2024 Volume 64 Issue 6 Pages 1037-1046
When analyzing the strip coiling process, the finite element (FE) method is closer to the actual working conditions compared to the analytical method. However, due to the large number of strip elements and contact elements, it often leads to problems such as long-time consumption and non-convergence. Meanwhile, traditional FE methods are still unable to solve the problem of additional contact deformation between layers. Therefore, in order to overcome the shortcomings of the above methods, the FE software MSC Marc is used to establish a strip coiling model. The distribution pattern of interlayer friction and contact stress are analyzed to propose a new step-by-step bonding FE model, which greatly reduces the computing time. Through laminated compression experiments, the variation curve between additional contact deformation and pressure is obtained. The curve is introduced into the gasket elements to consider the additional contact deformation between the layers, and the effect of additional contact deformation between the layers on the stress of the coil and the pressure on the mandrel is studied. Finally, the analytical solution is compared with the FE solution proposed in this paper, and the errors generated by the analytical method are analyzed.
Coiling is the most economical transportation and storage method for cold rolled sheet and strip.1,2,3,4) The coiling process, as the last step of the production line, also affects the final performance of the product.5,6) The stress state inside the coil directly determines whether defects will occur, such as surface quality damage caused by sliding between adjacent layers, collapse after unloading, and plastic deformation of the inner layers and sleeve caused by excessive radial stress.7,8,9,10) However, it is difficult to measure the internal stress state during the coiling process through experiments. Therefore, simulating the coiling process to predict the internal stress state of the coil is of great significance in reducing the occurrence of defects and improving the product quality.11,12,13,14)
The analytical method is the simplest and most efficient. Sims et al.15) simplified the coiling process into thin-walled cylinders nested layer by layer on a thick-walled cylinder and adopted the axisymmetric assumption. Considering the elastic deformation of the mandrel and the superposition state, Doghri16) used the pressure of the innermost layer and the tension of outermost layer as boundary conditions to calculate the stress distribution inside the coil:
| (1) |
| (2) |
where, σr,i, σθ,i and σT are respectively radial stress, circumferential stress of the i-th layer and coiling tension. N, t and h are the total number of layers, the number of coiled layers and the thickness of the strip, respectively; Ri, Rt and Rm are the radius of the i-th layer, the radius of the coiled layers and the outer radius of the mandrel, respectively; p1,t is the contact pressure of the innermost layer when the t-th layer strip has been rolled, which can be obtained from the elastic deformation of the mandrel.
In fact, the surface of the strip is not smooth and interlayer contact actually occurs on the rough surface.17) Under the pressure, the radial deformation of coil can be divided into two parts: own elastic deformation and additional contact deformation during interlayer rough contour compression.18,19,20,21,22) The anisotropy of coils is actually generated by interlayer additional contact deformation. However, additional contact deformation between layers cannot be considered by traditional analytical methods. Therefore, methods that modify the radial elastic modulus to reflect anisotropy have begun to be widely used. Altmann23) considered anisotropy in the assumption of plane strain and calculated the stress distribution inside the coil. In Hakiel’s24) research, the radial elastic modulus was set as a function of radial pressure to reflect nonlinearity accordingly. Benson25) used Hakiel’s method to select an exponential function about compressive stress to describe the variation of radial elastic modulus and proposed a calculation model for the internal stress of the coil. In the study of stress and deformation in coils by Li et al.,26) it is assumed that the radial elastic modulus is a quadratic function of compressive stress and the undetermined coefficient was determined by experiments. By modifying the radial elastic modulus to consider the additional contact deformation between layers, the calculation results can be optimized. However, due to the strong nonlinearity between additional contact deformation and contact pressure, the above methods are difficult to directly reflect the effect of additional contact deformation on the stress distribution. In addition, the above methods are all based on the axisymmetric assumption, which is significantly different from the continuous winding of strips in actual working conditions. Therefore, the above methods may cause certain errors.
The FE method can directly simulate the winding of strip, which can avoid the axisymmetric assumption so that fully consider factors such as interlayer slip, friction and bending.27) Hu et al.28) used FE method to simulate the tension coiling process and analyzed main factors that affect the pressure on the mandrel (tension, friction, outer diameter of the coil). The relationship between the pressure on mandrel and tension, friction, and coil diameter was obtained. To understand a force equilibrium for successful coiling, a FE model for coiling process with the belt wrapper was established by Park et al.29) Moreover, Park et al.30) established a FE model for coiling 1–3 mm thick stainless steel and divided the coiling process into two parts: initial coiling and stable coiling. After coiling a few layers, the analytical method was used to predict the radial pressure on the outer surface of the sleeve after coiling more layers. By the FE method, Park et al.31) further studied the influence of strip weight and mandrel on the stress distribution and concentration near the starting point of the inner layer. The maximum allowable stacking thickness and minimum sleeve thickness were determined to ensure that the steel coil and sleeve do not deform. However, the anisotropy caused by radial interlayer compression was not reflected in the above researches. Meanwhile, the coiling process involves large deformation of the strip and complex interlayer contact, and the length of the strip is often tens of thousands or even hundreds of thousands of times its thickness, which can lead to a large number of elements, especially contact elements. During simulation, these issues will ultimately result in long computing time and difficult convergence.
In order to solve the problems mentioned above, software MSC Marc is used to establish a FE model for coiling in this paper. Firstly, a step-by-step bonding model has been proposed, which not only ensures the accuracy of the calculation results but also significantly shortens the computing time. Then, the additional contact deformation between layers during coiling process is simulated by gasket elements, and the stress distribution law inside the steel coil considering the additional contact deformation between layers is analyzed. Finally, a comparison is made between the FE solution and the analytical solution and the errors caused by the neglect of friction and strip bending in the analytical method are analyzed.
This paper takes the coiler of a twenty high mill as an example for analysis. The physical image and structural schematic diagram are shown in Fig. 1. The mandrel is driven by the main motor through a gear reducer, with a speed of 6.2832 rad/s. The sleeve is connected to the mandrel through a flat key, with an outer radius of R=52 mm and an inner radius of r=32.5 mm, made of ordinary steel. There is a groove on the outer surface of the sleeve and the head of the strip is inserted into the groove before the coiling starts. During the coiling process, the tension between the mill and the coiler is 50 MPa.
The interlayer contact state and stress-strain will constantly change during the coiling process. Moreover, excessive length of strips will result in a large number of grids, ultimately leading to long computing time and difficulty in convergence. Therefore, in order to make the calculation feasible, the following simplifications and assumptions are introduced:
(1) The width of the strip is much greater than its thickness and the effect of inertia force is not considered. The coiling process can be simplified as a plane strain quasi-static model.
(2) To avoid unnecessary contact caused by the mandrel structure, the effect of the key is not considered. Assume direct contact between the mandrel and the sleeve. Ignore the deformation of the mandrel and make it a rigid body. The sleeve is considered a thick walled elastic cylinder.
(3) Assume that the coiling tension is uniformly distributed along the width direction of the strip.
(4) Assume that there is only elastic deformation in both the sleeve and strip. The elastic modulus is 210000 MPa and Poisson’s ratio is 0.28.
According to the actual working state of the coiler, the following boundary conditions are adopted in the FE model:
(1) The tail elements of the strip are applied to a distributed force of 50 MPa.
(2) The contact type between the strip head element and the surface of the sleeve is bonding.
(3) The contact type between the surface of the mandrel and the inner surface of the sleeve is bonding. The sleeve is driven by mandrel, with a speed of 6.2832 rad/s.
The grid division and boundary conditions are shown in Fig. 2. The number of elements and nodes is 49048 and 198213 respectively and the number of contact nodes is 98787. The model is iteratively solved by the full Newton Raphson iteration method and the convergence is judged according to the relative residual force. The friction model is a two-wire Coulomb model with a friction coefficient of 0.05 between the strip and the sleeve and between layers.
In order to obtain the contact state of each layer during the coiling process, the interlayer stress distribution is analyzed when the number of coiling layers is 10, 30 and 50. Figures 3, 4 and 5 show the comparison curves of friction stress σf and friction coefficient μ multiplied by normal contact pressure σn.
It can be seen from the figures that when coiling different layers, the data overlap of σf and μσn only occurs in the outermost layer part and the beginning and end of each layer, while σf is far less than μσn in other parts. In this paper, the strip coiling models with different thickness and roughness are simulated and the phenomenon is also obvious in every model. Based on the above interlayer state, it can be seen that sliding only occurs in the outermost layer and the local positions of the head and tail of each layer, while no sliding occurs in other parts.
When dealing with the contact problems with friction, the friction state of each contact point will be iterative to judge whether the contact state is sliding or bonding for each incremental step, which will consume much time and sometimes even cause the iteration non-convergence. Based on the above FE simulation results, the established model can be improved for the feature that all interlayer contacts except the outermost layer are in bonding state.
2.3. Establishment of the Step-by-step Bonding FE ModelIn order to reduce the determination and iteration of contact state and shorten the computing time, the following methods are used to improve the model:
Let n be the maximum number of layers with influence of interlayer friction on sliding in the coiling process. Assume that 2n layers have been coiled. Then, in the process of coiling 2n+1 to 3n layers, the influence of interlayer friction on sliding only exists in the n+1 to 2n layers and the subsequent new layers. The interlayer friction of the previous n layer has no influence on the relative sliding and the adjacent layers are in a bonding state. Similarly, in the process of coiling the 3n+1−4n layers, there is no relative sliding in the first 2n layers, and so on. This assumption can be implemented in the software by setting the contact property of the corresponding contact body to be bonding and then inserting the corresponding contact table in multiple cases.
Using the calculation parameters and boundary conditions mentioned above, the strip elements are segmented into different contact bodies according to the winding length. Define the elements of the first n layers of strip as contact body 1, named Strip 1, and define the elements of n+1 to 2n layers of strip as contact body 2, named Strip 2, and so on. Then define the contact relationship and contact table of different contact bodies under corresponding working conditions. For example, in the first working condition, Strip 1 (the first n-layer elements) is the contact state, while in the second working condition, Strip 1 is the bonding state. Establish multiple working conditions and insert the contact table into the corresponding working conditions. Finally, select the operating conditions in sequence in the analysis task. Figure 6 shows the distribution of each contact body after coiling 50 layers when n=5. The contact elements of strip1-strip8 are in the bonding state, while the contact elements of Strip 9-Strip 10 are in the contact state.
In order to verify the feasibility of the step-by-step bonding method, the original coiling model without this method and the improved models with n=5 and n=2 are calculated in this section. Figure 7 shows the variation curves of circumferential stress and contact pressure for three models, as well as the comparison of their computing time. In order to provide a more intuitive analysis of the effect of the step-by-step bonding method on the coiling results, additional contact deformation between layers is not considered in this section.
The calculation shows that the circumferential stress and contact pressure inside the coil have significant fluctuations near the inner layer. This is because there is interference at the head during the continuous winding process and elements are in the suspended state without direct contact with the sleeve. The circumferential stress increases from the inner layer to the outer layer and then begins to decrease to the coiling tension at the outermost layer. The internal contact pressure decreases in a stepped manner from the inner layer to the outer layer. At the beginning and end of each layer, there is a significant fluctuation and the fluctuation range gradually decreases with the increase of the number of layers.
By comparing the curves of the three models in Figs. 7(a) and 7(b), it can be seen that the results of the improved model and the original model are basically the same. Figure 7(c) shows the computing time of the original model and the improved models on the same computer. It can be seen that the computing time of the improved models is less than that of the original model. The effect of shortening the time is more obvious with the increase of the number of coiling layers. When coiling to 50 layers, the computing time of the original model is 232 hours. The time saved by the two improved models is 14% (32 hours, n=5) and 26% (60 hours, n=2) of the original model, respectively. It shows that the smaller the n is, the more obvious the effect of shortening the computing time is.
Through the above analysis, it is found that the improved models are basically consistent with the original model in the calculation results and the computing time is less. Therefore, the step-by-step bonding FE modeling method is feasible. It can be predicted that if more layers are coiled, the step-by-step bonding method can not only ensure the accuracy of the calculation results, but also greatly improve computational efficiency.
There is a kind of gasket elements in Marc’s material library. The elements have specific performance in the thickness direction, which show a change relationship between pressure and compression as shown in Fig. 8. After the initial nonlinear elastic response (segment AB), if the pressure p on the gasket exceeds the initial yield pressure, the gasket begins to yield. During further loading, the plastic deformation increases, accompanied by (possibly nonlinear) hardening, until the gasket is fully compressed (segment BD). At this stage, unloading occurs along the nonlinear elastic path (such as segment FG). When the gasket is fully compressed, loading and unloading follow a new nonlinear elastic path (curve CDE), while maintaining the permanent deformation generated during compression. Once the gasket is fully compressed, no additional plastic deformation will occur.
Based on the compression characteristics of the gasket element in the thickness direction, the additional compression deformation between the gasket layers can be measured experimentally to determine the loading and unloading path, that is, the relationship between pressure and compression.
3.2. Experimental Measurement of Additional Deformation between Layers during CompressionIn this section, the laminated compression experiment is conducted to investigate the relationship between the interlayer contact pressure and compression value. The material selected for the compression component is soft 304 stainless steel with a thickness of 0.4 mm and the strip is cut into ϕ 20 mm sized disk. The three-dimensional laser microscopy imaging system (VK-150K, KEYENCE, Japan) is used to scan the surface of the specimen and obtain the surface roughness Ra of 0.06 μm. Figure 9 shows the contour measurement results along a certain line on the surface of the specimen. To ensure the coaxiality of the laminations during compression, the hollow metal cylinder and cylindrical metal rod are used as the fixture and the pressure head. The inner diameter, outer diameter, and height of the metal cylinder are 21 mm, 38 mm and 30 mm, respectively. The metal rod has a diameter of 21 mm and a height of 30 mm.
In order to eliminate various possible factors such as shape defects, impurity mixing and improper operation that may affect the experimental results, 20, 30, 40, and 50 layers of discs are stacked for compression testing. Each stack conducts 3 repeated experiments. The tensile machine and specimen assembly for the experiment are shown in Fig. 10.
It is necessary to consider the deformation of the testing machine, metal rod, and disc substrate during the compression process. The thickness and deformation of each layer in the stack are considered uniform. First, the deformation displacement of the testing machine and metal rod is subtracted from the obtained displacement to obtain the overall compression displacement of the stack. Then, the stack displacement is divided by the number of layers to obtain the compression displacement of 1 layer. Furthermore, the displacement generated by the disc substrate is subtracted from displacement of 1 layer to obtain the additional compression displacement. Finally, the relationship between the pressure and the additional contact deformation is obtained. The corresponding curve is shown in Fig. 11.
The test results indicate that the pressure compression curve gradually stabilizes with the increase of the number of layers from 20 to 50. Therefore, in this paper, the curve obtained from the 50 layers laminated compression experiment is used as the loading path curve for the gasket elements. In the initial part (segment k1k2), the compression value increases rapidly as the pressure increases. When the pressure reaches a certain value, the rate of increase in compression rapidly slows down (segment k2k3). Finally, the rate of increase in compression gradually stabilizes (segment k3k4). This is because there are gaps between the laminated layers in the segment k1k2 and the gaps will be compressed first. Therefore, in the initial stage of compression, a small pressure will also cause a large amount of compression, which reflects extremely low stiffness. After the gaps disappear, the deformation is the additional deformation of rough contour contact and the stiffness gradually increases and tends to stabilize.
3.3 Establishment Method of the FE Model Considering Additional Contact Deformation between LayersAs shown in Fig. 12, the gasket elements can be attached to the upper and lower surfaces of the strip elements, with the thickness of 0.01 mm. The relationship between the pressure and compression value of the gasket can be defined based on the curve obtained from the previous experiment. The same boundary conditions as in Section 2 are used to establish the FE model considering additional contact deformation between layers.
In order to analyze the effect of additional contact deformation between layers on the internal stress of coils, calculations are conducted on the coiling models considering and not considering additional contact deformation between layers. In this section, the step-by-step bonding method mentioned above is used to ensure efficient calculation. The model that does not consider contact additional deformation is named Model 1, while the model that considers contact additional deformation is named Model 2. Figure 13 shows the equivalent stress cloud maps of Model 1 and Model 2. Through comparison, it is found that the equivalent stress of Model 2 is significantly lower.
Figure 14 shows the distribution of circumferential stress and contact pressure in Model 1 and Model 2. From Fig. 14(a), it can be seen that the circumferential stress variation trends of the two models are basically same and the difference is very small. This is because, under the same radius and thickness of the strip, the magnitude of circumferential stress depends on the difference in radial stress between the upper and lower parts of the strip, which is proportional to the derivative of the curve in Fig. 14(b). Therefore, the additional contact deformation between layers has a relatively small effect on the circumferential stress of the coil.
As shown in the Fig. 14(b), the contact pressure variation trends of the two models are basically the same. The contact pressure of the model considering additional contact deformation is smaller and the difference gradually narrows from the inside out. This is because considering additional contact deformation between layers reduces the radial stiffness of Model 2.
From Fig. 15, it can be seen that the radial pressure of the sleeve of the Model 2 is smaller and smoother. When coiling to 50 layers, the contact pressure of the Model 2 is 0.86 MPa lower than that of the Model 1. As the number of winding layers increases, the gap between the two models will further widen, which is because the decrease in interlayer contact stiffness also leads to a decrease in radial pressure on the surface of the sleeve. It can be predicted that if there are more layers of coiling, the difference between the results of the 2 models will further expand.
From the above analysis, it can be seen that due to the presence of additional contact deformation between layers, the radial contact stiffness inside the coil is reduced, resulting in a decrease in contact pressure, but the reduction is not significant. This is because the stiffness of segment k1k2 in Fig. 11 is very small, but it is caused by interlayer gaps. The gap will be eliminated when the outermost layer is taken. Therefore, the smaller stiffness of k1k2 segment is not reflected in the FE model. The applicable stiffness in the model is reflected in k2k3 and k3k4 segments, which is affected by rough contours.
During the coiling process, it is very difficult to measure the stress distribution inside the coil and on the surface of the sleeve. Therefore, it is not possible to prove the validity of the FE results through experiments. However, analytical methods have been accepted by scholars in academic research and practical production. Therefore, to verify the rationality of the FE method, the analytical methods of formulas (1) and (2) are adopted to compare with the FE solution. Then, the differences between the results of the two methods are also analyzed.
Figure 16 shows the circumferential stress and contact pressure inside the steel coil when the thickness of the strip is 0.4 mm and the number of coiling layers is 50. The FE method adopts the step-by-step bonding method mentioned above and considers additional contact deformation between layers, with friction coefficients of 0.05, 0.1, 0.2, and 0.3, respectively.
From Fig. 16, it can be seen that the FE solution is consistent with the analytical solution in trend, and the numerical values are also relatively close. This proves the correctness of the FE solution, in part. However, there are still differences between the two results. This is because the winding state, interlayer contact, friction and strip bending in the coiling process are comprehensively considered by the FE method, while the analytical method cannot consider these factors. The following will analyze the specific reasons for the differences:
(1) Compared with the analytical solution, the circumferential stress and contact pressure of the FE solution exhibit significant fluctuations near the inner ring, and the amplitude of the fluctuations gradually decreases from the inner ring to the outer ring. This is due to the interference at the head of the strip in the continuous winding model. Each layer of strip steel is not a thin-walled cylinder, and there is stress concentration near the inner ring due to its non-circular characteristics. As the number of layers increases, the curvature radius of each layer of strip also increases. The non-circular characteristics gradually weaken, and the amplitude of fluctuations decreases. The stress concentration can cause damage to the lifespan of the sleeve. However, analytical methods cannot reflect this problem.
(2) Neglecting friction in analytical solutions can cause certain errors. As the friction coefficient increases, the circumferential stress inside the coil increases and the error of the analytical solution also increases. This is because the circumferential stress during the coiling process is balanced with the influence of coiling tension, interlayer friction, and strip bending. When the friction coefficient increases, the interlayer friction force correspondingly increases, so the circumferential stress also increases. However, the analysis method cannot consider factors such as friction, which leads to an increase in circumferential stress error. In terms of contact pressure, the error also increases with the increase of friction coefficient. It is not satisfactory to obtain the same stress distribution by analytical methods when the friction coefficients are different. Therefore, based on the FE solution, it can be considered to modify the circumferential stress according to the friction coefficient, so that the analytical solution can take into account the influence of the friction coefficient.
(3) Near the core of the coil, the circumferential stress in the analytical solution is greater than that in the FE solution, while the contact pressure in the analytical solution is smaller. This is due to the difference between the axisymmetric model and the continuous winding model. In the FE model, each layer of strip is not a cylinder. When compressed, the strip steel near the core will not slide between layers but still produce a certain amount of tangential deformation. Compared with the axisymmetric cylinder of the analytical method, the stiffness decreases. Therefore, the FE solution is smaller in circumferential stress. Due to the high stiffness of the cylinder near the core in the axisymmetric analytical solution, the radial stress accumulated layer by layer during the coiling process is more difficult to transfer to the core. Therefore, the contact pressure of the analytical solution near the core is smaller.
(4) Unlike the analytical solution where the circumferential stress on the outermost layer is equal to the coiling tension, in the FE solution, the circumferential stress on the outermost layer is greater than the coiling tension and this trend becomes more pronounced as the friction coefficient increases. This is due to the influence of strip bending and interlayer contact friction on strip coiling. During the coiling process, the bending deformation of the strip causes the lower surface to compress and shrink, resulting in a sliding trend (some areas have sliding) opposite to the coiling direction between the lower surface of the outermost strip and the upper surface of the next layer in contact with it. Therefore, the outermost circumferential stress is greater than the given coiling tensile stress. As the friction coefficient increases, the difference between the maximum circumferential stress and tension in the outermost layer also increases. Therefore, in the analytical method, treating the coiling tension as the circumferential stress of the outermost layer can cause certain errors. It is worth noting that in previous analyses, the coiling tension was almost assumed to be equivalent to the circumferential stress on the outermost ring of the coil. However, from comparison, it is found that if the bending of the strip and interlayer friction are comprehensively considered, further correction of the outermost circumferential stress is necessary in the analytical method.
The above analysis explains the differences between the FE solution and the analytical solution, which also demonstrates the rationality of the finite element methods. The FE model takes into account factors such as winding state, interlayer contact, friction and strip bending during coiling. It is an improvement on the traditional analytical method, and the results will be further accurate compared to the analytical solution. However, in terms of computational efficiency, there is a significant gap between the FE method and the analytical method. Therefore, it is worth further research to make certain modifications to the analytical method based on the FE solution.
(1) A new step-by-step bonding method is proposed based on the FE software MSC. Marc. The method is summarized and proposed by analyzing the interlayer contact state during the coiling process. It can save computational costs while ensuring accuracy. Due to the excessive number of elements and iterative judgments, it is extremely difficult to simulate large deformation and complex contact like coiling by the FE method. Therefore, this method has certain reference significance for the FE simulation of coiling process in the future.
(2) For the additional deformation between layers, this article uses innovative gasket elements for reasonable treatment. The relationship between pressure and compression in the gasket is obtained through laminated compression experiments. It is found that additional contact deformation can cause a decrease in stiffness, thereby reducing contact pressure. This work also provides a new approach for future FE methods to deal with contact additional deformation.
(3) A comparison is made between the FE solution and the analytical solution in this paper. It is found that although the trend is same, compared with the FE solution, the circumferential stress in the analytical solution is higher and the radial stress is lower near the inner layer. The circumferential stress of the outermost layer in the FE solution is not equal to the coiling tension. This is due to the axisymmetric assumption and the neglect of friction and bending in the analytical solution. Therefore, the errors of the analytical solution are worth noting when the friction coefficient is too large.
This work was supported by the National Natural Science Foundation of China (No. 51474190).
On behalf of all authors, the corresponding author states that there is no conflict of interest.
