively calculate the defect-length factor. After that, cable models are made for MF examination, and results of the experiments are used to verify the proposed theoretical model of the defect-length factor. Then, parametric sensitivity is studied for parameters of the cable configuration and the defect based on the proposed model. Finally, conclusions are drawn.

Theory background

In this study, the relationship between the output of the area sensor and the defect length is to be deduced based on the equivalent magnetic circuit method, which is commonly used in the analysis of the MF examination [³,¹⁶]. This method simplifies the magnetic testing system into an electric circuit, namely the equivalent magnetic circuit. The circuit is composed of the magnetic motive force (MMF) provided by the magnets, and the magnetic conductors including the air gap, the steel cable and the magnetic yoke. The testing system might be simplified into different equivalent magnetic circuits due to the problem being focused on. In this work, the leakage flux in the air around the device is omitted, because the main problem considered here is how the magnetic conductance of the cable (i.e. G_c) is influenced by the defect, and the variation of the G_c is quite small compared with the magnetic conductance of the surrounding air. Thus, the equivalent magnetic circuit in Fig. 2 is used in the following analysis.

According to the Ohm's law for a magnetic circuit, the following equation is obtained

(1)\mathrm{2}{{\displaystyle \varphi }}_c/{{\displaystyle G}}_g+{{\displaystyle \varphi }}_c/{{\displaystyle G}}_c+{{\displaystyle \varphi }}_c/{{\displaystyle G}}_{yo}=\mathrm{2}{{\displaystyle F}}_m,

Thus, the MF in the magnetic circuit f_c (unit: Wb) is

(2){{\displaystyle \varphi }}_c=\mathrm{2}{{\displaystyle F}}_m(\mathrm{2}/{{\displaystyle G}}_g+\mathrm{1}/{{\displaystyle G}}_c+\mathrm{1}/{{\displaystyle G}}_{yo}),

where F_m is the MMF of the magnets (unit: Ampere or kA), G_c, G_yo and G_g are the magnetic conductance of the cable, the magnetic yoke, and the air gap between the magnets and the cable surface respectively(unit: Henry or H).

The magnetic conductance of an object is calculated via G = mS/L, where m, S and L are respectively the magnetic permeability(unit: H/m), the cross-sectional area(unit: m²) and the length of the object(unit: m). In addition, direction of S is perpendicular to that of L, while the direction of L is parallel to that of the flux. Then, G_yo can be calculated with Eq. (3)

(3){{\displaystyle G}}_{yo}={{\displaystyle \mu }}_{\mathrm{0}}{{\displaystyle \mu }}_{yo}\pi T({{\displaystyle D}}_{yo}+T)/{{\displaystyle L}}_{yo},

where m₀ is the magnetic permeability of vacuum(unit: H/m), m_yo is the relative permeability of the yoke(unit: 1), D_yo is the inner diameter of the yoke, T is the thickness of the yoke, and L_yo is the length of the yoke. The geometric parameters are marked in Fig. 3. In common practice, G_g and G_c are calculated with

(4)\{\begin{array}{l}{G}_g={\displaystyle \frac{{\mu }_{\mathrm{0}}\pi L_m}{\ln \left({\displaystyle \frac{r_c+\delta }{r_c}}\right)}}\\ G_c={\displaystyle \frac{{\mu }_{\mathrm{0}}{\mu }_s{S}_s}{L_c}}\end{array},

where L_m is the length of the magnet, r_c is the radius of the cable, d is the air gap distance of the magnets to the cable surface, m_s and S_s are respectively the relative permeability and the cross-sectional area of the steel wires. Eq. (4) treats the steel wires as a whole, but does not take into account the influence from the defects.

In this paper, G_g and G_c are recalculated in a novel way. The influence from the defects and the air in the cable is considered to obtain a detailed analysis of G_c. Further, the air gap between the magnets and the cable is considered as a part of the cable when calculating the magnetic conductance. Thus, G_g and G_c are combined into one parameter G_gc. Thus, Eq. (2) becomes

(5){{\displaystyle \varphi }}_c=\mathrm{2}{{\displaystyle F}}_m/(\mathrm{1}/{{\displaystyle G}}_{gc}+\mathrm{1}/{{\displaystyle G}}_{yo}).

In MF examination, f_c is measured to assess the steel corrosion since it is approximately proportional to the percentage of LMA. However, when the defect is short, f_c is also affected by the defect length as G_gc does. Accurate calculation of G_gc is needed to quantify the defect-length effect.

The proposed model to quantify the defect-length effect

The equivalent length of the MF line

To calculate G_gc in Eq. (5), the m and L of the mediums composed of air and steel shall be classified. For convenience, steel is designated as the base material of the composed medium, then, the length of the MF line in these two mediums is merged into an equivalent length. Then, G_gc can be calculated with the magnetic permeability of the steel m_s and the equivalent length. To compute the equivalent length, two questions need to be specified beforehand:

A. Through which path does the MF line pass the cable?

B. How much air is there on that path?

For question A, Yuan [¹⁷] suggests that the part inside the cable of a MF line, which passing through point i and point j, is composed of a line segment and two circle segments (Fig. 3(a)). Thus, length of the MF line in the cable is given by

(6){{\displaystyle L}}_{ij}={{\displaystyle L}}_c+(\pi -\mathrm{2}){{\displaystyle r}}_{ij},

where r_ij is the radius of the circular arc in the path. For each steel wire i in the cable, r_ij varies with the position of point j (Fig. 3(b)).

In Yuan’s work [¹⁷], the cable is magnetized with coils. In this study, Eq. (6) is applied to the testing system with magnets. Assume that:

The MF passing through the small area i (see Fig. 3(b)) is the superposition of the MF produced by all the magnet units.

According to this assumption, the magnetic conductance of a wire i can be calculated using the average L_ij. The average r_ij for wire i is defined as r* in Eq. (7), where r_e is the distance from the steel wire i to the cable axis, and q refers to the circumferential location of point P on the cable surface (see Fig. 3(b)).

r*={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\pi }}{\displaystyle {\int }_{\mathrm{0}}^{\mathrm{2}\pi }\sqrt{r_c^{\mathrm{2}}+r_e^{\mathrm{2}}-\mathrm{2}{r}_c{r}_e\cos \theta }}d\theta .

The relative value of r* is given by

(8)r\text{'}=r*/r_c={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\pi }}{\displaystyle {\int }_{\mathrm{0}}^{\mathrm{2}\pi }\sqrt{\mathrm{1}+{(r_e/r_c)}^{\mathrm{2}}-\mathrm{2}(r_e/r_c)\cos \theta }}d\theta .

Since no explicit solution can be obtained from Eq. (8), an approximate expression is proposed as

(9)r^{\text{'}}(\mathrm{4}/\pi -\mathrm{1}){({{\displaystyle r}}_e/{{\displaystyle r}}_c)}^{\mathrm{2}}+\mathrm{1.}

The value of Eq. (9) is slightly larger than that of Eq. (8) (see Fig. 4), with the relative error less than 1.03%. Thus, Eq. (9) is adopted in the following analysis. For a steel wire without defect, the average L_ij is proposed as

(10){{\displaystyle L}}_{\mathrm{0}}={{\displaystyle L}}_c+(\pi -\mathrm{2})r^{*}.

For question B, it should be noted that steel is ferromagnetic, while the protective grease or epoxy resin in the cable, and the cable sheath, which are usually made of polyethylene (PE) or high density polyethylene (HDPE), are all non-ferromagnetic. Since the magnetic permeability of non-ferromagnetic materials is close to that of air, it makes no difference to the MF examination no matter the materials is air or grease. Therefore, these non-ferromagnetic materials are all regarded as air hereafter.

In Eq. (10), the cable is simplified into a solid steel bar, and the influence of air, both inside and outside the cable, is omitted. In fact, there is considerable amount of air between the steel wires (see Fig. 5). The air-filling ratio, b, is proposed as in Eq. (11). Due to the parallel arrangement of the steel wires (or strands) in bridge cables, a triangle unit composed of three pieces of steel wires and the surrounded air is taken to calculate b.

(11)\beta ={\displaystyle \frac{S_{air}}{S_{air}+S_{steel}}}=\mathrm{1}-{\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{6}}}\pi {\left({\displaystyle \frac{d_w}{d_{ww}}}\right)}^{\mathrm{2}},

where, d_w is the diameter of the wire, and d_ww is the center-to-center distance between two adjacent wires. For cables composed of parallel steel wires, d_w = d_ww, b = 0.093. For cables composed of stranded wires, b depends on d_ww. For convenience of calculation, the hexagonal wires and the internal air are simplified into an equivalent round bar which has the same area as the wires and the air do (Fig. 5). The radius of that bar is written as r_c in this paper.

The relative magnetic permeability is m_s for steel and 1 for air, in other words, the magnetic conductance of a certain amount of air is 1/m_s times of that of steel. Therefore, the equivalent length of the magnetic flux line (ELMFL) should be multiplied by m_s when the MF line passes through air. For convenience, the internal air is assumed to be evenly distributed in the equivalent round bar. Then, two more items are added to Eq. (10) after considering the internal air and the external air.

(12){{\displaystyle L}}_{\mathrm{0}}={{\displaystyle L}}_c+(\pi -\mathrm{2})r^{*}+\pi \beta ({{\displaystyle \mu }}_s-\mathrm{1})r^{*}+\mathrm{2}{{\displaystyle \mu }}_s\delta .

Equation (12) provides the ELMFL of a steel wire in healthy status. Finally, G_gc in Eq. (5) is given by

(13)G_{gc}={\displaystyle {\int }_{\mathrm{0}}^{r_c}{\displaystyle \frac{\mathrm{2}\pi r_e{\mu }_{\mathrm{0}}{\mu }_s}{L_{\mathrm{0}}}}}d{r}_e={\displaystyle {\int }_{\mathrm{0}}^{r_c}{\displaystyle \frac{\mathrm{2}\pi r_e{\mu }_{\mathrm{0}}{\mu }_s}{L_c+[\pi -\mathrm{2}+\pi \beta ({\mu }_s-\mathrm{1})]r*+\mathrm{2}{\mu }_s\delta }}}d{r}_e.

Substituting Eq. (9) into Eq. (13) yields

(14)G_{gc}={\displaystyle \frac{r_c{\mu }_{\mathrm{0}}{\mu }_s}{[\pi -\mathrm{2}+\pi \beta ({\mu }_s-\mathrm{1})](\mathrm{4}-\pi )}}\ln \left\{\mathrm{1}+{\displaystyle \frac{[\pi -\mathrm{2}+\pi \beta ({\mu }_s-\mathrm{1})](\mathrm{4}-\pi )}{L_c+[\pi -\mathrm{2}+\pi \beta ({\mu }_s-\mathrm{1})]r_c+\mathrm{2}{\mu }_s\delta }}{\displaystyle \frac{S_c}{r_c}}\right\},

where S_c is the cross-sectional area of the equivalent round bar in Fig. 5.

Influence on the magnetic conductance from a defect

Once a steel wire is corroded, the magnetic permeability of the corrosion products will drop to a level near that of air [¹⁸], thus, the corrosion products is usually regarded as air in magnetic inspection [¹⁹]. Designate L_D as the defect length. When L_DL_c-2r*, the ELMFL of the corroded wire is

(15)L={{\displaystyle L}}_{\mathrm{0}}+(\mu -\mathrm{1}){{\displaystyle L}}_D.

When L_c-2r*<L_DL_c, the path of the MF line becomes too complicated to make a concise assumption. As an approximation, Eq. (15) is applied to all the L_D that is not greater than L_c. In the following discussion, the accuracy of this approximation turns out to be acceptable.

Let S_D be the corroded cross-sectional area of the steel wire, and substitute the magnetic conductance of air for the magnetic conductance of the corroded part of steel, then, the magnetic conductance of a partially corroded cable is obtained

(16){{\displaystyle G}}_{gc}^{\text{'}}={{\displaystyle G}}_{gc}-{{\displaystyle \mu }}_{\mathrm{0}}{{\displaystyle \mu }}_s{{\displaystyle S}}_D/{{\displaystyle L}}_{\mathrm{0}}+{{\displaystyle \mu }}_{\mathrm{0}}{{\displaystyle \mu }}_s{{\displaystyle S}}_D/[{{\displaystyle L}}_{\mathrm{0}}+({{\displaystyle \mu }}_s-\mathrm{1}){{\displaystyle L}}_D],

in which L₀ is calculated with Eq. (12).

The defect-length factor

By substituting Eqs. (14) and (16) into Eq. (5), the MF f_c is obtained. When the corrosion defect is long enough, the MFV in the yoke is proportional to the percentage of LMA but independent of L_D. For short defects, the relationship between f_c and L_D needs to be classified. Designate Df_c as the MFV caused by corrosion. A defect-length factor f_dl is proposed to quantize the influence of defect length on the MF.

(17)f_{dl}={\displaystyle \frac{\Delta {\varphi }_c}{\Delta {\varphi }_c(L_D=L{}_c)}},

where Df_c = f_c-f_c(L_D = 0). Apparently, f_dl depends on the defect length L_D, and f_dl<1 for L_D<L_c.

Experimental verification of the theoretical model

Experiment setup

A series of experiments were performed to check the defect-length effect in magnetic test. The test system is shown in Fig. 6. In these experiments, 103 high strength steel wires of f7mm were placed in an aluminum tube with an inner diameter of 100mm and a thickness of 5mm (see Fig. 7). In that tube, there are 35 other small aluminum tubes with inner diameter of 8mm, thus the steel wires can slide freely in the small tubes to simulate LMA defects of various lengths. This simulation is equivalent to the uniform corrosion because magnetic permeability of the corrosion products is quite low and the rust has almost no effect on the MF examination [¹⁸]. During the test, the device was driven by the electric winch at a steady speed, within a free distance of 2.5m along the cable model.

Consider 9 sets of experiments to simulate the defects at various locations or with different percentage of LMA (Fig. 7(b)). In Groups 1 to 5, the defect location is different from the center to the surface of the cable. In Groups 5 to 9, the percentage of LMA changes from 1.94% to 9.71%. In each group, the defect lengths are {5, 10, 20, 30, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, 260, 280, 300, 350, 400, 500, 600, 700} mm. Altogether 207 scenarios were simulated.

Results and discussion

By scanning the cable models, the device records the area sensor’s output for all 207 scenarios. The MF signal is de-noised with the soft-threshold wavelet de-noising. The defect-length effect can be seen directly from the typical MF signal shown in Fig. 8. In tests with the same area loss, when the defect length L_D is larger than L_C (265mm), the maximum MFV in each test is almost the same for different L_D; otherwise, the maximum MFV varies with L_D.

For long defects (L_D>L_C), the maximum value of MFV in a test shows a good proportional relationship with the percentage of LMA. This is shown in Fig. 9(a), in which the measured data refers to the average MFV from defects longer than L_C in each group. The adjust coefficient of determination in the linear fitting is 0.995. Therefore, the percentage of LMA caused by uniform corrosion can be directly estimated from the MFV in MF examination.

For short defects, an obvious defect-length effect is shown by the measured data (Fig. 9(b)). The maximum MFV in each test increases nonlinearly with the defect length. Besides, obvious magnetic shielding is also observed in the experiments. The wires at outer layer magnetically shield the wires at inner layer, thus, the superficial wires have a higher level of magnetic saturation, and the MFV caused by the same amount of LMA is larger for superficial defect.

To eliminate the influence of the magnetic shielding and focus on the defect-length effect, the MFV in each test is normalized with the average MFV caused by the long defects in each group. The defect length is normalized with L_c. Then, theoretical f_dl is computed for comparison with the measured data, and verification of the proposed model. The parameters of the device and the cable model used in the experiments are listed in Table 1. Comparison between the theoretical f_dl and the normalized MFV is shown in Fig. 9(c). The correlation coefficients between the theoretical f_dl (L_D/L_c<1) and measured data for Group 1 to Group 4 are respectively 0.979, 0.989, 0.984 and 0.997. In Fig. 9(c), the theoretical f_dl is almost the same for defects at different locations, thus, the r_e/r_c is fixed at 0.5 in the calculation. For internal defects, the defect-length effect can be quantized by the proposed model.

In the theoretical model, it is assumed that the wire at point i is magnetized by all the magnet units surrounding the cable (Fig. 3(b)). Actually, due to the magnetic shielding effect, the superficial wire is mainly magnetized by the magnet unit that is close to the wire, while the magnet units far away from the wire contribute little to the magnetization process. Therefore, the real r* for the superficial wire is quite small. By multiplying r* by 0.05 when calculating G'_gc, a modified f_dl can be obtained. After modification, f_dl is much closer to the measured data (see Fig. 9(d)). The correlation coefficients between the theoretical f_dl (L_D/L_c<1) and measured data for Group 5 to Group 9 are respectively 0.536, 0.847, 0.844, 0.906 and 0.892.

The magnetic shielding effect mainly dues to the fact that the cable is not made of solid steel, and the air under the superficial wires makes it difficult for the MF line to pass through and get to the internal wires. Thus, much MF gathers in the superficial wires. Besides, the defect-length effect for superficial defects is also partially compensated by the MFL aroused by the short defects with sharp broken ends. Therefore, it can be deduced that if the arrangement of the wires was more condense, the magnetic shielding effect would be weaker, and that if the cable diameter was smaller, the influence from the magnetic shielding and the MFL would both be weaker.

The deduction can be proved with another experiment conducted by the authors [¹¹], where the air gap distance is 25mm, d_ww is 8.3mm, and the total number of wires n = 66. In that experiment, a cable model similar to that in Fig. 7 is used, and 3 groups of defects are simulated with the percentage of LMA equals to 2/66= 3.03%. The correlation coefficients between the theoretical f_dl (L_D/L_c<1) and measured data for central defect, middle defect and superficial defect are respectively 0.990, 0.997 and 0.983. The defect-length effect of superficial defect is almost the same with that of internal defect (Fig. 11). In Fig. 11, the theoretical f_dl is slightly smaller than the measured data, thus, the percentage of LMA will be slightly overestimated after the nonlinear correction of the test signal via f_dl, providing a safe assessment of the steel cable corrosion.

These experiments show that the proposed theoretical model can predict the defect-length effect in a steel cable with a high accuracy. The nonlinear correction via LMA= LMA_tested/f_dl to reduce the defect-length effect helps to improve the quantitative evaluation accuracy. In the previous work of the authors [¹¹], f_dl is obtained from a series of finite element (FE) simulations. In Fig. 11, the FEM simulation results are normalized with the longest defect in the simulations. However, the gap surrounded by adjacent wires is essential magnetic medium, and should be simulated in the electromagnetic FE model. Due to the sharp shape formed by three concave arcs, the gaps must be densely meshed, which results in a large number of elements and time-consuming FE analysis. For instance, in case of f7-f61 steel cable, about 6 h is needed with the 1/6 FE model to obtain a coarse curve of f_dl in a 4-core 64 bit personal computer with 16GB primary memory. Notice that the computing time increases dramatically with the wire number. In contrast to the electromagnetic FE method, the proposed theoretical model is much more efficient in calculating the defect-length factor. For instance, a smooth curve of f_dl can be obtained within 1s via the Microsoft Excel with a high accuracy in formulating the defect-length effect.

Parametric study on the defect-length effect

A parametric study is further conducted based on the proposed model to inspect how the defect-length effect varies with the parameters of the cable and the defect. The parameters include the cross-sectional area of the steel wires, the radial location of the defect, the filling ratio of the internal air, and the corrosion ratio. In Table 1, the above parameters are substituted by n, r_e/r_c, d_ww, and percentage of LMA respectively. The four factors are set to be variable one by one in the following analysis, their default values are respectively designated as n = 91, r_e/r_c = 0.4, d_ww = 0.007m, and percentage of LMA= 10%.

The results are shown in Fig. 12. Figure 12(a) indicates that f_dl is smaller in cables with smaller diameter, owing to the difference in the air gap distance d in different cables. According to Eq. (5), f₂ varies with 1/G_gc along a hyperbola. As 1/G_gc denotes the magnetic resistance of the cable and the air gap, when the air gap is small, 1/G_gc is small too, then a short defect will cause large relative increase in 1/G_gc, thus obvious decrease of f₂. Although the magnetic resistance of a long defect is even larger, the further decrease of f₂ is not significant. Therefore, the defect-length effect for n = 223 in Fig. 12(a) is weaker compared with others, and small cable diameter always engender larger air gap and stronger defect-length effect.

In a bridge cable with protective sheath and hexagonal arrangement of the wires, the air gap is actually quite large. For instance, with a device having an inner diameter of 120mm, the largest cable that could possibly be inspected is the f7-f169, the outer diameter of which is 114mm with single layer sheath or 118mm with double layer sheath [²⁰]. Thus, the equivalent air gap distance is 12.2mm at least. Therefore, the defect-length effect in bridge cables is more significant than in other elongated objects, such as naked steel ropes or round steel bars.

Figures 12(b) and 12(c) show that the percentage of LMA and r_e/r_c bring almost no influence to the factor f_dl. Thus, the defect-length factor for bridge cables can be calculated with fixed corrosion ratio and radial location of the defect. Note that, the actual magnetization intensity of outer layer wires and inner layer wires might be different, due to the magnetic shielding effect. Since m_s = B/(m₀H), the permeability depends on the magnetic field intensity H and the magnetic induction density B in the steel wire, then, the m_s of steel wires of different layers are different. Therefore, the f_dl curves in Fig. 12(c) refer to the case where all the steel wires are uniformly magnetized or saturated.

Figure 12(d) implies that larger d_ww triggers more obvious defect-length effect. When L_D/L_c = 0.2, the f_dl is 0.57 for d_ww = 7mm and 0.43 for d_ww = 11.8mm, in other words, the magnitude of the MFV decreases by 43% and 57% respectively. Therefore, the defect-length effect is more significant in a parallel steel-strands cable than in a steel wire cable. Although the value of f_dl decreases when d_ww is increased, the f_dl curve will reach a limitation that is significantly greater than the linear model as applied in steel rope test [¹⁵]. Therefore, nonlinear defect-length effect in MF testing and evaluation should be considered when trying to update the stranded cable, which might be done by replacing the single strand using the defect location technique [¹⁷,²¹].

According to the above analysis, the air-filling ratio and the air gap distance are the main factors affecting the defect-length effect. For components with less external and internal air, such as round ropes or steel bars, the total magnetic resistance in the magnetic circuit is small, thus, area-loss defect will result in significant increase of the magnetic resistance even though the defect is short. As the MF is inversely related to the total magnetic resistance in the magnetic circuit, the percentage reduction of MF caused by short defect will be close to that caused by long defect. Thus, the defect-length effect is weak. In case of bridge cables, due to the protecting sheath and the hexagonal arrangement of the wires, the air gap distance and the air-filling ratio of the cable is obviously larger than that of a round bar. Therefore, the defect-length effect in bridge cables is more significant than that in round ropes or steel bars.

In another case, if the air gap is large or the steel wires in the cable are arranged loosely, a considerable amount of air will fill the gaps between wires, making the magnetic resistance of the cable quite large. Then, the increase of magnetic resistance caused by a short defect is obviously smaller in ratio than that by a long defect. Thus, the MFV caused by the defect will gradually increase with the defect length L_D, until L_D is larger than L_C, that is, the magnetized length of the steel wires. Therefore, the defect-length effect in a parallel wire cable is usually more obvious than that in a steel rope, because of the large air gap in bridge cable inspection. The defect-length effect in a steel-strand cable is even stronger than that in a parallel wire cable due to the loosely arranged wires.

Conclusions

In this work, the influence of defect length on the corrosion detection based on MF examination for bridge cables has been investigated. Main conclusions are as follows:

1) The MF signal aroused by the steel-wire defect is dependent on the defect length L_D if L_D is smaller than a critical value. This dependence causes underestimation of the area loss of the steel cable. The critical value of L_D is approximately the length of the magnetized part of the cable, i.e. L_c.

2) The defect-length factor is nonlinear with the relative defect length L_D/L_c, and can be quantified with the proposed theoretical model. The defect-length factor calculated by the experimental data agrees well with the theoretical prediction.

3) Both experimental tests and theoretical analysis show that the defect-length effect is not sensitive to corrosion ratio. If the magnetic shielding effect of the outer layer steel wires on the internal wires is not obvious, the defect-length effect is not sensitive to the radial location of the defects. The magnetic shielding effect weakens the defect-length effect of outer layer wires, but enhances that of the internal wires.

4) The defect-length effect in MF examination is associated with the air gap between the magnet and the outer wires as well as the arrangement of the cable’s steel wires. In cases of bridge cables where the air gap is usually quite large, the defect-length effect is much more significant than that in other elongated members, such as steel ropes or steel rods. On the other hand, such effect is more serious for steel-strand cables due to the relatively loose arrangement of the steel wires.

These outcomes help to understand the defect-length effect in the MF examination. The proposed analytical model for the defect-length factor may give a more accurate evaluation of steel wires’ corrosion for bridge cables. The defects considered in this research are with constant damage grade, more complicated defects, such as those with gradient damage grade, are being studied in the future work.

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