Bending experimental study of structural steel beam on magnetic field gradient based on modified Jiles-Atherton model

Abstract

Previous research of metal magnetic memory (MMM) primarily focused on uniaxial stress-magnetic field behaviors and the judgment criterion of stress concentration zone mainly depended on the tangential and normal component of MMM signals. Magnetic field gradient was only mentioned as an auxiliary condition. In this paper, the variation characteristic of magnetic field gradient under bending load was investigated. Four-point bending tests of Q235B I-steel beams were carried out. The results show that all magnetic field gradient curves are centrosymmetric about midpoint of measurement line. The position of concentrated load can be preliminarily predicted by the mutational point of magnetic field gradient slope on the flange and accurately judged by the zero-crossing position of magnetic field gradient on the web. Whether a specimen is under eccentric compression or not, can be determined by comparing average characteristic parameter values on both sides of specimen. The variation law of magnetic field gradient on both the flange and web can be well explained by the modified Jiles-Atherton model, which is constructed by replacing stress $\sigma$ with equivalent stress $\sigma_{\text{eq}}$ . The calculated results are in good agreement with measured data.

Keywords

Metal magnetic memory magnetic field gradient Jiles-Atherton model bending test characteristic parameter

1. Introduction

In recent years, structural steel buildings have developed rapidly and been widely used. However, early damage of steel members will inevitably occur due to the action of complex stress condition and will occupy the most part of the whole service life [1]. With the accumulation of early damage, the normal operation and safety of the whole structure will be affected due to the degradation of stiffness and stability [2]. Therefore, the detection for early damage under complex stress state is necessary. Metal magnetic memory (MMM) has been demonstrated to be capable of diagnosing early damage and stress concentration of ferromagnetic material. The physical basis of MMM is self-magnetization under the action of stress. MMM measures the stress-induced surface magnetic fields of ferromagnetic materials under the excitation of geomagnetic field [3]. Therefore, MMM is an effective method for evaluating early damage, stress concentration and even preventing the occurrence of abrupt structural failure [4].

Recently, MMM method has received much attention due to several advantages since its first report and scholars worldwide have conducted much research. Bao et al. [5] investigated the variation of stress-induced magnetic field under tension and found that the peaks in the tangential component curve can determine the local stress concentration caused by cracks and dislocation. Guo et al. [6] measured the surface magnetic field intensity of Q345R during tensile tests, the result showed that the normal component signals decreased with a rise in stress, and then reversed to the initial field when the stress was greater than 160 MPa. Dong et al. [7], Hu et al. [8] and Roskosz and Bieniek [9] also researched variation of magnetic field during uniaxial tensile experiments and obtained corresponding results. However, previous research of MMM primarily focused on the effect of uniaxial tensile stress on the MMM signals, which can’t be applied in structural steel buildings. Because most of structural steel members (i.e. beam and slab) are mainly subjected to bending loads during service. In addition, the criterion to distinguish the stress concentration zone mainly depended on the tangential and normal component of MMM signals, magnetic field gradient was only mentioned as an auxiliary condition, which would easily lead to missing and false detection. Actually, magnetic field gradient can effectively restrain the effect of uniform background magnetic field and enhance the detection precision [10]. Therefore, the change law of magnetic field gradient under bending stress state should be investigated.

To this end, four-point bending tests of Q235B I-steel beams are carried out in this paper. The object of this work is to explore the variation characteristic of magnetic field gradient induced by different bending loads. The possible reason underlying the distribution of magnetic field gradient is analyzed and discussed by modified Jiles-Atherton model.

Table 1
Chemical composition of Q235B low carbon steel (wt. %)

Chemical	C	Si	Mn	P	S
Content	0.22	0.27	0.61	0.035	0.017

Table 2

Mechanical properties of Q235B low carbon steel

Properties	Elastic modulus/GPa	Yield strength/MPa	Ultimate strength/MPa
Content	200	$\geqslant$ 235	375 $\sim$ 460

2. Experimental details

2.1 Experimental method

The tested material is Q235B low-carbon steel, which is widely applied in steel structure field due to its excellent mechanical properties. Its chemical composition and mechanical properties are listed in Tables 1 and 2. The Chinese 14# hot-rolled I-steel beam with a length of 1500 mm was employed in four-point bending test, with a section size of 140 mm $\times$ 80 mm $\times$ 5.5 mm $\times$ 9.1 mm. To prevent the specimen from sliding during loading process, each support was overhung by 150 mm. Therefore, the actual length of loading span was 1200 mm. The distributions of measurement lines on the flange and web are shown in Fig. 1.

Figure 1.

Sketch of specimen and measurement lines (unit: mm).

The bending tests were carried out on YAW5000 hydraulic servo-actuated machine, whose loading error was within $\pm$ 1%. To make the steel beam bear four-point bending load, two 80 mm rigid cubic blocks, which were connected by a steel bar to ensure a fixed distance of loading positions, were arranged on the points of the trisection of specimen. Therefore, the location ranges that concentrated loads acted over on the left and right sides were 360 mm–440 mm and 760 mm–840 mm, respectively. Owing to the hindrance of loading point and the effect of loading surface on the magnetic signal, accurately measuring the magnetic signal on the extreme compression fiber of steel beam was unable to be achieved. Therefore, the measurement of the compressed flange was not performed during the test. Figure 2 shows the bending results of force versus displacement at mid-span of steel beam. According to the results, the steel beam experienced both elastic and plastic deformation with the increase of load. The specimen yielded at approximately 102 kN, and then achieved instability at about 135 kN. The multi-stage loading was adopted, and each stage increased at 10 kN. The test method and apparatus used in the experiment were described in [11]. The magnetic field was measured along the measurement lines from left to right with a lift-off value of 0.5 mm. Each measurement point was positioned at intervals of 50 mm.

Figure 2.

Force versus displacement curve at mid-span of steel beam.

2.2 Data processing

Because the magnetic field is a vector field, the changes in position and environment have significant impact on magnetic field gradient values. Thus the measurement was always performed at the same location and in the same environment [12]. When the specimen was loaded to a pre-set value, the magnetic signal along each measurement line was measured three times. To reduce the experimental error, the average of the measurement was considered as the best estimate of magnetic field gradient and the standard uncertainty of magnetic field gradient was estimated by standard deviation. The distribution of the data was characterized by standard error bars.

3. Theoretical background

An applied stress can alter the macro-magnetic properties of ferromagnetic materials due to the magnetic moments rotation and domain wall motion. Such a phenomenon is called magneto-mechanical effect. In order to explain magneto-mechanical effect, the Jiles-Atherton model [13] was proposed based on the law of approach and effective field theory.

According to law of approach, the magnetization of ferromagnetic material would move irreversibly towards the anhysteretic magnetization, the relationship between the anhysteretic magnetization $M_{\textit{an}}$ and stress $\sigma$ for an isotropic material can be quantified by the modified Langevin equation [14]

$\displaystyle M_{\textit{an}}(H,\sigma)=M_{s}\left[{\coth\left(\frac{H_{% \textit{eff}}}{a}\right)-\frac{a}{H_{\textit{eff}}}}\right]$ (1)

where $M_{s}$ represents saturation magnetization, and the constant $a$ is the effective domain density.

Effective field theory means that the magnetization under magnetic field $H$ and stress $\sigma$ , is identical to the magnetization under an equivalent effective magnetic field $H_{\textit{eff}}$ . In metal magnetic memory test, the equivalent effective magnetic field induced by the combined actions of applied stress and the geomagnetic field can be calculated using the following formula [15]

$\displaystyle H_{\textit{eff}}=H+\alpha M+H_{\sigma}$ (2)

where $H$ is the geomagnetic field, $\alpha$ is dimensionless mean field coefficient representing the interdomain coupling, $M$ is magnetization, and $H_{\sigma}$ is component of the effective field due to applied stress $\sigma$ . In certain circumstances, $H$ , $\alpha$ and $M$ remain approximately constant values. Therefore, equivalent effective field $H_{\textit{eff}}$ is determined by magnetic field $H_{\sigma}$ .

For an isotropic material under uniaxial stress state, the magnetic field $H_{\sigma}$ caused by applied stress can be described by dependence

$\displaystyle H_{\sigma}=\frac{3}{2}\frac{\sigma}{\mu_{0}}\left({\frac{d% \lambda}{dM}}\right)_{\sigma}\left({\cos^{2}\theta-\nu\sin^{2}\theta}\right)$ (3)

where $\mu_{0}$ is permeability of free space, $\theta$ is the angle between the stress axis and the direction of external magnetic field, $\nu$ is Poisson’s ratio and $\lambda$ is bulk magnetostriction. Since the magnetostriction must be symmetric about $M=0$ , the bulk magnetostriction $\lambda$ can be described by a simple series expansion [16]

$\displaystyle\lambda=\sum\limits_{j=0}^{\infty}{\gamma_{j}}(\sigma)M^{2j}$ (4)

where $\gamma_{j}(\sigma)$ can be given using a Taylor series expansion

$\displaystyle\gamma_{j}(\sigma)=\gamma_{j}(0)+\sum\limits_{n=1}^{\infty}{\frac% {\sigma^{n}}{n!}\gamma_{j}^{n}(0)}$ (5)

where $\gamma_{j}^{n}(\sigma)$ is the $n$ th derivative of $\gamma_{j}$ with respect to stress at $\sigma=0$ . By taking approximation method and omitting its high-order term [17, 18], the magnetostriction $\lambda$ can be obtained by including the terms up to $j=1$ and $n=1$ , this gives

$\displaystyle\lambda=\gamma_{1}(\sigma)M^{2}=\left[{\gamma_{1}(0)+{\gamma}^{% \prime}_{1}(0)\sigma}\right]M^{2}$ (6)

Using the magnetostriction data measured by Kuruzar and Cullity on polycrystalline iron [19], the parameter values are given: $\gamma_{1}(0)=7\times 10^{-18}$ m ${}^{2}$ /A ${}^{2}$ , ${\gamma}^{\prime}_{1}(0)=-1\times 10^{-25}$ m ${}^{2}$ /A ${}^{2}$ /Pa ${}^{1}$ . Therefore, the equivalent effective field is given by

$\displaystyle H_{\sigma}=\frac{3\sigma M}{\mu_{0}}\left({7\times 10^{-18}-1% \times 10^{-25}\sigma}\right)\left({\cos^{2}\theta-\nu\sin^{2}\theta}\right)=A% \sigma-B\sigma^{2}$ (7)

where $A=\frac{2.1M}{\mu_{0}}\left({\cos^{2}\theta-\nu\sin^{2}\theta}\right)\times 10% ^{-17}$ and $B=\frac{3M}{\mu_{0}}\left({\cos^{2}\theta-\nu\sin^{2}\theta}\right)\times 10^{% -25}$ . As mentioned above, the measurement was always performed at the same location and in the same environment during the experiment. Therefore, the external magnetic field can be regard as constant. A simplified method for considering the angle between the stress axis and the direction of magnetic field was employed in literature [20], which considered the magnetic field acting only along the stress axes. Therefore, $A$ and $B$ can be considered as constant values.

Substituting Eq. (7) into Eq. (2), the final equivalent effective magnetic field is obtained

$\displaystyle H_{\textit{eff}}=H+\alpha M+A\sigma-B\sigma^{2}$ (8)

Thus, it can be concluded that the magnetic field $H_{\textit{eff}}$ are mainly affected in terms of applied stress $\sigma$ .

Although it has been widely employed in the research of MMM for its definite physical meaning and relative consistency with experiment results, Jiles-Atherton model still has some fundamental limitations. For example, this model is only valid under uniaxial stress state and can’t be applied directly to complex stress state.

4. Experimental results

Based on Euler-Bernoulli theory, bending loads will cause zero stress at the neutral axis, tensile stress in the lower region of the beam, and compressive stress in the upper region of the beam. Therefore, the studies on magnetic field gradient along measurement line 1 on the flange as well as lines 1 and 5 on the web were conducted to explore the effect of bending load on magnetic field gradient. Since the distributions of magnetic field gradients along each measurement line under different loads exhibit the similar trend, the gradients under F $=$ 20 kN, 40 kN, 60 kN, 80 kN, 100 kN and 120 kN are extracted to reflect regularity and improve readability.

Figure 3.

Distribution of gradient $K$ under different loads on the flange.

Figures 3 and 4 show the variation of magnetic field gradient $K$ along the measurement lines under different bending loads. The magnetic field gradient curve along each measurement line exhibits the similar trend under different loads, which is centrosymmetric about the midpoint of measurement line.

Figure 3 indicates that magnetic field gradient curve on the flange decrease monotonically along with measurement line. The variation of magnetic field gradient is slight, nearly close to zero in the pure bending region, whereas the magnetic field gradient in the non-uniform bending region shows an approximately straight line with negative slope. There exist mutational points of magnetic field gradient slope at interface of each region. Therefore, the mutational point can be used to preliminarily predict the position of concentrated load. However, this judgment criterion has a certain error. The mutational point is located outside of loading position. Further research should focus on magnetic field gradient curve of the web.

Figure 4.

Distribution of gradient $K$ under different loads on the web: (a) measurement line 1 and (b) measurement line 5.

Figure 4 shows that the magnetic field gradient curve exhibits double peak-peak changes and passes through zero. To research the positional relationship between concentrated load and zero-crossing point of magnetic field gradient on the web, the statistics of zero-crossing points close to loading positions are listed in Table 4. For the zero point not on measurement point, the zero-crossing position is identified by linear interpolation method. According to Table 4, the zero-crossing positions on the left and right sides of magnetic field gradient along lines 1 and 5 are not fixed, namely occur zero drift. The drift of zero-crossing position mainly concentrates in the range that concentrated loads act over (360 mm–440 mm and 760 mm–840 mm). In addition, the average zero-crossing positions under different loads are all within the range of concentrated loads. Therefore, the zero-crossing position of magnetic field gradient on the web can be used to judge the position of concentrated load.

Figure 5.

Characteristic parameters on the flange (F $=$ 60 kN).

Figure 6.

Characteristic parameters on the web: (a) measurement line 1 and (b) measurement line 5 (F $=$ 60 kN).

After the experiment, the deformations on both sides of specimen were observed. It is shown that the deformation of beam on the right in the stress concentration zone is greater than that on the left. Such a phenomenon is caused by eccentric compression of beam. The left and right loading values are not equal during the loading process. To explore the relationship between deformation degree of specimen and magnetic field gradient, several characteristic parameters of gradient curve are extracted as shown in Figs 5 and 6. Table 4 lists characteristic parameter values of magnetic field gradient under different loads. According to Table 4, it is obvious that the right average characteristic parameter values are greater than the left, which are consistent with the deformation degree of experimental result. However, the perform-

Table 3

Zero-crossing positions near loading points of magnetic gradient on the web

Zero-crossing	Load (kN)														Average
position		10	20	30	40	50	60	70	80	90	100	110	120	135	position
Line 1	Left	421.2	342.3	350.0	373.6	398.2	371.4	359.5	365.6	377.9	371.4	361.8	367.5	329.4	368.5
	Right	821.0	801.7	779.3	784.6	790.2	804.5	798.8	781.0	825.6	823.8	824.7	833.5	792.1	804.7
Line 5	Left	392.6	437.5	418.8	429.2	359.1	350.0	405.0	418.8	496.2	386.4	394.5	411.1	400.0	407.6
	Right	800.0	850.0	861.4	850.0	807.1	778.0	812.5	800.0	740.6	804.2	728.8	761.5	761.1	796.6

Table 4

Characteristic parameter values of magnetic field gradient under different loads ( $\times$ 10 ${}^{-3}$ )

Characteristic	Load (kN)													Average
parameter	10	20	30	40	50	60	70	80	90	100	110	120	135	value
$\Delta K_{1}(y)_{\text{f}}$	1.05 $\pm$ 0.24	1.61 $\pm$ 0.31	1.52 $\pm$ 0.25	1.92 $\pm$ 0.29	1.57 $\pm$ 0.29	1.97 $\pm$ 0.31	3.04 $\pm$ 0.31	4.35 $\pm$ 0.33	3.74 $\pm$ 0.31	4.33 $\pm$ 0.30	2.87 $\pm$ 2.95	3.12 $\pm$ 0.28	4.28 $\pm$ 0.25	2.72 $\pm$ 1.19
$\Delta K_{1}(y)_{\text{f2}}$	2.05 $\pm$ 0.30	2.69 $\pm$ 0.29	1.74 $\pm$ 0.29	2.09 $\pm$ 0.29	2.29 $\pm$ 0.33	2.96 $\pm$ 0.35	5.76 $\pm$ 0.31	4.88 $\pm$ 0.33	4.97 $\pm$ 0.29	4.20 $\pm$ 0.33	5.49 $\pm$ 0.27	6.52 $\pm$ 0.27	3.41 $\pm$ 0.28	3.77 $\pm$ 1.62
$\Delta K_{1}(y)_{\text{w1}}$	0.69 $\pm$ 0.12	0.54 $\pm$ 0.16	0.22 $\pm$ 0.15	0.52 $\pm$ 0.15	0.80 $\pm$ 0.13	0.53 $\pm$ 0.15	0.34 $\pm$ 0.16	0.51 $\pm$ 0.19	0.97 $\pm$ 0.16	0.51 $\pm$ 0.16	0.60 $\pm$ 0.16	1.00 $\pm$ 0.15	0.83 $\pm$ 0.17	0.62 $\pm$ 0.23
$\Delta K_{1}(y)_{\text{w2}}$	0.65 $\pm$ 0.17	0.94 $\pm$ 0.13	0.44 $\pm$ 0.16	0.62 $\pm$ 0.14	0.63 $\pm$ 0.14	0.93 $\pm$ 0.17	0.79 $\pm$ 0.18	0.63 $\pm$ 0.12	0.93 $\pm$ 0.15	0.52 $\pm$ 0.11	1.06 $\pm$ 0.12	1.21 $\pm$ 0.14	0.70 $\pm$ 0.09	0.77 $\pm$ 0.23
$\Delta K_{5}(y)_{\text{w1}}$	0.34 $\pm$ 0.12	0.24 $\pm$ 0.08	0.27 $\pm$ 0.10	0.38 $\pm$ 0.02	0.37 $\pm$ 0.07	0.40 $\pm$ 0.13	0.48 $\pm$ 0.13	0.26 $\pm$ 0.09	0.16 $\pm$ 0.10	0.43 $\pm$ 0.10	0.70 $\pm$ 0.13	0.48 $\pm$ 0.08	0.08 $\pm$ 0.10	0.35 $\pm$ 0.16
$\Delta K_{5}(y)_{\text{w2}}$	0.26 $\pm$ 0.12	0.18 $\pm$ 0.09	0.16 $\pm$ 0.12	0.39 $\pm$ 0.11	0.20 $\pm$ 0.10	0.37 $\pm$ 0.09	0.16 $\pm$ 0.10	0.19 $\pm$ 0.10	0.18 $\pm$ 0.09	0.16 $\pm$ 0.06	0.30 $\pm$ 0.06	0.38 $\pm$ 0.03	0.14 $\pm$ 0.11	0.24 $\pm$ 0.09

ance of magnetic field gradient along line 5 on the web is opposite, which is consistent with the phenomenon observed by Yi et al. [21]. The explanation for this phenomenon is that the supports were magnetized before the test, and the superposition between the magnetic signals along line 5 and the interference signals generated by the supports was occurred during loading process. Consequently, whether a specimen is under eccentric compression or not, can be determined by comparing average characteristic parameter values on both sides of specimen.

5. Analysis and discussion

According to the previous analysis, the magnetic signals along line 5 on the web were affected by the two supports during the test, which is inconsistent with the actual behaviors. Thus, there are no analysis and discussion about magnetic field gradient along line 5.

Magnetic field gradient $K$ characterizes the rate of change in the magnetic signals [22]. The magnetic field gradient $K$ and magnetic signals $H_{p}(y)$ can be quantified by the following dependence

$\displaystyle K=\frac{H_{p}(y)_{i+1}-H_{p}(y)_{i}}{\Delta L}$ (9)

where $H_{p}(y)_{i+1}$ and $H_{p}(y)_{i}$ are magnetic signals respectively at adjacent measurement points $i+1$ and $i$ , and $\Delta L$ is the distance between two adjacent measurement points, which is equal to 50 mm in this experiment. The resulting magnetic field gradient $K$ can be expressed by substituting Eq. (8) into this

$\displaystyle K=\frac{A(\sigma_{i+1}-\sigma_{i})-B(\sigma_{i+1}-\sigma_{i})(% \sigma_{i+1}+\sigma_{i})}{\Delta L}$ (10)

where $\sigma_{i+1}$ and $\sigma_{i}$ are stresses corresponding to magnetic signals $H_{p}(y)_{i+1}$ and $H_{p}(y)_{i}$ .

Given that the web of steel beams are subjected to both bending stress $\sigma_{x}$ and shear stress $\tau_{x}$ rather than uniaxial stress, Jiles-Atherton model is no longer applicable. Therefore, in order to apply Jiles-Atherton model to structural steel buildings, appropriate modification is required. To solve this problem, a fictive uniaxial equivalent stress is introduced, which would change the magnetic behavior in a similar manner to a multiaxial situation [23]. Schneider and Richardson [24] proposed the equivalent stress method as the following

$\displaystyle\sigma_{\text{eq}}=\sigma_{1}-\sigma_{2}$ (11)

where $\sigma_{1}$ and $\sigma_{2}$ are two principal stresses in plane stress state.

Therefore, Jiles-Atherton model is modified by replacing stress $\sigma$ with equivalent stress $\sigma_{\text{eq}}$ , and Eq. (10) is rewritten as

$\displaystyle K=\frac{A(\sigma_{\text{eq},i+1}-\sigma_{\text{eq},i})-B(\sigma_% {\text{eq},i+1}-\sigma_{\text{eq},i})(\sigma_{\text{eq},i+1}+\sigma_{\text{eq}% ,i})}{\Delta L}$ (12)

where $\sigma_{\text{eq},i+1}$ and $\sigma_{\text{eq},\,i}$ are equivalent stresses corresponding to magnetic signals $H_{p}(y)_{i+1}$ and $H_{p}(y)_{i}$ .

5.1 Analysis and discussion for flange

Since the measurement lines on the flange are located at the extreme tension fiber of steel beam, all the measurement points are only subjected to bending stress $\sigma_{x}$ . Therefore, the two principal stresses $\sigma_{1}$ and $\sigma_{2}$ in Eq. (11) are respectively equal to $\sigma_{x}$ and zero, which means that $\sigma_{\text{eq}}=\sigma_{x}$ .

Figure 7.

Stress diagram under bending load on the flange.

Figure 7 shows the stress diagram of the flange under four-point bending load (Because of symmetry, only left half is shown.). In the pure bending region, the stress value $\sigma_{x}$ keeps constant, which can be calculated by the following expression

$\displaystyle\sigma_{\text{eq},i}=\sigma_{x,i}=\frac{FLh}{12I}$ (13)

where $h$ and $L$ respectively represent the height and span of beam, and $I$ is the moment of inertia of the cross-section. In the non-uniform bending region, the relationship between stress $\sigma_{x}$ and position coordinate $L_{i}$ is linear, which can be expressed as

$\displaystyle\sigma_{\text{eq},i}=\sigma_{x,i}=kL_{i}+b$ (14)

where $k$ and $b$ respectively represent slope and intercept, which are dependent on load $F$ and span $L$ . For a certain load, $k$ and $b$ can be considered as constant values.

The resulting magnetic field gradient $K$ on the flange can be obtained by substituting Eqs (13) and (14) into Eq. (12)

$\displaystyle K={\begin{array}[]{ll}0\hfill\\ {-2Bk^{2}L_{i}+Ak-Bk^{2}\Delta L-2Bbk}\hfill\\ \end{array}}\quad{\begin{array}[]{l}{\text{(pure bending region)}}\\ {\text{(non-uniform bending region)}}\\ \end{array}}$ (15)

where $A$ , $B$ , $k$ , $b$ and $\Delta L$ can be considered as constant values as mentioned above.

In the pure bending region, the reason why the magnetic field gradient curve resembles a horizontal line, nearly close to zero is that magnetic field gradient $K=0$ . In the non-uniform bending region, magnetic field gradient $K$ is a linear function of position coordinate $L_{i}$ and the slope $-2Bk^{2}$ is negative, which can explain the reason why the variation of magnetic field gradient shows an approximately straight line with negative slope. Therefore, the change law of magnetic field gradient on the flange shown in Fig. 3 can be well explained by the modified Jiles-Atherton model.

5.2 Analysis and discussion for web

Different loads only change the magnitude and do not change the variation trend of equivalent stress. Here, the specimen under F $=$ 90 kN is taken as an example. Given that measurement line 1 on the web is subjected to compressive stress, the following calculated values all take the negative.

Since the web of steel beam is subjected to both bending stress $\sigma_{x}$ and shear stress $\tau_{x}$ , the principal stresses $\sigma_{1}$ and $\sigma_{2}$ can be calculated using

$\displaystyle\sigma_{1}=\frac{1}{2}\sigma_{x}+\frac{1}{2}\sqrt{\sigma_{x}^{2}+% 4\tau_{x}^{2}}$ (16) $\displaystyle\sigma_{2}=\frac{1}{2}\sigma_{x}-\frac{1}{2}\sqrt{\sigma_{x}^{2}+% 4\tau_{x}^{2}}$ (17)

where $\sigma_{x}$ and $\tau_{x}$ represent bending stress and shear stress of measurement point on the web, respectively. The resulting equivalent stress $\sigma_{\text{eq}}$ can be obtained by inserting Eqs (16) and (17) into Eq. (11)

$\displaystyle\sigma_{\text{eq}}=\sqrt{\sigma_{x}^{2}+4\tau_{x}^{2}}$ (18)

Figure 8.

Relationship between the calculated $K_{\text{eq}}$ and measured $K$ (F $=$ 90 kN).

By substituting Eq. (18) into Eq. (12) and normalizing the calculation results, the distribution of calculated magnetic field gradient $K_{\text{eq}}$ along line 1 on the web is presented in Fig. 8. Except both ends of beam, the main variation tendency of $K_{\text{eq}}$ is quite similar to that of measured $K$ . The extrema positions of $K_{\text{eq}}$ and $K$ are primarily the same. Therefore, the change law of magnetic field gradient shown in Fig. 4a can be well explained by the modified Jiles-Atherton model as well.

6. Conclusions

Four-point bending tests of steel beams were conducted to analyze the influence of different loads on the values of magnetic field gradient. The characteristic parameter of magnetic field gradient is extracted. The possible reason underlying the distribution of magnetic field gradient is analyzed and discussed by modified Jiles-Atherton model. The following conclusions were obtained:

The magnetic field gradient curve exhibits the similar variation trend during loading process, which is centrosymmetric about the midpoint of measurement line. In the pure bending region of steel beam, the variation of magnetic field gradient is slight, nearly close to zero. In the non-uniform bending region, the magnetic field gradient shows an approximately straight line with negative slope.

The position of concentrated load can be preliminarily predicted by the mutational point of magnetic field gradient slope on the flange and accurately judged by the zero-crossing position of magnetic field gradient on the web. Whether a specimen is under eccentric compression or not, can be determined by comparing average characteristic parameter values on both sides of specimen.

The modified Jiles-Atherton model is constructed by replacing stress $\sigma$ with equivalent stress $\sigma_{\text{eq}}$ . The variation law of magnetic field gradient on both the flange and web can be well explained by the modified Jiles-Atherton model. The calculated results is consistent with measured data.

Footnotes

Acknowledgments

The authors are grateful for the financial support received from the National Natural Science Foundation of China (No. 51578449, No. 51478383) and the General Programs of the Shaanxi Provincial National Science Foundation (No. 2015JM5192).

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Bending experimental study of structural steel beam on magnetic field gradient based on modified Jiles-Atherton model

Abstract

Keywords

1. Introduction

Table 1 Chemical composition of Q235B low carbon steel (wt. %)

2.1 Experimental method

3. Theoretical background

Footnotes

Acknowledgments

References

Table 1
Chemical composition of Q235B low carbon steel (wt. %)