1 Introduction

Steel structure bridges have low weight, high strength, and easy quality control, and their proportion in the whole bridge project is increasing year by year. Steel box girders are widely used in the construction of long-span bridges because of their large bending stiffness, high torsional stiffness, economic and practical characteristics, and convenient construction [1,2,3]. However, steel box girders can undergo invisible damage during usage, and the source of damage is referred to as the stress concentration zone (SCZ). When SCZ reaches the yield strength, the component will have local buckling. In history, there have been a large number of disastrous accidents caused by the instability of steel box girders due to local yielding [4]. Because the steel box girder is a thin-walled structure, it is prone to yield in the local SCZ, many scholars [5,6,7] have studied the methods and measures to improve the local stiffness and strength of the steel box girder to increase the yield strength in the local SCZ. Aiming at the collapse of the bridge steel structure caused by the yield of the local SCZ, if the non-destructive testing (NDT) methods are used to promptly locate the SCZs and conduct key monitoring, accurately judge their critical yield state, and conduct hazard warnings, it can effectively prevent the occurrence of sudden and major disasters. Commonly used NDT methods including ultrasonic flaw detection, radiography, magnetic particle flaw detection, acoustic emission, eddy current, magnetic leakage, and magnetic Barkhausen have been proven to be suitable for damage location and evaluation on bridge steel structures. However, these methods are only suitable for capturing macro defects or formed cracks; they are inadequate for analyzing unformed damage and locations where damage may still occur (such as the local SCZ) [8,9,10].

Metal magnetic memory testing (MMMT) method was formally proposed by Russian scholar Dubov at the 50th international welding academic conference in 1997 [11]. The method takes the geomagnetic field as an excitation source, and its essence lies in the magneto-mechanical effect. The detection principle can be described as follows: If the ferromagnetic material is continuous and uniform, the magnetic induction lines will be constrained in the material, almost no magnetic induction lines pass through the surface, and no magnetic field on the material surface. However, when the ferromagnetic material has the defect of cutting the magnetic induction lines, the defect on the material surface will change the permeability. Because the permeability of the defect is very small, the magnetic induction line will change the path. In addition to a part of the magnetic induction lines passing through the defect directly or bypassing the defect inside the material, and also parts leaving the material surface, bypassing the defect through the air, and entering the material again, forming a self-magnetic-leakage field (SMLF) at the defect on the material surface. Under the joint action of the working load and the earth's magnetic field, the ferromagnetic components will undergo directional and non-directional reorientation of the magnetic domain in the SCZ, which will lead to the change of SMLF strength on the material surface. By detecting the distribution of SMLF strength, the SCZs or defects can be accurately judged [12,13,14].

MMMT method is usually classified as electromagnetic NDT together with magnetic flux leakage and eddy current methods. Other electromagnetic NDT, such as the magnetic leakage method, requires an externally strengthened magnetic field as the excitation source to magnetize the tested piece to the magnetic saturation state. However, the MMMT method uses the weak magnetism properties of ferromagnetic materials that were ignored in the past. The geomagnetic field is the only external magnetic field. At the time of detection, the MMMT method is simple and convenient to operate, does not require surface cleaning, artificial magnetization, and attachment of sensors, and measures the residual magnetic field caused by stress. Therefore, the MMMT is considered to be an ideal method for evaluating early invisible damage [8].

On account of the unique advantages of MMMT, many scholars have conducted numerous studies and proved its feasibility in stress state identification, stress concentration assessment, fatigue damage characterization, and practical application in the field of engineering structure, etc. of ferromagnetic components. Kashefi [15] studied the variation law of stress concentration and SMLF of steel plates with different groove depths under different tensile stresses. Jesus [16] studied the relationship between defect depth, width, and tangential normal SMLF by prefabricating V-shaped notches in steel plates. Bao [17] studied the relationship between the maximum gradient Kmax and the stress concentration factors through the SCZ of prefabricated circular and rectangular holes. Anuar [18] studied the quantitative relationship between SMLF parameters and the tensile fatigue life of a prefabricated cracked steel plate under different tensile stress. Pospisil [19] used MMMT to accurately diagnose the fracture positions of prefabricated broken bars in concrete. Zhou [20] studied the relationship between the magnetic characteristic parameters, cable breakage damage, and steel strand corrosion. Liu [21] studied the quantitative relationship between natural gas pipeline cracks and the magnetic characteristic parameters.

Currently, MMMT is still an emerging NDT method. Previous experimental studies were mainly prefabricating the notch on the steel plate, and the applied stresses were mainly uniaxial tensile or compressive stress with uniform stress, and the structural styles were mostly limited to plates and the invisible damage in steel components was rarely considered. Previous engineering applications mainly focus on structures with simple stress forms such as metal pipes. Only a few studies have been conducted on the stress concentration caused by uneven stress under the combined action of bending and shear. To further study the feasibility of MMMT in monitoring invisible damage in complex steel structures, and to make up for the shortage that the existing NDTs can not pre-diagnose the invisible damage in bridge steel structures. In this paper, the MMMT method was combined with bridge steel structure to discuss the identification of early SCZ and early warning of critical yield state, which is of great significance for evaluating the reliability of bridge steel structure. The concrete work of this study was to conduct a four-point bending loading test on a scale model of a steel box girder, with practical significance. The distribution law of \(H_{{\text{p}}} (y)\) on the surface of the top flange, bottom flange, and web of the steel box girder and the force-magnetic relationship of magnetic signal measuring points that were symmetrical with the strain measuring points were studied. Furthermore, a magnetic parameter was extracted from the statistical point of view to qualitatively characterize the critical yield state of different force parts of the steel box girder. It should be noted that previous studies usually demagnetized the sample to reduce the interference of the initial residual magnetic field. However, there is no demagnetization condition for steel box girders with large sizes in practical engineering. To directly apply the test results to practice, the test sample was not demagnetized. The research results are expected to provide an experimental basis for the application of MMMT in bridge engineering.

2 Experiments

2.1 Material properties

The sample was made of Q345qC steel, which is the most widely used steel for bridge structures due to its high strength and toughness, good fatigue resistance and impact resistance, and Tables 1 and 2 lists the chemical composition and mechanical properties of the steel, respectively.

Table 1 Mechanical properties of the Q345qC steel sample /%
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Table 2 Chemical composition of the Q345qC steel sample
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2.2 Sample details

The thickness of the steel was 6 mm. The height of the web was 250 mm, the width of the bottom and top flanges was 500 and 540 mm, and the total length was 3000 mm. Two and five diaphragms were uniformly arranged on the webs and flanges, respectively, to ensure the local stability of the test beam. The three-dimensional model and cross-section size of the sample are shown in Figs. 1, 2, respectively. Nine diaphragms were arranged along the length of the beam from the support and welded with the top flange and webs, and the opening rate is 30%. To prevent local instability, diaphragms at the loading end on both sides were replaced by two diaphragms at a spacing of 100 mm. The top view of the sample is shown in Fig. 3, where the vertical dashed lines were the diaphragms.

Fig. 1
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3D model diagram of steel box girder (unit: mm)

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Fig. 2
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Cross-section size of steel box girder (unit: mm)

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Fig. 3
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The top view of the steel box girder (unit: mm)

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2.3 Layout plan of inspection points

Six, five, and six detection lines were arranged on the top, bottom flange, and web surfaces, respectively, and denoted as U1–U6, L1–L5, and W1–W6. In particular, one detection line was arranged on the lower surface of the overhanging part of the top flange, and denoted U7. The interval between the detection lines on the same part was 50 mm, and the interval between the detection points on the same detection line was also 50 mm.

14 strain gauges, denoted as US1–US14 and LS1–LS14, were placed on the top and bottom flanges, respectively. Six 45°-strain rosettes were evenly arranged on both sides of the bending shear section of the web, and eight strain gauges were arranged in the pure bending section, denoted as WS1–WS20. The strain gauge model was BX120-3AA with a resistance of 120 Ω ± 1%, an effect size of 3 mm × 2 mm, and a sensitivity factor of 2.00. In particular, to reduce the interference of the strain gauges on the SMLF intensity, the strain gauges were arranged on one side of the sample, and the detection points were symmetrically arranged on the other side. The schematic diagram of the detection points and strain gauges of the sample are shown in Fig. 4.

Fig. 4
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Detection line and strain gauge layout diagram (unit: mm): a top flange; b bottom flange; c web detection line; d web strain gauge

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2.4 Experimental instruments and test setup

The test equipment was the YAJ20000 electro-hydraulic servo compression shear testing machine, and Fig. 5 shows the loading diagram and test device. The force generated by the hydraulic test cylinder was divided into two uniform loads of equal width of 200 mm applied on the sample. Therefore, both ends of the sample were simply supported and bore a four-point bending load. The center area was a pure bending region, and the end region was a bending shear region. The actual loading levels in the test were 0, 150, 300, 450, 600, 650, 750, 850, 950, 1100, and 1184 kN. The yield load of the sample was 600 kN, and the ultimate load was 1184 kN.

Fig. 5
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Loading diagram and test device of steel box girder (unit: mm)

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During the test, the preset load was first added, and the EMS-2003 MMM instrument (Fig. 6) was used to measure the magnetic signal intensity along the detection line. The test used a two-channel pen probe, keeping the pen probe perpendicular to the detection plane, as a point-to-point measurement method (Fig. 7). To reduce the test error, each measurement point was repeated thrice, and their average was taken as the final value. After further loading to a higher preset value, the above steps were repeated until the sample broke. During instrument detection, the obtained magnetic field intensity is converted by the host through the probe composed of hall elements and finally displayed digitally. The measurement range was ± 1000 A/m, and the measurement error was lesser than ± 5%. Before the test, the probe sensor having a sensitivity of 1 A/m, based on the hall sensor, was calibrated in the geomagnetic field, having a value assumed at 40 A/m.

Fig. 6
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EMS-2003 metal magnetic memory detector

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Fig. 7
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Pen probe detection

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3 Theoretical background

3.1 Relationship between effective field and applied stress based on the Jiles–Atherton model in the elastic stage

Due to magnetic moment rotation and domain wall motion, applied stress can change the macroscopic magnetism of ferromagnetic materials. This phenomenon is called the magneto-mechanical effect. To explain the effect, the Jiles–Atherton model [22] was proposed based on proximity law and effective field theory. The Jiles–Atherton model was widely used in the study of the MMMT method because of its clear physical significance and relative agreement with the experimental results.

After a load is applied to a ferromagnetic sample, the magnetization M can be expressed by the Langevin function under the combined influence of the external magnetic field, load, and other factors [23]:

$$M = M_{{\text{s}}} \left[ {\coth \left( {\frac{{H_{{{\text{eff}}}} }}{{\mu_{0} \alpha }}} \right) - \frac{{\mu_{0} \alpha }}{{H_{{{\text{eff}}}} }} } \right],$$
(1)

where \(M_{{\text{s}}}\) is the saturation magnetization, \(\mu_{0}\) is vacuum permeability, \(H_{{{\text{eff}}}}\) is the effective magnetic field strength, \(\alpha\) is the interaction effect between magnetic field molecules, and expand the series of hyperbolic cosine functions, as follows:

$${\text{coth}}\;x = \frac{1}{x} + \frac{x}{3} - \frac{{x^{3} }}{45} + \frac{{2x^{5} }}{945} \cdots .$$
(2)

Omitting the higher-order term in Eq. (2), M can be simplified as,

$$M = M_{{\text{s}}} \frac{{H_{{{\text{eff}}}} }}{{3\mu_{0} \alpha }},$$
(3)

According to the Jiles–Atherton model, the expression for \(H_{{{\text{eff}}}}\) in the elastic stage is shown in:

$$\begin{aligned} H_{{{\text{eff}}}} & = H + \alpha M + H_{\sigma } \\ & = H + \alpha M + \frac{3}{2}\frac{\sigma }{{\mu_{0} }}\left( {\frac{{{\text{d}}\lambda }}{{{\text{d}}M}}} \right)\left( {\cos^{2} \theta - \nu \sin^{2} \theta } \right), \\ \end{aligned}$$
(4)

where H is the external magnetic field intensity, namely the geomagnetic field intensity, σ is the applied stress, \(\lambda\) is the magnetostrictive coefficient. θ is the angle between the stress and the magnetic field strength, ignoring the geomagnetic declination and the included angle is assumed 900. v is Poisson’s ratio.

The \(\lambda\) of a polycrystal can be represented by a series, as follows [24]:

$$\lambda = \mathop \sum \limits_{i = 0}^{\infty } \gamma_{i} \left( \sigma \right)M^{2i} ,$$
(5)

Expand \(\gamma_{i} \left( \sigma \right)\) by Taylor series as shown in,

$$\gamma_{i} \left( \sigma \right) = \gamma_{i} \left( 0 \right) + \mathop \sum \limits_{n = 1}^{\infty } \frac{{\sigma^{n} }}{n!}\gamma_{i}^{n} \left( 0 \right),$$
(6)

where \(\gamma_{i}^{n} \left( 0 \right)\) is the n-th derivative of \(\gamma_{i }\) when \(\sigma\) = 0. To simplify the calculation, the higher-order term of the series is omitted, and take i = 1, n = 1, that is,

$$\lambda = \gamma_{1} (\sigma )M^{2} ,$$
(7)
$$\gamma_{1} (\sigma ) = \gamma_{1} (0) + \gamma_{0}^{^{\prime}} (0)\sigma ,$$
(8)

where reference [23], \(\gamma_{1} (0) = 7 \times 10^{ - 18} \;{\text{m}}^{2} /{\text{A}}^{2}\), \(\gamma_{0}^{\prime } (0) = - 1 \times 10^{ - 25} \;{\text{m}}^{2} /{\text{A}}^{2} /{\text{Pa}}\).

Bringing Eq. (7) into Eq. (4), the simplified expression of \(H_{{{\text{eff}}}}\) is expressed as follows

$$H_{{{\text{eff}}}} = H + \alpha M - 3\nu \frac{\sigma }{{\mu_{0} }}\gamma_{1} (\sigma )M.$$
(9)

Bring Eq. (3) into Eq. (9) to obtain the final expression of \(H_{{{\text{eff}}}}\), as follows:

$$H_{{{\text{eff}}}} = \frac{{3\mu_{0}^{2} \alpha H}}{{3\mu_{0}^{2} \alpha - M_{s} \alpha \mu_{0} + 3M_{s} \nu \sigma \gamma_{1} (\sigma )}}.$$
(10)

When the material and environment of the test piece are fixed, the \(\mu_{0}\), \(\alpha\), H, \(M_{{\text{s}}}\), v, and \(\gamma\) are constants, and the value of the effective field \(H_{{{\text{eff}}}}\) is only related to the stress \(\sigma\).

3.2 Relationship between effective field caused by plastic deformation and strain

The mechanism of SMLF strength induced in the plastic phase was different from that in the elastic stage. Plastic deformation can produce magnetoplastic energy and cause dislocation aggregation, which hinders the movement of magnetic domains in the form of pinning points. The dislocation density, N is approximately linear with the plastic deformation \(\varepsilon_{{\text{p}}}\). Therefore, it can be considered that the magnetoplastic energy, \(E_{{\upsigma }}^{{\text{p}}}\) is equal to the pinning energy \(E_{{{\text{pin}}}}\) that the domain wall motion needs to overcome [25], that is,

$$E_{\sigma }^{{\text{p}}} = E_{{{\text{pin}}}} = \int_{0}^{M} {N\frac{{\left\langle {\varepsilon_{\pi } } \right\rangle }}{2m}{\text{d}}M} = \int_{0}^{M} {b\varepsilon_{{\text{p}}} \frac{{\left\langle {\varepsilon_{\pi } } \right\rangle }}{2m}{\text{d}}M} ,$$
(11)

where m is the magnetic moment per unit volume, \(\left\langle {\varepsilon_{\pi } } \right\rangle\) is the 180。-domain wall pinning energy, and b is the material-related parameter.

The effective field caused by plastic deformation \(H_{\sigma }^{{\text{p}}}\) could be obtained by the first-order differentiation of magnetization, M, as follows:

$$H_{\sigma }^{{\text{p}}} = - \frac{1}{{\mu_{0} }}\frac{{{\text{d}}E_{\sigma }^{{\text{p}}} }}{{{\text{d}}M}} = - k^{\prime } \varepsilon_{{\text{p}}} ,$$
(12)

where, \(k^{\prime } = \frac{1}{{\mu_{0} }}\frac{{b\left\langle {\varepsilon_{\pi } } \right\rangle }}{2m}\).

3.3 Classical characteristics of the MMMT method at damaged locations

When there is no external physical field, the magnetic domains and domain walls in ferromagnetic materials are irregular and do not show magnetism macroscopically. Under the joint action of the stress and the geomagnetic field, the rotation and movement of the domain wall occur in the ferromagnetic material, which shows the magnetism externally on the macro level. The magnetization state of the stressed component “remembers” the stress concentration and defect position. Its classical discriminative feature is shown in Fig. 8. The normal component of the SMLF, \(H_{{\text{p}}} (y)\), at the macroscopic defect location has the characteristics of changing sign and zero crossing and produces peak and trough on both sides of the defect.

Fig. 8
figure 8

Schematic diagram of magnetic signal characteristics in SCZ

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4 Test results and analysis

The damaged photo of the sample as shown in Fig. 9. From Fig. 9, the main damaged area of the sample was the midspan part of the top flange, and the overhang part, that is, the position of the U7 detection line, was warped.

Fig. 9
figure 9

Damaged photo of the sample

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4.1 Analysis of strain data

The flanges of the samples at the pure bending section and loading points were the most stressed. From the strain data, the loading point, mid-span of the flanges, and both sides of the web bending-shear sections reached the yield strain earlier, and until the end of the loading, the bending-shear regions on both sides of the top flange did not reach the yield strain. According to the value of the strain gauge, the relationship curves of the load F-strain ε at the maximum strain of each part of the sample are shown in Fig. 10. The strain gauges were arranged on one side of the cross-section of the sample, and the magnetic signal detection points were arranged on the other side of the symmetry. To simplify the calculation, it was considered that the cross-section of the sample and the loading mode were strictly symmetrical; therefore, it was considered that the strains at the symmetrical positions on both sides of the cross-section were equal.

Fig. 10
figure 10

F − ε relation curves: a top and bottom flanges; b web

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As shown in Fig. 10, for the US7 and US8 curves of the top flange and the LS6, LS7, and LS8 curves of the bottom flange, the slopes of the curves dropped significantly when they loaded to 600 kN. Hence, it could be determined that the yield strength of the strain measuring points of the top and bottom flange was 600 kN. Therefore, the yield load of the magnetic signal measuring point, symmetrical to the strain measuring point, was approximately 600 kN. Similarly, the yield strength of the magnetic signal measuring points, symmetrical to the web strain measuring points, was 650 kN.

4.2 The normal component of SMLF intensity, \(H_{{\text{p}}} (y)\)

Owing to abundant detection lines and tedious data, one representative detection line was selected at the top flange, bottom flange, and web for the analysis. The main obstacle in the online measurement of magnetic signals in the laboratory was the influence of a strong magnetic field caused by the long-term use of testing machines [12]. Therefore, the U7 detection line that was least affected by metal equipment was selected for the top flange. According to the strain data, the strain gauges LS6 and LS8 at the bottom flange had the maximum strain, and it could be approximated considering that the maximum strain of the bottom flange also appeared on the L4 detection line-symmetric with these two strain gauges. Similarly, the strain gauges WS4 and WS16 at the web had the maximum strain, and the W4 detection line was symmetrical to the two strain gauges. Therefore, the L4 and W4 detection lines were selected as representatives of the bottom flange and web to analyze the correlation between magnetic signal and SCZ.

According to the stress, the sample was divided into three sections: bending shear sections on both sides and pure bending sections. Because the pure bending section of the flanges and the bending shear section of the web were subjected to the largest force, we analyzed only the distribution law of \(H_{{\text{p}}} (y)\) in the sections with the maximum stress, as shown in Figs. 11, 12, where the left bending shear section was selected as an example. The dotted line in the figure was the position of the diaphragm, and the x-coordinate value was the distance from the detection point to the left support.

Fig. 11
figure 11

\(H_{{\text{p}}} (y)\) curves of different stress parts in the elastic loading stage: a L4 detection line; b U7 detection line; c W4 detection line

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Fig. 12
figure 12

\(H_{{\text{p}}} (y)\) curves of different stress parts in the plastic loading stage: a L4 detection line; b U7 detection line; c W4 detection line

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As shown in Fig. 11a, because the sample did not undergo demagnetization and stress relief treatment, the magnetic domain structures of the measuring points at different positions were pinned, and the pinned nodes exhibited different SMLF intensities at the macroscopic level, resulting in the magnetic field intensity of each measuring point not being zero when not loaded. There are many reasons for the formation of domain-fixed nodes, such as inclusions inside ferromagnetic materials, grain-size direction, uneven internal stress, and past magnetization history [9]. The change law of the \(H_{{\text{p}}} (y)\) curves was similar under different loads. Obviously, the classical features of diagnosing SCZ by the MMMT method were not applicable to this experiment. The curves showed extreme values at 1200 and 1500 mm. However, the 1200 and 1500 mm were just the SCZs caused by the diaphragms. This indicated that the extreme value of the \(H_{{\text{p}}} (y)\) curves had a good correlation with the position of stress concentration.

As shown in Fig. 11b, obviously, the curves had an extreme value at 1200 mm, while the extreme value at 1500 mm was not obvious. This is because the initial magnetic state of each detection point was random. When the magnetic domain structure at the detection point was severely pinned due to the past magnetization history, the external stress difficulty causes the rotation of the magnetic domain and the displacement of the domain wall, and the SMLF intensity of the detection point almost did not change with the change of the external load [8]. Compared with the bottom flange, the top flange was affected by the loading equipment, resulting in a more complex distribution and multiple extreme values on the curves. According to Fig. 9, the buckling damage source of the U7 detection line was about 1300 mm, and the damage range was about 1250–1350 mm, indicating that there was a large stress concentration in this area. The \(H_{{\text{p}}} (y)\) curves had a minimum value at a position of about 1300 mm, which further proved the correlation between the extreme value of the \(H_{{\text{p}}} (y)\) curves and the position of stress concentration. Different from the bottom flange, the \(H_{{\text{p}}} (y)\) curves of the top flange showed sawtooth fluctuation at 1000 and 1700 mm, which was caused by the uneven stress and SMLF strength generated by the loading end edge. 900 and 1800 mm were located in the center of the loading end, due to the influence of the loading end, the \(H_{{\text{p}}} (y)\) values had a large dispersion. The external interference of irrelevant ferromagnetic materials should be eliminated as far as possible when the MMMT method is applied in practical engineering.

As shown in Fig. 11c, note that 300 and 600 mm were the SCZs caused by the diaphragms. Obviously, the extreme value of the \(H_{{\text{p}}} (y)\) curves was created at 600 mm, which further verified the correlation between the extremum of the \(H_{{\text{p}}} (y)\) curves and the position of stress concentration. It was worth noting that unlike the pure bending section of the flanges, the 300 mm at the web was no obvious characteristic point in the early stage of loading, that is, before 450 kN. As the load further increased, the curves showed a minimum value here. This is because the force at 300 mm was relatively small, resulting in the magnetic characteristic being covered by the residual magnetic field. With further increased load, the stress accelerated the movement of the domain wall and the rotation of the magnetic moment at the detection point [8], leading to a significant increase in the SMLF strength at the SCZ, at this time, the effect of the stress on the \(H_{{\text{p}}} (y)\) values was greater than that of the residual magnetic field, magnetic characteristics began to appear. This indicated that the magnetic characteristics were clearer under the action of larger elastic stress. Compared with the pure bending section, the \(H_{{\text{p}}} (y)\) curves on the bending shear section fluctuated greatly. This may be because this region bore obvious composite stress, different kinds of stress led to different \(H_{{\text{p}}} (y)\) values, and the superposition of multiple \(H_{{\text{p}}} (y)\) values led to the obvious fluctuation of the \(H_{{\text{p}}} (y)\) curves. But the overall trend of the \(H_{{\text{p}}} (y)\) curves was still that the extreme value had a good correlation with the position of the stress concentration.

According to Fig. 11a–c, when the flanges were loaded to 600kN and the web to 650KN, the values of the \(H_{{\text{p}}} (y)\) curves were significantly larger, which indicated that the \(H_{{\text{p}}} (y)\) values were particularly sensitive to the yield behavior of the steel box girder. Because the extreme values of \(H_{{\text{p}}} (y)\) curves of the top, bottom flanges, and web were well correlated with the stress concentration position, the extreme values could be used to detect the stress concentration position of the steel box girder. However, this was obviously different from the classical diagnostic features of the MMMT method (Fig. 7), and this could be attributed to this test detecting an early stress concentration without evident appearance defects. The larger stress accelerated the movement of the domain wall and the rotation of the magnetic moment, resulting in a relatively larger SMLF intensity at the SCZ. While previous studies focused on stress concentration using a prefabricated notch [15,16,17, 19,20,21]. The permeability at the interface between the notch and matrix was close to the vacuum permeability. The magnetic-force line penetrates from the steel into the notch, and from the notch into the steel, causing two abrupt changes in the material parameters, and consequently, two abrupt changes in the magnetic field strength. Yin [26] reported that the normal magnetic field intensity is not necessarily zero in the SCZ if only the zero value is considered; specifically, the diagnosis of stress concentration position during the online test may be incorrect. In [27], a new method of using the peak value of \(H_{{\text{p}}} (y)\) to detect the SCZ was proposed. Zhang [28] showed that the \(H_{{\text{p}}} (y)\) curve has extreme points where the defects exist, so it can be judged that the defects and residual stress inside a steel plate at the extreme points of the curve are more severe than at other positions. The results of this experiment were consistent with the conclusions from previous literature [26,27,28].

As shown in Fig. 12a–c, after the elastic stage, the sample was subjected to greater stress, and stress redistribution occurred in the SCZ. The direction change and separation of the magnetic domain structure inside the sample distorted the magnetic signal. The \(H_{{\text{p}}} (y)\) distribution curves fluctuated greatly, and the \(H_{{\text{p}}} (y)\) values were significantly reduced, particularly after 1000 kN, and the accuracy of determining the stress concentration position by the \(H_{{\text{p}}} (y)\) curves decreased. Because the advantage of the MMMT method was to detect invisible damage, the plastic deformation stage of the sample was not discussed too much.

4.3 Analysis of the force-magnetic relationship of different parts of steel box girder

Currently, in bridge design, the yield strength is regarded as the limit condition for structural safety. Therefore, elastic stress is the primary working stress of steel under normal working conditions. If the yield strength of the steel box girders can be detected qualitatively, an early warning can be provided before the bridge is damaged. MMMT is a type of micromagnetic inspection, and its biggest advantage is that it can detect SCZ without evident defects to realize the early diagnosis of ferromagnetic components. With a change in the imposed load, the SMLF intensity on the sample surface also changed. An early warning was issued before the steel box girder was damaged by qualitatively judging the yield strength based on the changed characteristics of the SMLF.

4.3.1 Relationship between force and \(H_{{\text{p}}} (y)\)

Because there were mass detection points on the sample, the detection points with the largest strain were selected from different parts of the sample as representatives to analyze the relationship between external load F and \(H_{{\text{p}}} (y)\), and the selection basis is as follows: as shown in Fig. 10, the US7 and US8 strain measuring points were at the mid-span position, that is, 1350 mm, and their symmetrical magnetic signal measuring points were the 1350 mm measuring points on the U2 and U4 detection lines. However, the measuring point was close to the loading beam, distribution beam, and strain gauge, and the strong magnetic field generated by the metal equipment caused strong interference in the magnetic signal at this point. Therefore, the 1350 mm measuring point on the U7 detection line at the same cross-section and least affected by the metal equipment was selected for analysis. It is worth noting that the stress at each point on the flange cross-section was basically the same in the elastic stage; therefore, we selected the measuring points on the U7 detection line of the same cross-section that did not affect the judgment of the magnetic characteristic points. The magnetic signal measuring point symmetrical to the LS6 strain measuring point was at the 900 mm position on the L2 detection line, that is, the midpoint of the left loading regions. The magnetic signal measuring points symmetrical to the LS7 and LS8 strain measuring points were 1350 mm measuring points on the L2 and L4 detection lines, that is, the mid-span point. The magnetic signal measuring points symmetrical to the WS4 and WS16 strain measuring points were close to the 550 and 2150 mm measuring points on the W4 detection line.

Figure 13 shows the \(H_{{\text{p}}} (y)\)–F and ε–F relationship curves of the magnetic signal measuring point symmetrical to the maximum strain measuring point on the top flange, bottom flange, and web of the sample. To reduce the randomness of the \(H_{{\text{p}}} (y)\) distribution law of the measuring point, the measuring points on both sides of the target measuring point as shown in the figure.

Fig. 13
figure 13

\(H_{{\text{p}}} (y)\)-F relation curves: a top flange; b bottom flange; c web

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As shown in Fig. 13a, broadly, the three curves showed the same change trend as the load increases. In the elastic loading stage, the \(H_{{\text{p}}} (y)\) value increased with an increase in the load. When loaded to the yield load, that is, 600 kN, the curves had a peak value. Combined with the characteristics of the strain curves under this load, the peak could be used as the magnetic characteristic point of the top flange yield limit, representing the limit of the safe state of the structure, to achieve the purpose of identifying early damage, which was also the value of MMMT. When loaded to 1000 kN, a second peak appeared in the curves, the strain started to increase rapidly under this load, so the emergence of this peak indicated that the top flange had entered the plastic stage, and this peak could be used as a sign of an early warning of top flange failure. Generally, when the metal reaches the ultimate load, the opening of the macrocrack is only a fraction of a millimeter, which is a dead zone in most NDT methods [8]. MMMT has a unique function: It can evaluate the risk degree based on the magnetic anomaly parameters measured in the SCZ, as shown by the two peaks in the curves. After loading to 1000 kN, the \(H_{{\text{p}}} (y)\) value decreased rapidly with increasing load, and the top flange entered the failure stage. The most severely damaged regions of the sample were the mid-span regions of the top flange, and a photo after failure is shown in Fig. 9.

After the elastic loading stage, unlike the bottom flange and web, the \(H_{{\text{p}}} (y)\) value of the top flange regularly increased and then decreased with an increase in the load, which was caused by different stress mechanisms. In the elastic–plastic deformation stage, that is, between 650 and 1000 kN, Pitman [29] reported that after the measuring point was under pressure, a large plastic flow occurred inside it, the original lattice structure was damaged, and a new lattice structure was formed. At this time, the gap cracks at the measuring point were compressed, the structure was more uniform and denser, the deflection of the magnetic domain was further developed, and the magnetic signal was further increased. In the plastic deformation stage, that is, after 1000 kN, Leng [30] reported that with a further increase in plastic strain, the displacement and slip rate of the slip surface also increased, and a large number of dislocations appeared and gathered to form strong pinning, which hindered the unified arrangement and orientation of magnetic domains. Therefore, the magneto-elastic effect was suppressed, resulting in a decrease in magnetization at the measurement point.

As shown in Fig. 13b, broadly, the \(H_{{\text{p}}} (y)\) curves at the left loading point and the mid-span point on the bottom flange L2 and L4 detection lines had a similar variation law, that is, the \(H_{{\text{p}}} (y)\) values first developed in the negative direction and then increased in the positive direction with increasing load. Combined with the characteristics of the strain curves under this load, all 12 curves showed a unique peak at the yield load, that is, 600 kN. Similar to the top flange, the peak value of the force-magnetic relationship curve could also be used as the magnetic characteristic point of the yield limit of the bottom flange, representing the limit of the safe state of the structure. Additionally, the change law of the force-magnetic relationship curve of the bottom flange was different from that of the top flange, and the changing trend of the \(H_{{\text{p}}} (y)\) value was the opposite, which was mainly due to the different mechanisms of tensile stress and compressive stress on the SMLF on the surface of the sample. In this test, whether a certain part of the sample was under compression or tension could be clearly determined by the changing trend of the force-magnetic relationship curve, which was consistent with the test results of Yi et al. [9].

As shown in Fig. 13c, the \(H_{{\text{p}}} (y)\) values of the detection points W4-450, W4-500, and W4-550 mm in the left bending shear section and W4-2000, W4-2100, and W4-2150 mm in the right bending shear section changed in the opposite trend with the increase in load. Under the action of a four-point bending load, the shear force of the left bending shear section was positive and that of the right bending shear section was negative. The different shear directions on both sides resulted in different directions of the \(H_{{\text{p}}} (y)\) curves on the web surface. Therefore, the direction of the shear force could be reflected by the changing trend of the force-magnetic relationship curve. The \(H_{{\text{p}}} (y)\) curves in the bending shear section on both sides of the web had a unique peak value at a load of 650 or 750 kN, whereas the yield load of the target measuring point was approximately 650 kN. Combined with the characteristics of the strain curves under this load, the load value corresponding to the peak value of the curves was approximately equal to the yield strength of the measuring point. Similar to the top and bottom flanges, the peak value of the force-magnetic relationship curves could also be used as the magnetic characteristic point of the web yield strength to identify early damage. In the compression zone, the range of \(H_{{\text{p}}} (y)\) was 13–94 A/m, whereas in the tension zone, it was 106–54 A/m, and the values significantly increased. In other words, The magnetization was different under tensile and compressive stress, and the \(H_{{\text{p}}} (y)\) value was higher under tensile stress, which was consistent with the test results reported in the literature [12, 31]. The bending shear section of the web bore all of the shear force and part of the bending moment of the sample, and the amplitude of \(H_{{\text{p}}} (y)\) was superimposed under the action of different types of stresses, resulting in the maximum \(H_{{\text{p}}} (y)\) amplitude.

4.3.2 Discussion of the laws of the relationship between force and \(H_{{\text{p}}} (y)\)

The magnetic field intensity is a vector, and the positive and negative signs of the \(H_{{\text{p}}} (y)\) only represent the direction of the SMLF intensity, not the size. From Fig. 13a–c, in the elastic stage, the \(H_{{\text{p}}} (y)\) values of different parts of the steel box girder increased with the load increased. When loaded to yield force, the \(H_{{\text{p}}} (y)\) curves had a sharp peak. In the plastic deformation stage, that is, after 1000 kN, the \(H_{{\text{p}}} (y)\) values decreased with the load increased. However, In the elastic–plastic stage of the intermediate loading process, with the load increased, the \(H_{{\text{p}}} (y)\) values first increased and then decreased on the top flange, approximately fluctuate horizontally on the bottom flange, and decreased on the web, and different parts showed different magnetic characteristics. These magnetic phenomena are explained as follows:

  1. 1.

    Elastic loading stage According to the Jiles–Atherton model, the expression of stress σ and effective field \(H_{{\text{eff }}}\) in the elastic loading stage was obtained, as shown in Eq. 10. Solved the equation and got the following result: when σ > 35 MPa, then \(H_{{\text{eff }}}\) > 0. In this stage, the effective magnetic field strength increased with increasing stress, that is, there was a positive correlation between the effective magnetic field and stress. The steel used in this test was Q345qC, and its yield strength was greater than 345 MPa, indicating that the effective field and stress of the sample in the elastic stage mainly showed a positive correlation, that is, \(H_{{\text{p}}} (y)\) gradually increased as the stress increased. The theoretical results were consistent with the test results, implying that the theory could also be used for steel box girder structures encountering complex stresses.

  2. 2.

    Yield moment When the measuring point reaches the yield stress, the equivalent magnetic field generated by the stress is relatively large and the displacement of the domain wall can overcome the maximum value of the domain resistance. At this time, a slight increase in the stress caused the domain wall to move instantaneously, and the magnetization changed sharply, resulting in a sharp increase in the value and change rate of the magnetic signal near the yield stress of the measuring point [30]. Additionally, many scholars have found that the coercivity changes sharply under a very small deformation of approximately 0.5% [32,33,34]. Moreover, hysteresis loss [32] also exhibited an initial sharp step that was dependent on the plastic strain.

  3. 3.

    Elastoplastic loading stage In the elastic–plastic stage, that is, between 650 and 1000 kN, the deformation produced dislocations, dislocation entanglements, and even dislocation cell-like structures, which made the magnetic behavior more complex. Relevant studies have shown that the magnetic characteristics in the elastic–plastic deformation stage showed contradictory results; that is, with an increase in strain, the magnetic field intensity on the sample surface either increased [35] or decreased [30], or remained unchanged [36]. In this test, the different parts of the sample also exhibited different magnetic characteristics.

  4. 4.

    Plastic deformation stage × Equation 12 showed a relationship between the effective field caused by plastic deformation and strain, according to the literature [25], assuming the k′ = 200 A/m. From Eq. (12), the \(H_{{\text{p}}} (y)\) value decreased with increasing strain after the flanges and web entered the plastic deformation stage. The test results were consistent with the theoretical results.

4.4 Damage warning analysis based on magnetic characteristic parameter

The \(H_{{\text{p}}} (y)\) value of a single measurement point may result in measurement errors. If a statistical method was used, the measurement error would have been smaller, and the test results would have been easier to analyze. To verify the qualitative correlation between the F and \(H_{{\text{p}}} (y)\) at the critical yield state of different parts of the steel box girder, statistical methods were used to obtain the average value of the magnetic signal, \(\Delta H_{{\text{p}}} (y)_{a}\). To eliminate the influence of the initial residual magnetic field on the amplitude of a magnetic signal, the initial residual magnetic field value was subtracted from the \(H_{{\text{p}}} (y)\) value under different loads, and the equation is expressed as follows:

$$\Delta H_{p} (y)_{a} = {{\left\lfloor {\sum {H_{{\text{p}}} (y)_{{F_{i} }} - H_{{\text{p}}} (y)_{{F_{0} }} } } \right\rfloor } \mathord{\left/ {\vphantom {{\left\lfloor {\sum {H_{{\text{p}}} (y)_{{F_{i} }} - H_{{\text{p}}} (y)_{{F_{0} }} } } \right\rfloor } N}} \right. \kern-0pt} N},$$
(13)

where, \(H_{{\text{p}}} (y)_{{F_{i } }}\) and \(H_{{\text{p}}} (y)_{{F_{0} }}\) are the \(H_{{\text{p}}} (y)\) values of a detection point in the pure bending section of the flanges or the bending shear section of the web when the load is i and 0 kN, respectively. For flange, N is the number of detection points on an inspection line, and for web, N is the number of inspection lines. In other words,\({\Delta }H_{{\text{p}}} (y)_{a }\) represents the average value of the flanges along the length direction and the average on the same cross-section of the web along the height direction. It should be noted that the flanges were subjected to normal stress and the web was subjected to shear stress. They bore the largest stress in the pure bending section and bending shear section, respectively, and the stress magnitudes of all test points on the same test line in the corresponding section were almost the same. Therefore, the force-magnetic relationship between different detection lines on the flanges and different detection points on the web could be compared and analyzed under the same load. U6 was located at the edge of the sample, and L1 was close to the strain gauges (as shown in Fig. 4). The edge effect and strong external magnetic field caused no obvious characteristic in its \({\Delta }H_{{\text{p}}} (y)_{a }\) curves. From the strain data, the WS4 strain on the left bending shear section of the web was the largest, so the detection points symmetrical to WS4 were analyzed. Figure 14 shows the \({\Delta }H_{{\text{p}}} (y)_{a }\)-F and ε-F relationship curves at different force parts of the sample.

Fig. 14
figure 14

\(\Delta H_{{\text{p}}} {\text{(y)}}_{a }\)-F relationship lines: a top flange; b bottom flange; c web

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According to Fig. 14a, the distribution pattern of each \({\Delta }H_{{\text{p}}} (y)_{a }\)-F curve was similar and completely consistent with the law of the force-magnetic relation curves, which verified the accuracy of the force-magnetic relationship of the top flange. In the elastic loading stage, the \({\Delta }H_{{\text{p}}} (y)_{a }\) value of each curve increased approximately linearly with the increasing load. When loaded to about 650 kN, the curves had a maximum value. According to the characteristics of the strain curves under this load, the pure bending section just reached the yield load. With the further increasing load, the \({\Delta }H_{{\text{p}}} (y)_{a }\) values decreased rapidly, and then the curves trend changed inversely. In engineering, the reversal of the trend of the \({\Delta }H_{{\text{p}}} (y)_{a }\) curves could be used as a warning sign of the critical yield state of the top flange. In addition, the position of the U7 detection line was significantly different from that of the others. The other detection lines were more affected by the loading end and strain gauges, which led to the obvious difference in the amplitude of the \({\Delta }H_{{\text{p}}} (y)_{a }\) curves between U7 and others.

From Fig. 14b, similar to the force-magnetic curves, the \({\Delta }H_{{\text{p}}} (y)_{a }\)–F curves first decreased, then increased, and then fluctuated with the increasing load, which verified the accuracy of the qualitative relationship between F and \({\Delta }H_{{\text{p}}} (y)_{a }\) of the bottom flange. Combined with the strain curves, when loaded to about 600 kN, the curves created a minimum value, and the pure bending section just reached the yield strain. Similar to the top flange, the trend of the \({\Delta }H_{{\text{p}}} (y)_{a }\) curves changed in reverse with the further increasing load, In engineering, the reversal of the \({\Delta }H_{{\text{p}}} (y)_{a }\) curves trend could be used as a warning sign of the critical yield state of the bottom flange.

From Fig. 14c, the \({\Delta }H_{{\text{p}}} (y)_{a }\)–F curves distribution laws of different detection points were similar. In particular, unlike the \(H_{{\text{p}}} (y)\)–F curves of the web, the \({\Delta }H_{{\text{p}}} (y)_{a }\) curves showed a downward trend at the early stage of loading, that is, before 300 kN. According to Eq. (10), when σ = 35 MPa, the positive and negative signs of the effective field changed, and the experimental results were qualitatively similar to the theoretical results. Similar to the analysis method of the flanges, combined with the strain curve, when loaded to around 750 kN, the curves showed an extreme value and the web reached the yield state. In engineering, the reversal of the \({\Delta }H_{{\text{p}}} (y)_{a }\) curves trend could be used as a warning sign of the critical yield state of the web.

As seen in Fig. 14a–c, under the same load, \({\Delta }H_{{\text{p}}} (y)_{a }\) values at different positions had great dispersion, which made it difficult to establish a quantitative relationship between SMLF strength and external load. The reasons could be divided into controllable factors and uncontrollable factors. First, controllable factors. The data in this test were all manually detected, and the lifting value and angle of the detector probe were inevitably different in the detection process, which directly led to quantitative errors. Second, the uncontrollable factor, the MMMT method was based on the geomagnetic field as the excitation source and assumes that it remains constant in the detection project. However, the geomagnetic field was weak, ranging from 40 to 60 A/m, which was easily disturbed by the external magnetic field, such as the magnetic field generated by the loading types of equipment and strain gauges, led to the change of the environmental magnetic field (excitation magnetic field), and then affected the magnitude of SMLF intensity. In addition, the initial residual magnetic field, temperature, loading speed, material chemical composition, and size effect all had different degrees of influence on the SMLF intensity [9]. Therefore, applying the MMMT method to the quantitative evaluation of bridge steel structures requires a lot of tests to study the influence of various factors on the SMLF intensity and requires measures to improve the signal-to-noise ratio.

5 Conclusions

In this study, we investigated the distribution law of \(H_{{\text{p}}} (y)\) at the maximum stress position on the top flange, bottom flange, and web of the steel box girder under different bending loads. Furthermore, we analyzed the force-magnetic relationship of magnetic signal measuring points corresponding to the maximum strain measuring points on the different force-bearing parts, and the characteristic parameters of the magnetic signal were extracted to qualitatively characterize the critical yield state of different parts of the steel box girder. The following conclusions were drawn:

  1. 1.

    In the elastic loading stage, the peak value of the \(H_{{\text{p}}} (y)\) distribution curves could accurately determine the stress concentration position at the maximum stress position of the flanges and web of the steel box girder, however, the strong magnetic field such as the surrounding loading equipment will interfere with the detection accuracy of the MMMT method.

  2. 2.

    The load value corresponding to the peak value of the force-magnetic relationship curve of the top flange, bottom flange, and the web was the yield load of the corresponding part, which represented the limit of the safety state of the steel box girder structure. This characteristic point could be used as an early warning sign before the steel box girder was damaged. In particular, the appearance of the maximum \(H_{{\text{p}}} (y)\) value on the top flange indicated that it had entered the plastic failure stage, and this characteristic point could be used as an early warning sign before its failure. Additionally, the trend of the force-magnetic relationship curve reflected the type of tensile and compressive stresses experienced by the flange and the direction of the shear force experienced by the web.

  3. 3.

    The average SMLF intensity \(\Delta H_{{\text{p}}} (y)_{a }\) of different parts of the steel box girder and external load F was approximately linearly correlated in the elastic loading stage. While in the elastic–plastic or plastic deformation stage, the \(\Delta H_{{\text{p}}} (y)_{a }\) curves distribution trend suddenly changed in reverse. In engineering, the reverse change of the trend of the \(\Delta H_{{\text{p}}} (y)_{a }\)–F curves could be used as an early warning sign that the different parts of the steel box girder had reached the critical yield state.