Improvement of the coiling stress calculation model for a mandrel-sleeve-coil

Abstract

The steel rolling process employs a coiling-uncoiling process in which a steel sheet is wound and unwound in a coil shape using a coiler to efficiently produce a long steel sheet with a constant thickness. As front and rear tension is required when the steel sheet enters and exits the rolling mill, the coiler introduces tension in the steel sheet through the control of the rotational speed. As the coil is produced, coiling tension accumulates, and pressure is applied to the inside of the coil. Finite element analysis and stress calculation analysis were derived from previous studies to prevent such pressure increases in the sleeves and coils. However, the radial and circumferential stresses at arbitrary positions inside the coil cannot be accurately determined by considering without the stresses’ difference in the thickness direction based on the assumption that the coil’s thickness is thin. In this study, an analytical model that can accurately calculate the sleeve and coil stress during elastic deformation was established by improving the internal circumferential stress generated when the steel sheet is bent into a coil and the radial stress equation associated with the beam bending theory. In addition, by comparing the finite element analysis model results reflecting the same coiling condition, this model’s validity was verified by confirming the consistency of the results.

Introduction

The steel rolling process employs a coiling-uncoiling process in which a steel sheet is wound and unwound in a coil shape using a coiler to efficiently produce a long steel sheet with a constant thickness. The steel sheet is also wound into a concentric coil and supplied to consumers for ease of transportation and storage. The steel sheet is wound in a state where tension is imposed to uniformly wind the steel sheet; if the coiling tension is weak, the coil formation fails due to the insufficient bending of the strip coil and slipping between the strip layers. Conversely, when the coiling tension is excessive, excessive radial stress occurs inside the sleeve and strip coil, causing the coil’s material to buckle and severe deformation. To apply the coiling tension to a strip from rollers exactly, several analytical solutions have been suggested mainly to estimate the coiling tension according the variation of contact width, roll shape, roll forces.^1,2 In the case of a thick steel plate, coiling is possible by directly connecting the steel plate and a mandrel, but in the case of a thin steel plate, the steel plate is wound with a sleeve inserted into the mandrel. As the thickness of the steel plate decreases, the reaction forces required maintain the coil shape against the external pressure caused by coiling tension decrease, and severe deformation to the steel plate occurs locally around the part where the mandrel and the steel plate are not in contact. ³ At this time, the pressure on the sleeve and the coil increase through the coiling process, and additional processes are required for separating the sleeve and the coil, resulting in additional costs. From these phenomena, there is a need for an approach that is capable of improving the process by accurately predicting the stress distribution in the sleeve and coil according to the coiling process conditions, and many studies related to this topic have been conducted.

Since a study on the verification of the radial and circumferential stress calculation formula using the pressure value applied inside and outside the hollow cylinder and the calculation formula using a strain gauge have been suggested, ⁴ stress calculation solutions in which a steel sheet is wound directly on a mandrel using the assumption of a two-dimensional plane stress problem have been developed.^5,6 In the analytical solution structure, a pressure value based on the coiling tension is applied to the Nth steel plate, the other pressure values between the mandrel and the N-1th steel due to coiling of the Nth steel plate using the interference-fit equation are calculated and accumulated from the 1st coiling when the N th steel plate is wound on the mandrel, and the N-1th steel plate is wound accordingly. The structure of the analytical solution calculates and adds the pressure on the cylinders by force fitting. Based on this basic model, the solution is developed by analyzing the internal stress distribution of the coil through the coupling between the thick and thin cylinders, ⁷ and the elastic stress model is expanded to three dimensions to calculate the stress distribution according to the flatness. ⁸ Application studies such as model improvement ⁹ for the stress calculation have been carried out. In addition, based on the multi-cylinder interface fitting method,^10–12 a stress analysis solution ¹³ was developed to calculate the stress for mandrel-sleeve-coil coiling introduced for coiling thin steel sheets, and a finite element analysis model for the structure was developed. According to the coiling condition, a study on the parameters, ¹⁴ such as the change in the stress concentration zone due to coiling, was conducted. The previous models have been developed to predict the internal status of the mandrel, sleeve, and coiling in the engineering. For instance, when thin steel plates is hard to coil on the mandrel directly, it is hard to maintain the homogenous coil’s shape without internal forces along to the radial direction. Therefore, the sleeve is used to maintain the coil’s shape. In this case, we should know how much the sleeve endures coiling pressures, how much frictional forces needs to remove the sleeve from the coil before shipping, and so on in the engineering.

To predict the coiling process whether the coiler’s internal part is on the plastic deformation or not, and to alarm the statue to an operator immediately, the internal stress calculation model must be improved according to the change in coiling conditions. It is essential to accurately predict the state of internal stress by predicting nonlinearity such as contacts and plastic deformations in the mandrel-sleeve-coil coiling. However, the nonlinearity is hard to be mastered due to several iteration processes to check states variables each stage, so it is more efficient to find out a simple analytical model that estimate the state variables similarly. In this study, the coil stress calculation assumption in the previous study was improved by changing the model to accurately express the stress within the elastic deformation. The developed analytical model was verified by comparing the results with a finite element analysis model that has the same coiling conditions and design variables.

Computational model

Analytical model of the coiling process with a mandrel and sleeve

Figure 1 shows a coiling process consisting of a mandrel-sleeve coil after the second steel sheet coil lamination is completed. R_J and R_J+1 are the inner and outer radius of the J-th layer of the coil. In the previous stress analysis, ¹³ the calculation process below is performed to calculate the internal stresses of the sleeve and coil.

Figure 1.

Schematic diagram of coiling.

First, the internal stress of the outermost layer of the coil was calculated. In previous studies, it is assumed that the coil’s circumferential stress (σ_θθ) is equal to the coiling tension (σ_T) because the thickness of the steel sheet (h) is skinny (equation (1)). In other words, there is no internal stress due to bending. When the method of sections is applied to obtain the internal force of the outermost coil as the outermost coil has a contact force in contact with the sleeve or the previously stacked coil, the internal force and the external force due to the contact pressure must be the same in the plane (Figure 2). When we compute a summation of the internal and external forces in the vertical direction using equations (2)–(3), equation (4), which is the balanced force relationship between the circumferential stress inside the outmost layer coil and the internal pressure applied to the outermost layer coil from the previously stacked coil, is derived. In addition, the internal pressure corresponding the radial stress (σ_rr) given in equation (5) is calculated using equation (4). Finally, the radial stress applied to the inside is calculated using Lame’s cylinder theory. ⁴ At this time, there is no external pressure applied to the third layer of the coil (equation (6)); therefore, J = 2 in equations (4)–(6).

$$\begin{array}{c}\multicolumn{1}{c}{{\sigma }_{\theta \theta }\approx {\sigma }_T}\end{array}$$

(1)

$$\sum F_y=-2\times {\sigma }_{\theta \theta }\times h+{\int }_0^{\pi }p\times \sin \theta \times R_{J}d\theta =0$$

(2)

$$\sum F_y=-2\times {\sigma }_{\theta \theta }\times h+2\times p\times R_J=0$$

(3)

$${\sigma }_{\theta \theta }\approx \frac{R_J}{h}[-{{\sigma }_r}_r\left(R_J\right)]$$

(4)

$${\sigma }_{\mathrm{rr}}\left(R_J\right)=-{\sigma }_T\frac{h}{R_J}$$

(5)

$${\sigma }_{\mathrm{rr}}\left(R_{J+1}\right)=0$$

(6)

Figure 2.

The outermost layer coil with the method of sections.

After calculating the internal stress of the outermost coil layer, the pressure existing on the mandrel-sleeve-coil layers or the interface is calculated. Basically, the fundamental equation to calculate interface contact pressures among the mandrel, the sleeve and the coil is based on the interference fit theory of the composite hollow cylinder under inner and outer pressures. ¹⁵ First, in the inner and outer portions, there is no inner pressure in the inside of the mandrel (R_i,man), and the pressure applied to the inside of the second layer of the coil calculated above (equation (5)) is applied to the outside of the first layer of the coil (R₂). The pressure applied to the interface between the mandrel’s outer diameter and the sleeve’s inner diameter (R_o,man) and the interface between the outer diameter of the sleeve and the inner diameter of the first layer of the coil (R₁) are unknown. To calculate the two unknown pressures, a force-fit equation using the condition that the pressure on several cylinders exists and the displacement at each intermediate interface must have continuity was used. First, the relationship between the mandrel and the sleeve, the internal pressure applied to the inner diameter of the mandrel (R_i,man), the pressure applied to the outer diameter of the mandrel and inner diameter of the sleeve (R_o,man), and the pressure applied to the outer diameter of the sleeve (R₁) must be calculated by the interference-fit. The pressures cause deformations, and the outer diameter of the mandrel and inner diameter of the sleeve must have the same displacement. Based on this, the same pattern is applied between the sleeve and the first layer of the coil; as a result, two simultaneous equations that can solve two unknown pressures are derived (equations (7) and (8)).

$$\begin{array}{c}{p}_{o,\mathrm{man}}\left\{\begin{array}{c}\frac{1+{\nu }_{\mathrm{man}}}{E_{\mathrm{man}}}\frac{[(1-2{\nu }_{\mathrm{man}}){\left(\frac{R_{o,\mathrm{man}}}{R_{i,\mathrm{man}}}\right)}^2+1]}{{\left(\frac{R_{o,\mathrm{man}}}{R_{i,\mathrm{man}}}\right)}^2-1}\hfill \\ \multicolumn{1}{c}{+\frac{1+{\nu }_{\mathrm{sleeve}}}{E_{\mathrm{sleeve}}}\frac{[(1-2{\nu }_{\mathrm{sleeve}})+{\left(\frac{R_1}{R_{o,\mathrm{man}}}\right)}^2]}{{\left(\frac{R_1}{R_{o,\mathrm{man}}}\right)}^2-1}}\end{array}\right\}\hfill \\ =2p_{i,\mathrm{man}}\frac{1-{{\nu }_{\mathrm{man}}}^2}{E_{\mathrm{man}}}\frac{1}{{\left(\frac{R_{o,\mathrm{man}}}{R_{i,\mathrm{man}}}\right)}^2-1}+2p_1\frac{1-{{\nu }_{\mathrm{sleeve}}}^2}{E_{\mathrm{sleeve}}}\hfill \end{array}$$

(7)

$$\begin{array}{c}{p}_1\left\{\begin{array}{c}\frac{1+{\nu }_{\mathrm{sleeve}}}{E_{\mathrm{sleeve}}}\frac{[(1-2{\nu }_{\mathrm{sleeve}}){\left(\frac{R_1}{R_{o,\mathrm{man}}}\right)}^2+1]}{{\left(\frac{R_1}{R_{o,\mathrm{man}}}\right)}^2-1}\hfill \\ \multicolumn{1}{c}{+\frac{1+{\nu }_{\mathrm{coil}}}{E_{\mathrm{coil}}}\frac{[(1-2{\nu }_{\mathrm{coil}})+{\left(\frac{R_2}{R_1}\right)}^2]}{{\left(\frac{R_2}{R_1}\right)}^2-1}}\end{array}\right\}\hfill \\ =2p_{o,\mathrm{man}}\frac{1-{{\nu }_{\mathrm{sleeve}}}^2}{E_{\mathrm{sleeve}}}\frac{1}{{\left(\frac{R_1}{R_{o,\mathrm{man}}}\right)}^2-1}\hfill \\ +2p_2\frac{1-{{\nu }_{\mathrm{coil}}}^2}{E_{\mathrm{coil}}}\frac{{\left(\frac{R_2}{R_1}\right)}^2}{{\left(\frac{R_2}{R_1}\right)}^2-1}\hfill \end{array}$$

(8)

where p_o,man, p_i,man, ν_man, E_man, R_o,man, and R_i,man are the outer pressure, the inner pressure, Poisson’ ratio, modulus of elasticity, the outer radius and the inner radius of the mandrel, p₁, ν_sleeve, E_sleeve, and R₁ are the outer pressure, Poisson’ ratio, modulus of elasticity and the outer radius of the sleeve, and p₂, ν_coil, E_coil, R₂ are the outer pressure, Poisson’ ratio, modulus of elasticity, the outer radius of the coil.

The process of equations (7) and (8) calculates the pressure (p₁) at the interface between the sleeve and the coil using the changed radius each time the coiling layer is stretched. When R₂ increases like R₃, R₄…R_N, a new p_1,N can be calculated for each case. The calculated pressure can be derived by calculating the accumulated pressure on the mandrel and sleeve and then calculating the sleeve’s internal stress at each lamination based on the Lame’s cylinder theory applying the accumulated pressure. When a new coiling layer is stacked on the outermost coiling layer, the coil’s stress change due to the pressure applied by the new coiling layer to the radial and circumferential stress when the coil is first stacked is calculated using cylinder theory (equations (9) and (10)).

$$\begin{array}{c}{\sigma }_{\theta \theta }\left(r\right)={\sigma }_T+\sum _{K=J+1}^N\hfill \\ [-{\sigma }_T\frac{h}{R_K}+\frac{{\left(\frac{R_K}{r}\right)}^2+1}{{\left(\frac{R_K}{R_1}\right)}^2-1}\times (p_{1,k}-{\sigma }_T\frac{h}{R_K})]\hfill \end{array}$$

(9)

$$\begin{array}{c}{\sigma }_{\mathrm{rr}}\left(r\right)=-\frac{{\sigma }_T}{2}[{\left(\frac{R_{J+1}}{r}\right)}^2-1]+\sum _{K=J+1}^N\hfill \\ [-{\sigma }_T\frac{h}{R_K}-\frac{{\left(\frac{R_K}{r}\right)}^2-1}{{\left(\frac{R_K}{R_1}\right)}^2-1}\times (p_{1,k}-{\sigma }_T\frac{h}{R_K})]\hfill \end{array}$$

(10)

where J ≥ 1, R_J ≤ r ≤ R_J+1

It is not difficult to calculate the stress of the mandrel and sleeve using the method proposed in the previous study, but for steel plates other than web products such as paper, tissue paper, and aluminum foil, the bending stress in the thickness direction is critical. The assumption that the bending stress is the same as the coiling tension is incorrect, as shown in equation (1). Accordingly, a process of calculating the bending stress(σ_b) was added by calculating the curvature that the steel sheet has when it is wound into a coil using the bending theory of the beam and then converting it into the length deformation (Figure 3). The calculation formula for calculating the outermost coil stress in equation (1) was changed to equation (11) by adding coiling tension and bending stress (equation (12)) to the existing coil circumferential stress calculation. The final coil circumferential stress calculation equation in equation (13) was derived by applying the circumferential stress from equation (9).

$${\sigma }_{\theta \theta }\left(r\right)={\sigma }_b\left(r\right)+{\sigma }_T$$

(11)

$${\sigma }_b\left(r\right)=\frac{E(r-(R_J+h/2))}{R_J+h/2}$$

(12)

Figure 3.

Internal stress of the outermost coil.

where E is modulus of elasticity of the coil

$$\begin{array}{c}{\sigma }_{\theta \theta }\left(r\right)={\sigma }_b(r)+{\sigma }_T+{\sum }_{K=J+1}^N\hfill \\ [-{\sigma }_T\frac{h}{R_K}+\frac{{\left(\frac{R_K}{r}\right)}^2+1}{{\left(\frac{R_K}{R_1}\right)}^2-1}\times (p_{1,k}-{\sigma }_T\frac{h}{R_K})]\hfill \end{array}$$

(13)

In the case of the radial stress calculation formula, the bending stress of the steel sheet has symmetrical bending stress in the compressed state on the inside and in the tensile state on the outside based on the neutral axis, so the average circumferential stress is the same as the coiling tension; therefore, the calculation formula should be changed. However, in deriving the radial stress equation of the outermost coil in the previous study, the final coil radial stress equation, including the initial derivation equation, was derived by approximating equation (14) to a value of 0.5 for ease of calculation. Therefore, the radial stress equation was revised as shown in equation (15) without the approximation.

$${\sigma }_{\mathrm{rr}}\left(r\right)=-{\sigma }_T\frac{R_J}{(R_J+R_{J+1})}[{\left(\frac{R_{J+1}}{r}\right)}^2-1]$$

(14)

$$\begin{array}{c}\multicolumn{1}{c}{{\sigma }_{\mathrm{rr}}\left(r\right)=-{\sigma }_T\frac{R_J}{(R_J+R_{J+1})}[{\left(\frac{R_{J+1}}{r}\right)}^2-1]}\\ +{\sum }_{K=J+1}^N[-{\sigma }_T\frac{h}{R_K}-\frac{{\left(\frac{R_K}{r}\right)}^2-1}{{\left(\frac{R_K}{R_1}\right)}^2-1}\times (p_{1,k}-{\sigma }_T\frac{h}{R_K})]\hfill \end{array}$$

(15)

Finite element analysis model of the coiling process with a mandrel and sleeve

A finite element analysis model for calculating the mandrel, sleeve, and coil stress was established to verify the previously defined analytical model (Figure 4). The difference of them is that the finite element analysis model calculate several physical quantities such as stress, position, frictional force in time by solving algebraic equations including a constitutive equation and equilibrium equation. It has lots of strength compared to the analytical model, but it takes lots of computing time to get a numerical solution that is near an exact solution per each time. The finite element analysis model has two parts such as the sleeve and the strip that are deformable. These have linear elastic behavior with density, modulus of elasticity and Poisson’s ratio. The mandrel is regarded as a rigid body by giving kinematic coupling the rotational center and the inner radius of sleeve. Regarding the relationship between the parts in the finite element analysis model, the mandrel rotates without deformations even when coiling proceeds as the mandrel is extended and fixed in the inner radius of the sleeve by a hydraulic drive. So, the sleeve’s inner radius and the center of rotation are defined as rigid bodies to implement this mechanism. Regarding the rotational motion of the sleeve; for the boundary condition, the local distribution of the coiling tension is not homogeneous along the strip, if the resultant torque is applied to the sleeve. In case of the constant rotational motion, ABAQUS can give proper resultant torque which gives a resultant traction force. According to the center’s rotation, a constant rotational motion of 4π rad/s in a counterclockwise direction is defined without deformation.

Figure 4.

Finite element analysis model.

Regarding other simulation conditions, the sleeve has an equally spaced element in the circumferential direction and the radial direction. The sleeve consists of 4800 elements. The strip is set to be wound according to the sleeve rotation by applying the condition that the top node of the outer diameter of the sleeve and the strip’s left edge are constrained in the circumferential direction. For the contact between the sleeve and the strip, the hard contact condition is applied to the vertical direction. The penalty method is implemented in the tangential direction with a friction coefficient of 0.05. ¹⁶ About the solver, dynamic implicit method was chosen to get exact numerical solution for verifying the previously defined analytical model. The maximum number of increments is 5e+8 and increment size is selected as initial 1e−5 and minimum 1e−15.

Since the strip has five elements distribution in the thickness direction, the bending stress distribution can be checked. In the longitudinal direction, the bias function is used so that the nodes between the coiling layers with the same angular phase are in contact with each other to stack the strip into a coil. Finally, coiling tension is applied to the right corner, while coiling tension is applied to the opposite side of the strip to wind the coil tightly. Table 1 shows the coiling analysis conditions in detail.

Table 1.

Coiling condition.

Item	Value
Inner radius of sleeve	0.304 m
Outer radius of sleeve	0.424 m
Thickness of strip	1 mm
Length of strip	1486 m
Coiling tension	50 MPa
Density	7850 kg/m3
Elastic modulus	214 GPa
Poisson’s ratio	0.28
Angular velocity of sleeve	4π rad/s
Frictional coefficient between sleeve and strip	0.05
Self frictional coefficient of strip	0.05

Analysis

Stress distribution of the elastic deformation of the sleeve and coil

The finite element analysis model and the stress result were compared to verify the validity of the analytical model. When the sleeve and the strip experienced elastic deformation, the validity was examined by comparing the radial stress and the circumferential stress change during coiling at the same location for each object. In the analysis solution, the coil is expressed as a hollow cylinder. A constant amount of radial and circumferential stress is calculated regardless of the circumferential position inside the coil. However, in the finite element analysis model, different results can be obtained depending on the circumferential position as both the internal stress distributions of the sleeve and coil are caused by the coiling of the spiral coil with the discontinuous connection at which the sleeve and strip contact begins (Figure 5). For instance, compared to the magnitude of the internal stress of the coil, the internal stress of the sleeve is small and hard to figure out the difference along the circumferential position in the left side of Figure 5. However, when we remove the coil temporarily in the stress plot of the assembly, you can see the difference of stress from red to blue along the circumferential position due to the stress concentration by the contact point of the sleeve and the coil. Accordingly, the stress at the location where the stress concentration portion’s influence is insignificant, such as the opposite side of the contact point where the stress concentration occurs, is analyzed and compared with the stress result. Table 2 summarizes the comparison positions of the stress results according to the coiling process.

Figure 5.

Radial stress of the sleeve during coiling.

Table 2.

The position of the stress comparison.

Item	Stress	Position
Sleeve	Radial stress	Inner radius ~ Outer radius
	Circumferential stress
The first coil	Radial stress	Inner radius~Outer radius
	Circumferential stress

Stress distribution of the elastic-plastic deformation of the sleeve and coil

In this section, nonlinear material information for the elastoplastic stress-strain is entered into the strip of the finite element analysis model. In general, it is somewhat difficult to establish a non-linear analysis solution including strain-based nonlinear material information. Therefore, a coiling analysis solution considering elastic deformation has been mainly developed to apply specific coiling process that has a high yield strength material or web deformation without the plastic deformation. However, as various types of materials are wound into a coil, the need for improvement of the analysis model was reviewed by examining the differences in the stress analysis results considering elastoplastic deformation from the existing analytical solutions. The position of the stress comparison between the elastic-based analysis solution and the finite element analysis model considering elastoplastic deformation is the same as that in Table 2, and the true stress and plastic strain for plastic deformation in Table 3 are applied to the finite element analysis. ¹⁷

Table 3.

True stress-plastic strain of the plasticity.

True stress, σt [MPa] (σt = σnom(1+εnorm))	Plastic strain, εpl = [-] (εpl = εt−εel = εt−σt/E)
200.2	0.0
246	0.0235
294	0.0474
374	0.0935
437	0.1377
480	0.18

Results and discussion

Stress distribution of the elastic deformation of the sleeve and coil

Figures 6 and 7 represent the radial stress σ_rr and the circumferential stress σ_θθ according to the radial position between the inner and outer diameters of the first layer of the coil and the sleeve. About the sleeve, the absolute values of σ_rr and σ_θθ increase as the pressure increases as new coil layers are stacked. In detail, when comparing the two stress values at the inner and outer diameters, the inner diameter’s σ_rr is higher than the outer diameter’s σ_rr, and the outer diameter’s σ_θθ is higher than the inner diameter’s σ_θθ. When the stress distribution results using the proposed analytical solution are compared with the finite element analysis, the calculation reliability of the proposed analysis solution can be confirmed by showing the same distribution trends of σ_rr and σ_θθ at the inner and outer diameters.

Figure 6.

Stress distribution of the sleeve.

Figure 7.

Stress distribution of the first layer of the coil.

At this time, the mandrel-sleeve-coil analysis solution was partially improved to improve the accuracy of the calculation of state variables such as the pressure in each coiling state, but a problem was found in that little difference occurred in the calculation of σ_θθ. This is due to the torsional deformation caused by the contact force that cannot be considered in the analysis solution, and it is difficult to implement an existing model that calculates the stress value based on the interlayer pressure value. Accordingly, based on the pressure applied to the sleeve, the Lame equation (equation (16)), that calculates the internal stress of a thick walled cylinder under outer and inner pressure, is applied to calculate σ_rr. The unknowns A and C in σ_rr can be determined by using the boundary conditions the outer and inner pressure. While σ_θθ is calculated by using equation (17) in which the weight α is applied to the Lame equation. ¹⁵ We plan to review the changes in α according to the coiling conditions in the future.

$${\sigma }_{\mathrm{rr}}\left(r\right)=\frac{A}{r^2}+2C$$

(16)

$${\sigma }_{\theta \theta }\left(r\right)=-\frac{A}{r^2}+\frac{2C}{\alpha }$$

(17)

In the case where the stress is generated inside the first stacked coil, tensile stress due to coiling tension and bending stress due to bending deformation occur during stacking, and then external pressure due to new coil stacking is applied, resulting in changes in σ_rr and σ_θθ (Figure 7). Relatively, σ_θθ has a higher σ_rr value, and the stress change due to coil stacking is insignificant. That is, σ_rr demonstrates that the stress value is influenced by the initial bending deformation. In the case of σ_rr, it can be seen that the absolute value of stress increases with the subsequent coil stacking, and the inner diameter has a relatively high stress value compared to the outer diameter. The outcome of comparing the results with the finite element analysis model shows the same stress distribution. For the coil in the finite element analysis model, it is very difficult to maintain the element aspect ratio because it has a very small thickness compared to the total length so the analysis result may be somewhat inaccurate compared to the sleeve stress calculation. When comparing the stress results of the analysis solution for the same position with the stress at the inner and outermost element positions, the same stress values were obtained so the reliability of the calculation results was adequate.

Stress distribution of the elastic-plastic deformation of the sleeve and coil

In this section, the sleeve and coil’s stress distribution between the analytical model for the elastic deformation and the finite element analysis model considering the plastic deformation of the coil is compared. As in Section 4.1, the results of the two models for σ_r and σ_θ occurring between the inner and outer diameters of the sleeve and occurring between the inner and outer diameters of the first stacked coil. (Figures 8 and 9) The basic analysis model of the elastic deformation cannot simulate the yield and plastic deformation caused by coil bending deformation and the elastoplastic stress state caused by the stacked coil layers, so the stress distributions of the two analysis models are different. However, for the sleeve, plastic deformation occurs due to bending in the coil, but the tensile stress in the vicinity of the outer diameter and compressive stress in the vicinity of the inner diameter are based on the neutral axis, so the coiling tension applied to the coil creates external pressure in the sleeve as they cancel each other. There was no change in the two analysis models, showing similar stress distributions for σ_rr and σ_θθ of the sleeve. Therefore, the elastic deformation basic analysis model developed in this study can be used to derive suitable operating conditions and predict the service life of the sleeve through strength evaluation based on the yield strength of the sleeve and fatigue analysis of the sleeve undergoing elastic deformation for a long time.

Figure 8.

Stress distribution of the sleeve.

Figure 9.

Stress distribution of the first layer of the coil.

Conclusion

In this paper, the analytical model for calculating the mandrel, sleeve, and coil stress based on elastic deformation was improved. The stress calculation formula for the coil was detailed by improving the assumption that there is no stress change depending on the radial position when coiling due to the thin thickness of the existing strip, and the assumption that the bending stress is not considered caused by the bending deformation of the steel plate. The validity of the analysis was verified by comparing the stress applied to the sleeve and coil calculated using this model against that of the finite element analysis results. The main conclusions are drawn as follows:

1. The analytical model developed in this paper is good to predict the sleeve’s internal stress, therefore critical coiling conditions that makes the plastic deformation of the sleeve, the amount of frictional force is need to remove the sleeve from the coil can be investigated with this model.

2. In addition, the analytical solution was compared with finite element analysis, including plastic deformation of the coil caused by strip bending. The limit of the model is due to the limit to predict the coil’s internal stress.

3. The current model could be further developed to consider the coil’s elastic-plastic deformation in the future.