Equivalent stress for steel box girder with corrugated web under bending based on magnetic memory method

Abstract

Metal magnetic memory testing (MMMT) is a nondestructive testing technique that can detect early damage of component. In order to distinguish the effect of normal stress and shear stress on magnetic field, the flexural loading testing of inclined steel box girder with corrugated web is carried out by four-point bending pattern, and the normal component of the magnetic field at different position of the beam under different load level are collected. It is found that the law of normal component of the magnetic field changing with load is similar under different stress states, but the amplitude is quite different. The stress in Jiles–Atherton model can be replaced by equivalent stress when calculating the magnetic field under shear stress.

Keywords

Metal magnetic memory testing (MMMT)bridge steel Q345qC four-point bending shear stress equivalent stress

1. Introduction

Steel structure bridge has the advantages of lightweight and fast processing in bridge construction. For the steel box girder with corrugated web, which is one type of steel structure bridges, the high shear capacity and high stability give it more opportunities to be applied. Steel structures are usually served in the elastic phase because they are prone to damage after yield. So it is necessary to have research on the yielding behavior to avoid sudden structural damage.

In the late 1990s, Russian scholar Doubov proposed the metal magnetic memory detection technology [1,2], stress concentration and various microscopic defects can be found by which. In order to catch the yielding phenomenon, the research on the change law of magnetic value during the whole loading process is needed.

Many researchers have studied the variation of the magnetic field with stress in the geomagnetic field. For the elastic stage, Yao [3] studied the variation of magnetic field strength under uniaxial tension and found that magnetic memory detection technology was more sensitive to the tensile test. Dong [4], Guo [5] studied the variation of the normal magnetic field with stress on tensile specimens and had a conclusion that the gradient of absolute value increases with stress in the elastic stage. Bao [6] found that the magnetic field of different directions showed different characteristics when the specimen was destroyed. For plastic stage, Leng [7], Li [8] studied the effect of plastic deformation on the magnetization process of ferromagnet material and found that the strength of the magnetic field will decrease with the increasing of plastic deformation. All the study above just focus on a particular stage during the loading process and the stress along the detection lines are the same. However, the stress distribution of the specimen is more complicated for flexural member.

For the research of complicated stress state, Ma [9] has proposed a modified model of magneto-mechanical coupling for steel wire under torsion, and Su [10] have studied the changing trend of the magnetic signal with stress along the measurement line and found that the normal component H _p(y) shows reasonable qualitative and quantitative correlations with the stress. However, neither of them has mentioned the difference effects of normal stress and shear stress on magnetic field. There are several types of bridge loads, such as tension, compression, bending, shear, and torsion. For thin-walled members, both bending failure and shear buckling will occur. Once shear buckling occurs to the web, the box girder loses its shear bearing capacity, which is also considered as failure. Therefore, it is necessary to study the effect of shear stress.

In this paper, the four-point bending test of Q345qC inclined steel box girder with corrugated web is carried out. The object of this work is to find out the difference effects of the normal stress and the shear stress on magnetic field by analyzing the change law of the normal component H _p(y) during the whole loading process in different stress condition. The equation of equivalent stress is analyzed afterward.

2. Experimental details

Table 1 and Table 2 exhibit the chemical composition and mechanical properties of the specimen shown in Fig. 1 separately. In order to ensure the local stability of the steel box girder, the flange of the steel box girder was equipped with transverse and longitudinal stiffening rib.

Table 1
Chemical composition of the Q345qC steel specimen

Material %

Name C Si Mn P S

Q345qC ≤0.20 ≤0.60 1.0--1.6 ≤0.035 ≤0.035

	Material %
Q345qC	≤0.20	≤0.60	1.0--1.6	≤0.035	≤0.035

Table 2

Mechanical properties of the Q345qC steel specimen

Name	Position	Yield stress/MPa	Tensile stress/MPa	Elongation/%
	Upper flange	≥345	503	≥21
Q345qC	Web	≥345	498	≥21
	Bottom flange	≥345	510	≥21

Fig. 1.

The steel box grider for experiment.

The steel box girder used for testing was welded by Q345qC steel. The thickness of the flange is 8 mm, and the thickness of the web is 6 mm. The web was designed in accordance with the bridge design model of The combined folding web and Specifications for Design of Highway steel bridge (JTG d64-2015). The cross section of the steel box girder is shown in Fig. 2.

Fig. 2.

Steel box girder cross section.

As shown in Fig. 3, the signal detection lines were arranged along the length of the steel box girder. The distance between the horizontal test points was 50 mm, and the intersection of the two lines was the location of the test point. In order to avoid the interference of strain gauge to the magnetic signal, the detection lines were arranged on one side of the steel box girder, and the strain gauges were stuck on the other side symmetrically. The upper flange had a test line of u1, and four detection lines of b1, b2, b3, b4 were installed on the bottom flange. For the web, five detection lines w1, w2, w3, w4, w5 were installed.

Fig. 3.

Magnetic signal detection line layout: (a) Detection lines on the web. (b) Detection line on the upper flange. (c) Detection lines on the bottom flange.

2.1. Experimental procedure

The test was conducted using YAJ20000 electro-hydraulic servo loading instrument, and the maximum test force is 2000 kN. Figure 4 presents an image of the experiment system. The EDDYSUN EMS-2003 intelligent magnetic memory/eddy current detector was used to detect the normal component signal H _p(y). A pre-test is indispensable so that the instrument was calibrated according to sensor 1 in the instrument probe, and the background magnetic field suppression was selected as “-ch1” to suppress the magnetic field strength. As a result, the real magnetic signal can be obtained. while detecting, the double-channel pen probe was used and the detector display mode of the detector was transformed into digital data.

Fig. 4.

The experiment machine.

The loading position of the specimen is shown in Fig. 3(a). The total length of the steel box girder is 1800 mm. The outstretched 150 mm from each side of the beam was needed to form a simply supported beam with a length of 1500 mm. A Rectangular plate and a vertical load distribution component of welded round steel were set at the two loading points. The loading points were set at the 1/3 length of the beam, so the two ends of the steel box girder were bending-shear section and the area between two loading points is the pure bending section. To ensure the accuracy of the load point and the concentrated force applied, the plate and the load distribution were welded and fixed with two steel bars. The vertical load distribution members were added to the steel box beam before loading.

Preloading the test instruments is also needed to ensure normal work. The material characteristic test was conducted before the formal test. According to the stress-strain curve of the material test shown in Fig. 5, the specimen yielded at approximately 798 kN and then achieved instability at about 1250 kN. The multi-stage loading was adopted, and each stage increased at 200 kN. The initial magnetic signal value is collected at the unloaded time, and the normal component value H _p(y) was detected by adopting the on-line method after each level of loading.

Fig. 5.

Force versus displacement curve at mid-span steel beam.

2.2. Error reduction measure

Some measures were taken for the sake of the accuracy of the results. In order to reduce the impact of accidental errors, three sets of data at each detection point were collected, but the data who was significantly different from others would be discarded. By taking the average of the rest, the relatively accurate result can be obtained. In addition, normalizing the magnetic signal detector before every detection of each test line can help to reduce unnecessary distractions. To normalize, the probe with the instruments was calibrated with the earth magnetic field so that the instrument and probe can match each other.

3. Experimental result

According to the force of the specimen, the beam can be divided into two regions: the pure bending area and the bending-shear area. The longitudinal direction of the beam is set as the direction of the x-axis, and the single x value is corresponding to each measuring point of the beam. The origin of the x-axis is set at the leftmost end of the beam, as shown in Fig. 3(a).

The stressed section of the beam is divided into the pure bending section and the bending shear section. As shown in Fig. 6, there are four detective lines on the bottom flange and each cross-section corresponds to an x value. Take the bottom flange as an example, detection line b4 was selected for data analysis, and the analysis content was the law of the change of magnetic signals with load at different detection points on detection line b4. Decection points along the x-axis on Detective line u1 for upper flange, w3 for web and b4 for bottom flange were chosen for analysis separately. The variation of normal component value with force are plotted in Fig. 7.

Fig. 6.

The detective points of a cross section.

Fig. 7.

Variation of normal component value with force: (a) Line u1 in pure bending section. (b) Line u1 in bending shear section. (c) Line b4 in pure bending section. (d) Line b4 in bending shear section. (e) Line w3 in pure bending section. (f) Line w3 in bending shear section.

The diagram at the top left in Fig. 7 shows the location of the selected test line and the stress area of the detection points, where the position circled by the dotted line represents the position of the selected detection line, and the shaded part in the trapezoid area represents the stress area of the detection points. For example, the detection points in Fig. 7(a) are selected from the detection points of detection line u1 on the upper flange and they are located at the pure bending section under force. Similarly, the detection points selected in Fig. 7(b) are the detection point of line b4 on the bottom flange detection, which is located at the bending shear section under force.

It can be seen from the Fig. 7(c) and Fig. 7(e) that the variation of normal magnetic value under the normal stress and the shear stress is different. The variation law of H _p(y) with force under normal stress is similar to the variety curve of peak to peak amplitude and gratitude of MMM signals [11], and it also has commom with the change law of absolute minimum amplitude of H _p(x) signals with applied tensile force [12], where H _p(x) represents the tangential component of self-magnetic flux leakage.

The H _p(y) − F curve under normal stress can be divided into random stage, ascent stage and declining stage, and the change of each stage is obvious. The variation under the shear stress can be only divided into the rising and falling stages, and the amplitude of variation is relatively small.

4. Discussion

4.1. Discussion of the laws of magnetic signals

The variation curve of the normal component H _p(y) with force is divided into three regions: the initial stage, the ascent stage, and the declining stage. The curve partition is shown in Fig. 8.

Fig. 8.

Different stage during loading.

4.1.1. Analysis of the initial stage

During the initial loading period, the magnetic signals will decrease slightly [5]. The intensity of the hysteresis field under the magnetic field is fixed with the change of the stress [13].

According to the theory of magnetomechanical effect based on the “effective field theory” and a, “law of approach” developed by Jiles, a differential equation to describe the change in magnetism with applied stress under a constant magnetic field is obtained as follows [14] $\begin{eqnarray}\displaystyle \frac{dM}{d{\sigma}}=\frac{1}{{\varepsilon}^{2}}{\sigma}(M_{an}-M)+c\frac{dM_{an}}{d{\sigma}} & & \displaystyle\end{eqnarray}$ (1) Where ϵ and c are constants, M is the magnetization. It can be seen from the equation that the rate of change of magnetization depends not only on stress σ but also on the anhysteretic magnetization M _an. Since the anhysteretic magnetization is in the lowest energy state of the domain [13,14], when the initial magnetization intensity is greater than that of the unhysteresis magnetization, the magnetization tends towards the anhysteretic curve on the application of stress. Therefore, the magnetization will turn to the direction of the non-hysteresis curve with the stress, in that situation, the magnetization will change based on the magnetic state of the test point.

4.1.2. Analysis of the ascent stage

When the component is subjected to force, the internal effective field is changed by the magnetostrictive coefficient, which can be replaced by an equivalent magnetic field H _σ [10,15]. Based on the proximity principle proposed by Jiles and Atherton [17], Jiles established the theoretical model of the magnetic mechanical effect of ferromagnetic materials under unidirectional stress in 1995 [15,17]. For uniaxial stress polycrystalline bulk ferromagnetic materials, the outer magnetic field under the action of magnetization including reversible magnetization caused by bending and irreversible magnetization caused by domain wall displacement [19]. The relationship between internal effective field H _eff and external stress is given as follow: $\begin{eqnarray}\displaystyle H_{\mathit{eff}}=H+{\alpha}M+\frac{3{\sigma}}{2{\mu}_{0}}\left(\frac{d{\lambda}}{dM}\right)_{{\sigma}}(\text{cos}^{2}{\theta}-{\upsilon}\text{sin}^{2}{\theta}) & & \displaystyle\end{eqnarray}$ (2) Where 𝛼 is the dimensionless quantity of a single magnetic unit within the material to combine the magnetization strength, 𝜆 is the magnetostriction coefficient, H represents the applied magnetic field, M is the magnetization, μ₀ represents the vacuum permeability. σ denotes the applied stress, 𝜐 denotes Poisson’s ratio, and θ is the angle between the axis of the applied stress σ and the axis of the magnetic field H.

The description of the bulk magnetostrictio 𝜆 depends on the domain configuration throughout the material, which cannot be known in advance. An empirical model according to the literature [18] can describe the relationship between the bulk magnetostriction 𝜆 and bulk magnetization M: $\begin{eqnarray}\displaystyle {\lambda}=\mathop{\sum }_{k=0}^{\infty }{\gamma}_{k}({\sigma})M^{2k}. & & \displaystyle\end{eqnarray}$ (3) The 𝛾_k(σ) reflects the stress-dependence of the magnetostriction, and can be described using a Taylor series expansion, $\begin{eqnarray}\displaystyle {\gamma}_{k}({\sigma})={\gamma}_{k}(0)+\mathop{\sum }_{n=1}^{\infty }\frac{{\sigma}^{n}}{n!}{\gamma}_{k}^{n}(0) & & \displaystyle\end{eqnarray}$ (4) Where ${\gamma}_{k}^{n}({\sigma})$ is the n derivative of 𝛾_k with respect to stress at σ = 0. According to the literature [18,19], the approximation to the magnetostriction can be obtained by including the terms up to k = 1, n = 1 and, this gives $\begin{eqnarray}\displaystyle {\lambda}=[{\gamma}_{1}(0)+{\gamma}_{1}^{\prime }(0){\sigma}]M^{2}. & & \displaystyle\end{eqnarray}$ (5)

By substituting Eq. (4) into Eq. (1), the relationship between the magnetic field caused by stress H _σ and applied stress σ can be found: $\begin{eqnarray}\displaystyle H_{{\sigma}}=\frac{3{\sigma}M}{{\mu}_{0}}[{\gamma}_{1}(0)+{\gamma}_{1}^{\prime }(0){\sigma}](\text{cos}^{2}{\theta}-{\upsilon}\text{sin}^{2}{\theta}). & & \displaystyle\end{eqnarray}$ (6)

In Eq. (2), some variables can be regarded as constants in this experiment, so the variables in the equation need to be processed to facilitate the calculation. Since the test is carried out in the same direction under the same environment, and the measure points are the same after loading at each level H and M can be regarded as constants.

Because the material is fixed, 𝛼, μ₀ are not affected by the environment. 𝛾₁(0) and ${\gamma}_{1}^{\prime }(0)$ can be obtained by experiment, so they can be regarded as constants. 𝛾₁(0) = 7 × 10⁻¹⁸ m²∕A² and 𝛾₁ ^′(0) = −1 × 10⁻²⁵ m²∕A²∕Pa. According to the literature [15]. So the modified equationtion of an effective magnetic field can be obtained $\begin{eqnarray}\displaystyle H_{\mathit{eff}}=H+{\alpha}M+a{\sigma}+b{\sigma}^{2}. & & \displaystyle\end{eqnarray}$ (7)

In certain circumstances, H, 𝛼 and M remain approximately constant values. Where $a=\frac{2.1M}{{\mu}_{0}}(\text{cos}^{2}{\theta}-{\nu}\text{sin}^{2}{\theta})\times 10^{-17}$ , $b=\frac{3M}{{\mu}_{0}}(\text{cos}^{2}{\theta}-{\nu}\text{sin}^{2}{\theta})\times 10^{-25}$ .

It can be seen from the simplified equation (7) that the effective magnetic field increases with stress in the elastic phase. In order to study the equation of strain characterization of the magnetic field, the strain-force curve of the elastic stage was selected for analysis. The relationship between stress and strain is approximately linear in the elastic stage. For the detection point under small stress, it can be assumed that σ = kϵ where k is a constant. The effective magnetic field can be shown as $\begin{eqnarray}\displaystyle H_{\mathit{eff}}=H+{\alpha}M+ak{\varepsilon}+bk^{2}{\varepsilon}^{2}. & & \displaystyle\end{eqnarray}$ (8)

It is obvious that the magnetic field increases with the change of strain, which is consistent with the trend of stress.

4.1.3. Analysis of the declining stage

J.C. Leng [7] have modified the Jiles–Atherton model, and present the variation model of the magnetic field in the plastic stage. It is assumed that the deformation of the specimen at a certain moment in the plastic phase is ϵ, the deformation while yielding is ϵ_y, and the deformation of the plastic phase is ϵ_p. The relationship among these three is $\begin{eqnarray}\displaystyle {\varepsilon}_{p}={\varepsilon}-{\varepsilon}_{y}. & & \displaystyle\end{eqnarray}$ (9)

In the plastic stage, a large number of dislocation density act as the pinning point to impede the magnetic domain displacement. Similar to the magnetoelastic energy in elastic deformation, magnetoplastic energy can be produced in plastic deformation, and it is considered to be equal to the nailing energy which will impede of the magnetic domain movement [17], that is $\begin{eqnarray}\displaystyle E_{{\sigma}_{p}}=E_{pin}=\frac{n\langle e_{{\pi}}\rangle }{2m}\int _{0}^{M}dM & & \displaystyle\end{eqnarray}$ (10) Where 〈e _π〉 = 2mB denotes the pinning energy of the site for 180° domain walls, m is the magnetic moment, and n represents pinning sites density. According to the literature [22], the pinning sites density has a linear relationship with |ϵ_p|, that is n = d|ϵ_p|, where d is a material constant. The effective component due to plastic deformation is given by [23] $\begin{eqnarray}\displaystyle H_{{\sigma}_{p}}=-\frac{1}{{\mu}_{0}}\frac{\partial E_{{\sigma}_{p}}}{\partial M}=-k|{\varepsilon}-{\varepsilon}_{y}| & & \displaystyle\end{eqnarray}$ (11) Where $k=\frac{1}{{\mu}_{0}}\frac{d\langle e_{{\pi}}\rangle }{2m}$ .

4.1.4. Fitting results of the changing rule of the magnetic field

The variation curve of dectective point in pure bending section on upper flange is chosen for fitting. The fitting result is shown in Fig. 9.

Fig. 9.

The fitting result of point on upper flange.

The elastic stage: H _p(y) = −8374.50ϵ² + 2477.67ϵ − 136.55 (0.05 ≤ ϵ ≤ ϵ_y).

The plastic stage: H _p(y) = 108.99 − 462.80ϵ (ϵ_y < ϵ < 0.25).

Comparing the result to the equation (8) and Eq. (11), the calculation results fit well with the experimental values. The calculation result proves that the normal component H _p(y) of the test point will drop abruptly when the component yielded.

4.2. Numerical simulation of stress distribution

In this paper, Abaqus software of finite element analysis was chosen to simulate the stress distribution of the steel beam. The corrugated web, the flanges, and the stiffeners were modeled using the eight-node linear hexahedral element C3D8. The constitutive relation of the model is obtained from the material test. The results of the material test are shown in Fig. 10. The load is applied by static loading method. In order to prevent local failure caused by stress concentration, the load and boundary conditions are applied by adding rigid pad. The load is applied by point-surface coupling.

Fig. 10.

The constitutive relation of the steel Q345qC: (a) 6 mm standard plates. (b) 8 mm standard plates.

The distribution of normal stress and shear stress on steel box girder with corrugated web is shown in Fig. 11 and Fig. 12. It can be seen that almost all the normal stress is generated on the flange, and almost all the shear stress is generated on the web. It can be simplified that the upper flange and lower flange only produce normal stress and the web only produces shear stress. Therefore, the change of magnetic signal on the web can be regarded as the change under shear stress, and the change of magnetic signal on the flanges can be considered as the change under normal stress.

Fig. 11.

The distribution of normal stress: (a) Result of the finite element analysis. (b) The normal stress distribution of the upper flange obtained from the finite element analysis results.

Fig. 12.

The distribution of shear stress: (a) Result of the finite element analysis. (b) The shear stress distribution of the web obtained from the finite element analysis results.

4.3. Difference effect between the normal stress and the shear stress

In the experimental result, the normal stress of the pure bending section is constant, and the shear stress of the bending shear section is constant, too. Therefore, Fig. 7(c) and Fig. 7(f) can respectively represent the variation of magnetic memory signal with the normal stress and the shear stress. The following analysis is conducted for these two cases.

It can be seen from the figure that the normal stress has a significant influence on H _p(y), which conforms to the change rule in Section 4.1. However, the change range of H _p(y) under shear stress is small, indicating that the normal stress and the shear stress have different influences on H _p(y). Since the magnetic field is a vector, the magnitude of the magnetic field detected when the detection direction is different varies a lot, so it is speculated that the different influences are related to the detection direction. Besides, when the normal stress and the shear stress are produced, the direction of principal stress is different, so it is speculated that different influences are also related to the direction of principal stress.

4.3.1. Discussion about the equivalent stress

Experiments in the literature [24] show that the stress has different effects on the normal component of the magnetic field when the direction of principal stress is different. Therefore, the direction of the principal stress should be considered when calculating the equivalent stress. The x-axis in the literature [24] is the tension axis of the plate. The x-axis in this test is shown in the Fig. 3, which is also the tension direction of the component.

In order to obtain the relationship between H _p(y) and the angle of the principal stress, the experimental data are quoted as Table 3.

Table 3
The H _p(y) value under different stress states

H _p(y)/(A/m)

Stress states/MPa Plate 35A210 Plate 35A300 Plate 35A400

σ_x = 0, σ_y = 0, τ_xy = 0 46.166 59.2508 70.758

σ_x = 30, σ_y = 0, τ_xy = 0 284.892 278.7 212.274

σ_x = 0, σ_y = 30, τ_xy = 0 63.3094 44.766 33.213

σ_x = 0, σ_y = 0, τ_xy = 30 23.0216 21.6606 44.766

	H _p(y)/(A/m)
σ_x = 0, σ_y = 0, τ_xy = 0	46.166	59.2508	70.758
σ_x = 30, σ_y = 0, τ_xy = 0	284.892	278.7	212.274
σ_x = 0, σ_y = 30, τ_xy = 0	63.3094	44.766	33.213
σ_x = 0, σ_y = 0, τ_xy = 30	23.0216	21.6606	44.766

Suppose θ_σ is the angle between the direction of the principal stress and the direction of the tension axis (x-axis), as shown in Fig. 13. The change rule of H _p(y) with Angle of the principal stress is shown in Fig. 14, it is obvious that as the θ_σ increases, H _p(y) tends to decrease.

Fig. 13.

The stress component of the element.

Fig. 14.

The change rule with θ_σ.

When the component is subjected to uniaxial tension in the x direction, the angle of the principal stress θ_σ = 0°, and θ_σ = 45° when the member is subjected to pure shear stress. The normal magnetic values H _p(y) under two stress states are extracted from the Fig. 14, for θ_σ = 0°, H _p ⁰(y) = 228.29 A∕m, and H _p ⁴⁵(y) = 44.69 A∕m for θ_σ = 45°. Take the ratio of these two value, this gives $\begin{eqnarray}\displaystyle \frac{H_{p}^{{\tau}}(y)}{H_{p}^{{\sigma}}(y)}=0.196 & & \displaystyle\end{eqnarray}$ (12) Where $H_{p}^{{\tau}}(y)$ is the H _p(y) value under the pure shear stress, and $H_{p}^{{\sigma}}(y)$ is the H _p(y) value under the normal stress.

The intensity of magnetization varies little with the stress under shear stress [25]. In the literature [26], equivalent stress is expressed as σ_∥−σ_⊥, Where σ_∥ represents the stress parallel to the direction of the magnetic field, and σ_⊥ represents the stress perpendicular to the direction of the magnetic field. However, this method does not consider the influence of the direction of principal stress on the magnetic field. Considering the influence of principal stress direction, the equivalent stress is assumed as $\begin{eqnarray}\displaystyle {\sigma}_{eq}=\text{cos}(k{\theta}_{{\sigma}})\times ({\sigma}_{1}-{\sigma}_{2}) & & \displaystyle\end{eqnarray}$ (13) where σ₁ and σ₂ are the principal stress components, and θ_σ is the angle of principal stress. According to material mechanics method, the principal stress can be expressed as $\begin{eqnarray}\displaystyle & \displaystyle {\sigma}_{1}={\sigma}_{x}\text{cos}^{2}{\theta}_{{\sigma}}+2{\tau}_{xy}\text{cos}{\theta}_{{\sigma}}\text{sin}{\theta}_{{\sigma}}+{\sigma}_{y}\text{sin}^{2}{\theta}_{{\sigma}} & \displaystyle\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle & \displaystyle {\sigma}_{2}={\sigma}_{x}\text{sin}^{2}{\theta}_{{\sigma}}-2{\tau}_{xy}\text{cos}{\theta}_{{\sigma}}\text{sin}{\theta}_{{\sigma}}+{\sigma}_{y}\text{cos}^{2}{\theta}_{{\sigma}} & \displaystyle\end{eqnarray}$ (15) $\begin{eqnarray}\displaystyle & \displaystyle {\theta}_{{\sigma}}=\frac{1}{2}\tan ^{-1}\frac{2{\tau}_{xy}}{{\sigma}_{x}-{\sigma}_{y}}. & \displaystyle\end{eqnarray}$ (16)

In order to find the value of k in the equation (16), it is necessary to select the point where the stress state of the bottom flange is σ_x = A, σ_y = 0, τ_xy = 0, and the change value of H _p(y) relative to the state of σ_x = 0, σ_y = 0, τ_xy = 0 (expressed as ΔH _p(y)) in this state was compared with ΔH _p(y) of the detection point on the web, whose stress state is σ_x = 0, σ_y = 0, τ_xy = A, where A is a constant.

The stress tensors components σ_x, σ_y, τ_xy were evaluated from the mechanical strain, which was measured using strain gauge. The strain gauge in the bottom flange span is a single strain gauge, and the one on the web is a three-axis strain gauge. Figure 15 shows the structure of the three-axis strain gauge. The angles of the gauges are 0°, 45°, and 90° from the x-axis. The three-axis strain gauges were used to measure the three strain components ϵ₀, ϵ₄₅, and ϵ₉₀. When the specimen is loaded, the value of strain gauge is recorded. After measuring the three strain components, the stress tensor components were calculated from Hooke’s law under the plane stress assumption using the following equations $\begin{eqnarray}\displaystyle & \displaystyle {\sigma}_{x}=\frac{E}{1-{\upsilon}^{2}}({\varepsilon}_{0}+{\upsilon}{\varepsilon}_{90}) & \displaystyle\end{eqnarray}$ (17) $\begin{eqnarray}\displaystyle & \displaystyle {\sigma}_{y}=\frac{E}{1-{\upsilon}^{2}}({\varepsilon}_{90}+{\upsilon}{\varepsilon}_{0}) & \displaystyle\end{eqnarray}$ (18) $\begin{eqnarray}\displaystyle & \displaystyle {\tau}_{xy}=\frac{E}{2(1+{\upsilon})}(2{\varepsilon}_{45}-{\varepsilon}_{0}-{\varepsilon}_{90}) & \displaystyle\end{eqnarray}$ (19) Where E is the Young’s modulus, and 𝜐 is the Poisson’s ratio.

Fig. 15.

Structure of three-axis strain gauge.

The detection point at x = 900 mm on the bottom flange and the point at x = 400 mm on the web were selected to draw the change curves of H _p(y) with normal stress and shear stress respectively, as shown in Fig. 16. The stress state of the detection point on the web when F = 600 kN is selected for analysis. Assuming that the web is subject to pure shear stress, the stress state of the web is: σ_x = 0, σ_y = 0, τ_xy = 162 MPa.

Fig. 16.

Variation of H _p(y) with stress: (a) x = 900 mm on the bottom flange (b) x = 400 mm on the web.

For the bottom flange, the ΔH _p(y) value of the stress state σ_x = 162 MPa, σ_y = 0, τ_xy = 0 is extracted. According to the Fig. 16, and get the result of ΔH _p(y) = 291 MPa. According to Eq. ((12)), ΔH _p ^τ(y) = 0.2 × 291 = 58.2 MPa. Extracting the value from Fig. 17, it can be obtained that the equivalent stress on the web is 82 MPa. Put σ_eq = 82 MPa, σ₁ = τ _xy = 162 MPa, σ₂ = −τ_xy = −162 MPa and σ₂ = −τ_xy = −162 MPa into the equation ((13)) and get the k = 5∕3, so the equation of the equivalent stress can be expressed as $\begin{eqnarray}\displaystyle {\sigma}_{eq}=\text{cos}\left(\frac{5}{3}{\theta}_{{\sigma}}\right)\times ({\sigma}_{1}-{\sigma}_{2}). & & \displaystyle\end{eqnarray}$ (20)

In order to verify the applicability of the equation ((20)), it is applied to other detection points. The detection point at x = 1400 mm on the web is selected, and the curve of H _p(y) changing with the shear stress is plotted, as shown in Fig. 17. It can be seen from Fig. 12 that the shear stress on the bending shear section is opposite to the sign, so the strain at 1400 mm is taken as the absolute value.

Fig. 17.

Variation of H _p(y) with shear stress at x = 1400 mm on the web.

The stress state when F = 600 kN is also selected, σ_x = 0, σ_y = 0, τ_xy = 165 MPa. For the upper flange, when σ_x = 162 MPa, σ_y = 0, τ_xy = 0, ΔH _p ^σ(y) = 299 MPa. For the web σ_eq = 85.4 MPa, ΔH _p ^τ(y) = 63.7 MPa, and the relationship between $H_{p}^{{\tau}}(y)$ and $H_{p}^{{\sigma}}(y)$ can be expressed as $\begin{eqnarray}\displaystyle \frac{{\rm\Delta}H_{p}^{{\tau}}(y)}{{\rm\Delta}H_{p}^{{\sigma}}(y)}=0.21. & & \displaystyle\end{eqnarray}$ (21)

By comparing the equation ((21)) with Eq. ((12)), it can be concluded that the equation of equivalent stress has applicability. Since the change of H _p(y) with the load in the elastic stage follows the rule of Jiles–Atherton model, the virable σ in Jiles–Atherton model can be replaced by σ_eq when calculating the magnetic field under shear stress.

5. Conclusions

Four-point bending test of inclined steel box girder with corrugated web were conducted to analyze the difference between the effect of normal stress and shear stress on magnetic field. The variation curves of the the normal component of the magnetic field H _p(y) with force are plotted. The variation law of H _p(y) is analyzed and the equation of equivalent stress under different stress states is discussed. The following conclusions were obtained:

(1) The law of H _p(y) changing with load can be explained by Jiles–Atherton model and the theory of plastic stage, and the curve under normal stress can be divided into random stage, ascent stage and declining stage.

(2) The law of H _p(y) changing with load under the shear stress is similar to that of normal stress, but the amplitude of H _p(y) changes little under shear stress.

(3) The intensity of magnetic field is affected by the angle of principal stress. The equivalent stress ${\sigma}_{eq}=\text{cos}(\frac{5}{3}{\theta}_{{\sigma}})\times ({\sigma}_{1}-{\sigma}_{2})$ can replaced the virable σ in Jiles–Atherton model when calculating the magnetic field under shear stress, where θ_σ is the angle of principal stress and σ₁, σ₂ are the principal stress components.

Footnotes

Acknowledgements

The authors are grateful for the financial support from the National Science Foundation of China (No. 51578449, No. 51878548) and the First-Grade Science Foundation of China Post-Doctor (No. 20060400176).

References

Doubov

A.A.

, A study of mental properties using the method of magnetic memory, Mental Science and Heat Treatment 39(9–10) (1997), 401–402.

Doubov

A.A.

, Screening of weld quality using the metal magnetic memory effect, Welding in the World 41 (1998), 196–198.

Yao

Wang

Z.D.

Deng

and Shen

, Experimental research on metal magnetic memory method, Experimental Mechanics 52 (2012), 305–314.

Dong

L.H.

B.S.

Dong

S.Y.

, Variation of stress-induced magnetic signals during tensile testing of ferromagnetic steels, NDT&E International 41 (2008), 184–189.

Guo

P.J.

Chen

X.D.

Guan

W.H.

, Effect of tensile stress on the variation of the magnetic field of low-alloy steel, Journal of Magnetism and Materials 323 (2011), 2474–2477.

Bao

Liu

and Zhang

, Variation of residual magnetic field of defective U75V steel subjected to tensile stress, Strain 51 (2015), 370–378.

Leng

J.C.

Liu

Zhou

G.Q.

and Gao

Y.T.

, Metal magnetic memory signal response to plastic deformation of low carbon steel, NDT&E International 55 (2013), 42–46.

J.W.

M.Q.

Leng

J.C.

and Xu

M.X.

, Modeling plastic deformation effect on magnetization in ferromagnetic materials, Journal of Applied Physics 063909 (2012), 111–114.

X.P.

S.Q.

Wang

Yang

Y.Y.

S.C.

and Zhao

X.R.

, Damage location and numerical simulation for steel wire under torsion based on magnetic memory method, International Journal of Applied Electromagnetics and Mechanics (2019), 1, In Press.

10.

S.Q.

S.C.

and Wang

, Bending experimental study of structural steel beam on magnetic field gradient based on modified jiles-Atherton model, International Journal of Applied Electromagnetics and Mechanics 55 (2017), 409–421.

11.

Shen

G.T.

Gao

G.X.

and Li

Y.T.

, Investigation on metal magnetic memory signal during loading, International Journal of Applied Electromagnetics and Mechanics 33 (2010), 1329–1334.

12.

Shui

G.S.

C.W.

and Yao

, Non-destructive evaluation of the damage of ferromagnetic steel using metal magnetic memory and nonlinear ultrasonic method, International Journal of Applied Electromagnetics and Mechanics 47 (2015), 1023–1038.

13.

Leng

J.C.

M.X.

and Zhang

J.Z.

, Magnetic field variation induced by cyclic bending stress, NDT & E International 42 (2009), 410–414.

14.

Jiles

D.C.

and D.L.

Atherton

, Theory of ferromagnetic hysteresis, J. Magnetism and Magnetic Materials 61(1–2) (1986), 48–60.

15.

Kuruzar

M.E.

and Cullity

B.D.

, The magnetostriction of iron under tensile and compressive stress, International Journal of Magnetism 1(14) (1971), 323–325.

16.

Jiles

D.C.

, Theory of the magnetomechanical effect, Journal of Physics-Applied Physics 28(2) (1995), 1537–1546.

17.

Jiles

D.C.

and D.L.

Atherton

, Theory of the magnetization process in ferromagnets and its application to the magneto-mechanical effect, Journal of Physics-Applied Physics 17(6) (1984), 1265–1281.

18.

Jiles

D.C.

, A new approach to modeling the magnetomechanical effect, Journal of Applied Physics 95(11) (2004), 7058–7060.

19.

and Jiles

D.C.

, Modeling of the magnetomechanical effect: Application of the Rayleigh law to the stress domain, Journal of Applied Physics 93(10) (2003), 8480–8482.

20.

Huang

H.H.

Yang

Qian

Z.C.

Han

and Liu

Z.F.

, Magnetic memory signals variation induced by applied magnetic field and static tensile stress in ferromagnetic steel, Journal of Magnetism and Magnetic Materials 416 (2016), 213–219.

21.

Sablik

M.J.

Yonamine

and Landgraf

F.J.G.

, Modeling plastic deformation effects in steel on hysteresis loops with the same maximum flux density, IEEE Transactions on Magnetics 40(5) (2004), 3219–3226.

22.

Huang

Z.M.

, Mechanical Properties of Metal, Xi’an Jiaotong University Press, 1986, (In Chinese).

23.

Wang

Z.D.

Deng

and Yao

, Physical model of plastic deformation on magnetization in ferromagnetic materials, Journal of Applied Physics 083928 (2011), 109–115.

24.

Kaia

Enokizonoa

and Kidob

, Influence of shear stress on vector magnetic properties of non-oriented electrical steel sheets Bixial in steel, International Journal of Applied Electromagnetics and Mechanics 44 (2014), 371–378.

25.

Rekik

Huberta

and Daniel

, Influence of a multiaxial stress on the reversible and irreversible magneticbehaviour of a 3%Si-Fe alloy, International Journal of Applied Electromagnetics and Mechanics 44 (2014), 301–315.

26.

Schneider

C.S.

and Richardson

J.M.

, Biaxial magnetoelasticity in steels, Journal of Applied Physics 53 (1982), 8136–8138.

	H _p(y)/(A/m)
Stress states/MPa	Plate 35A210	Plate 35A300	Plate 35A400
σ_x = 0, σ_y = 0, τ_xy = 0	46.166	59.2508	70.758
σ_x = 30, σ_y = 0, τ_xy = 0	284.892	278.7	212.274
σ_x = 0, σ_y = 30, τ_xy = 0	63.3094	44.766	33.213
σ_x = 0, σ_y = 0, τ_xy = 30	23.0216	21.6606	44.766

Equivalent stress for steel box girder with corrugated web under bending based on magnetic memory method

Abstract

Keywords

1. Introduction

2. Experimental details

Table 1 Chemical composition of the Q345qC steel specimen Material % Name C Si Mn P S Q345qC ≤0.20 ≤0.60 1.0--1.6 ≤0.035 ≤0.035

3. Experimental result

4.1. Discussion of the laws of magnetic signals

4.3.1. Discussion about the equivalent stress

Footnotes

Acknowledgements

References

Table 1
Chemical composition of the Q345qC steel specimen

Material %

Name C Si Mn P S

Q345qC ≤0.20 ≤0.60 1.0--1.6 ≤0.035 ≤0.035