Tube end closing is a metal forming process that replaces welding processes while closing tubes ends. It depends on deforming a rotating tube using a roller, and therefore, it is also called tube end spinning. The process involves many parameters like contact depth, roller inclination angle, roller diameter, mandrel curvature, and tube rotational speed. This study develops a finite element model (FE-model) for this process and validates it through experimental results. The numerical and experimental results have shown minor deviation of 1.87%. The FE-model is then employed to carry out a statistical analysis based on the response surface method (RSM). The analysis of variance (ANOVA) and regression analysis have proved the accuracy of the obtained mathematical model. The contact depth has proved to have the most significant effect in the process responses, while the roller diameter has the least effect. Finally, an optimization analysis is carried out to select the finest conditions for the process.

Metal forming technology has proved over decades; it is the backbone of an enormous range of industries. There are countless applications involved in this field, such as deep drawing, blanking, coining, hydroforming, die forming, spinning, and tube forming. There are various topics that serve the technology of metal forming such as plasticity, forming limits, yield and failure criteria, strain rate, die design, lubrication, and process simulation [

Basic idea of tube end closing process.

The greatest merit of the tube end closing is that it replaces the welding processes for closing the tube ends, which makes it classified as a clean production process. The concept of clean production has attracted many research institutions over the last decades. One of the valuable researches that have focused on the environmental effects of welding processes is introduced by [

Free bending of tubes is one of the recent topics of interest. Murata and Kuboki [

On the other hand, optimization techniques have been employed to find the optimum conditions of the metal forming process that satisfies a specific objective function. Wang et al. [

Based on the literature, it is revealed that a statistical analysis of the parameters affecting the tube end closing process is not sufficiently covered. Therefore, this study aims first to develop a FE-model that can realistically simulate the real-life process. This can be achieved by validating the numerical results, specifically the tube closing diameter, to those obtained experimentally. The analysis is carried out on a standard size tube having an outer diameter of 31.75 mm and thickness of 1.778 mm and manufactured out of AL6061-T6. Moreover, this study introduces a statistical analysis to study the effect of various input parameters such as contact depth, roller inclination angle, roller diameter, mandrel curvature, and tube rotational speed. The selected responses for the analysis are the tube closing diameter and the resultant load. Besides, in this study, the response surface method (RSM) is used to develop a surrogate model that can replace the original numerical model. Analysis of variance (ANOVA) and regression analysis are carried out to evaluate the accuracy of the mathematical model. Finally, this study introduces an optimization analysis for the tube end closing by which the finest process conditions can be determined.

The structure of the research work is as follows: (1) the principles of the tube end closing process are investigated, (2) the FE-model is developed, (3) the numerical results are presented, (4) the tube end closing experiment is carried out and the numerical and experimental results are compared to each other, and (5) a statistical and optimization analysis of the process is carried out. The structure of the research work is illustrated in Figure

Structure of the research work.

Similar to the metal spinning process, the tube end closing depends mainly on moving a free rotating roller linearly towards a rotating tube supported by an inner mandrel. The process is categorized as one of the tube end forming processes, which depends on deforming the end of the tube into a desired shape rather than using caps and welding. By definition, the purpose of the tube end closing process is to convert the open end of a tube into a closed end, yet with a predefined closing diameter. Hence, it is required to decide the proper process parameters that can purposely result in closing the diameter of the tube as desired. Accordingly, this section is concerned with defining the parameters involved in the tube end closing process, which paves the way for further numerical analysis. The parameters of interest are divided into three categories: geometric, motion, and material parameters.

The geometric parameters can be classified into fixed and variable parameters. The fixed parameters are those related to the manufacturing of tube, roller, and mandrel. Hence, they cannot be changed after manufacturing. As an example, parameters such as tube outer diameter (

Geometric input parameters of the tube end closing process.

Symbol | Description |
---|---|

Roller inclination angle | |

Depth of contact at | |

Depth of contact at | |

Depth of the mandrel measured from tube’s face [mm] | |

Center distance between roller and tube for | |

Inclined center distance between inclined roller and tube for | |

Inner tube diameter [mm] | |

Outer tube diameter [mm] | |

Roller diameter [mm] | |

Mandrel diameter [mm] | |

Initial tube thickness [mm] | |

Radius of mandrel’s curvature/fillet [mm] | |

Tube closing diameter of the final deformed tube [mm] |

The contact depth (

Geometric parameters involved in tube end closing process.

As shown in Figure

On the other hand, the offset of the mandrel from the tube’s face (

Determination of mandrel position for (a) mandrel design 1 and (b) mandrel design 2.

From the equations, it can be noticed that the mandrel depth (

There are two basic motions involved in the tube end closing process: the tube rotation and the roller translation. Similar to metal spinning process, the tube end closing can be executed on a turning machine by simply holding the tube to the spindle and mounting the roller attachment to the carriage. One of the interests of this research work is to study the effect of the tube rotational speed (

Feed rate of the roller throughout the tube end closing process.

The properties of the tube material are undoubtedly one of the basic parameters that influence the tube end closing process. Obtaining the properties of AL6061-T6 is essential to provide the FE-solver with the material information required for the numerical analysis. Therefore, an experimental tensile test is carried out on a standard specimen out of an AL6061-T6 alloy as shown in Figure

True stress and true

Properties of AL6061-T6-T6 as obtained from tensile test.

Young’s modulus ( | Elongation to failure | Yield stress ( | Ultimate stress ( | Poisson’s ratio ( |
---|---|---|---|---|

68.94 GPa | 13% | 287 MPa | 370 MPa | 0.3 |

However, the FE-solver only accepts true effective stress and true equivalent plastic strain (

Studying the complicated mechanics involved in tube end closing process analytically is insufficient. Moreover, experimental technique is also limited because of the presence of multiple parameters that influence the process, which makes studying the process each time a parameter changes very costly. Instead, a FE-model that can accurately simulate the real-life process can provide an efficient tool for study. In this section, the FE-model of the tube end closing is developed using LS-Dyna 4.3©. Basic items involved in developing the FE-model, such as mesh settings, material modeling, and boundary conditions, are all discussed.

As the material behaviour in the tube end closing process is extremely nonlinear, the explicit dynamic analysis is employed. Generally, the time step for explicit analyses depends mainly on the mesh quality. On other words, the least local time step usually corresponds to the element of the least quality, and in turn, the global time step of the system is limited to this value [

In general, shell elements are suitable for cases of plane stress (membrane stress), which makes them the common choice for most of metal forming process simulations. As shown in [

Comparison of solid and shell elements for modeling the tube.

Based on all above, as the stress variation through the tube thickness cannot be considered negligible, solid elements are selected to model the tube. As the end of the tube is the region exposed to the highest deformation, it is modeled using a finer mesh size. Moreover, the circumference of the tube is divided into 100 divisions to obtain a mapped mesh of solid quadrilateral elements. The tube thickness is divided into 4 elements after a mesh-independent test is carried out as discussed later. The meshing of tube, roller, and mandrel is presented in Figure

Meshing of tube, roller, and mandrel using LS-Dyna©.

In the previous section, the properties of AL6061-T6 are obtained experimentally, and the data is transformed to suit the standards of the FE-solvers. This data is then used to provide the selected material model for the tube, which is the piecewise linear plasticity model (MAT_024), with all material data required such as density, young’s modulus, Poisson’s ratio, yield stress, and the isotropic hardening curve. It should be noted that the failure criteria available in this material model is only suitable for cases of uniaxial tension, which is not the case of this study. Instead, the localised necking criteria are employed to obtain the forming limit diagram FLD, as discussed later.

As both mandrel and roller are assumed rigid, the material model (MAT_RIGID) is found adequate as suggested by Maker and Zhu [

The larger the number of elements through the tube thickness, the more accurate the results, but also the larger the size of the model. To select the optimum number of elements through the tube thickness, a mesh-independent test has been carried out as shown in Figure

Effect of number of elements through tube thickness on the max

To evaluate the tube end closing process, a failure criterion should be clearly defined. The variation of the state of stress makes it insufficient to depend on only one failure criterion [

Forming limit curve, the safety margin curve, and the strain path for the tube end closing process.

The numerical simulation of the tube end closing process is carried out under the following conditions: [

Results obtained from the numerical simulation. (a) Distribution of equivalent plastic strain

In this study, tube end closing process is executed experimentally on a turning machine. The tube and mandrel are both fixed to a three-jaw chuck, and the rotational speed is controlled by a DC motor, as is usually the case in turning operations. To control the contact depth and the feed rate of the roller, the roller is mounted to the carriage of the turning machine, but via a special attachment that permits the rotational motion. Besides, by attaching the roller to the carriage, the inclination angle of the roller can be controlled.

The roller is a disk shaped of 140 mm diameter and is manufactured out of steel K110 coated with nickel chrome to obtain a higher hardness for the outer surface. As for the tube, a 2 m long bar of tube is used, and the protruding part is renewed after each experiment. According to the tube standards, the tube is manufactured out of AL6061-T6 and has 31.75 mm outer diameter and 1.778 mm thickness. The protruding length of the tube outside the chuck is set to be 30 mm to eliminate the bending effect on the tube as much as possible. A lubricant is used to provide a smooth contact between the tube and the roller throughout the process. The experimental setup of the tube end closing process is shown in Figure

Experimental setup of the tube end closing process.

In this study, five experiments have been carried out each at different contact depths

Deformed tubes obtained by five experiments each carried out at a specific contact depth while keeping the rest of parameters the same [

As the basic purpose of the experiments is to validate the FE-model results, the results of both numerical and experimental analyses are plotted against each other as shown in the graph in Figure

Numerical and experimental results of five experiments each carried out at a specific contact depth while keeping the rest of parameters the same [

Comparison of numerical and experimental results of five experiments each carried out at a specific contact depth while keeping the rest of parameters the same [

Contact depth ( | Numerical | Experimental | Error% |
---|---|---|---|

9.5 | 2.48 | 2.62 | 5.34 |

8 | 9.62 | 9.82 | 2.04 |

6.5 | 14.66 | 14.82 | 1.08 |

5 | 19.62 | 19.72 | 0.51 |

3.5 | 23.92 | 24.02 | 0.42 |

In this section, a statistical analysis is carried out to study the parameters involved in the tube end closing process. Through previous sections, a validated FE-model that can simulate the real-life process has been developed successfully. Hence, this FE-model can be employed to analyze the effect of various input parameters on a set of process responses of interest. However, due to the presence of various input parameters, studying the relation between input and output variables using the FE-model is costly. Instead, a statistical method called response surface method (RSM) is used to develop a surrogate model that can replace the original one. This surrogate model is simply a mathematical relation that provides an approximation for the real relation between the independent input variables and the process responses within a minimum error. Hence, the solving process depends on substituting in the mathematical model instead of solving a complete finite element problem. The mathematical relation consists of three main terms: linear, quadratic, and interaction terms, as shown in [

where

To create the desired mathematical relation, a design of experiment DOE is first carried out. DOE is used to create a set of experiments each has specific set of values for the controllable variables. There are two main methods for the DOE, the central composite design CCD and the Box-Behnken design BBD. In this study, the CCD is selected with face centered option and no replications, which means a total of 27 experiment to be solved. Although there may be many input parameters involved in the tube end closing process, in this analysis, only five parameters are chosen as the controllable variables, which are roller inclination angle (

The design levels of the selected input variables.

Controllable variable | Notation | Unit | -1 | 0 | 1 |
---|---|---|---|---|---|

Roller inclination angle | — | 0 | 7.5 | 15 | |

Depth of contact point between inclined roller and tube | mm | 3 | 6 | 9 | |

Radius of mandrel’s curvature/fillet | mm | 40 | 50 | 60 | |

Roller diameter | mm | 100 | 140 | 180 | |

Tube rotational speed | rpm | 400 | 800 | 1200 |

The selected response variables in this study are the tube closing diameter

Design table based on the DOE with the calculated results.

Test number | |||||||
---|---|---|---|---|---|---|---|

1 | -1 | -1 | -1 | -1 | 1 | 30.95 | 7.24 |

2 | 1 | -1 | -1 | -1 | -1 | 25.25 | 7.64 |

3 | -1 | 1 | -1 | -1 | -1 | 5.95 | 34.1 |

4 | 1 | 1 | -1 | -1 | 1 | 5.72 | 26.16 |

5 | -1 | -1 | 1 | -1 | -1 | 26 | 10.4 |

6 | 1 | -1 | 1 | -1 | 1 | 26.44 | 7.42 |

7 | -1 | 1 | 1 | -1 | 1 | 10.24 | 24.5 |

8 | 1 | 1 | 1 | -1 | -1 | 6.34 | 31.97 |

9 | -1 | -1 | -1 | 1 | -1 | 25.77 | 10.49 |

10 | 1 | -1 | -1 | 1 | 1 | 26.74 | 7 |

11 | -1 | 1 | -1 | 1 | 1 | 10.84 | 23.5 |

12 | 1 | 1 | -1 | 1 | -1 | 6.32 | 31.6 |

13 | -1 | -1 | 1 | 1 | 1 | 30.46 | 7.74 |

14 | 1 | -1 | 1 | 1 | -1 | 25.03 | 7.77 |

15 | -1 | 1 | 1 | 1 | -1 | 6.03 | 35.1 |

16 | 1 | 1 | 1 | 1 | 1 | 5.06 | 28 |

17 | -1 | 0 | 0 | 0 | 0 | 17.25 | 16.1 |

18 | 1 | 0 | 0 | 0 | 0 | 16.16 | 15.3 |

19 | 0 | -1 | 0 | 0 | 0 | 25.86 | 8.27 |

20 | 0 | 1 | 0 | 0 | 0 | 4.6 | 29 |

21 | 0 | 0 | -1 | 0 | 0 | 17.95 | 14.15 |

22 | 0 | 0 | 1 | 0 | 0 | 16.31 | 15.76 |

23 | 0 | 0 | 0 | -1 | 0 | 16.48 | 15.54 |

24 | 0 | 0 | 0 | 1 | 0 | 16.42 | 15.78 |

25 | 0 | 0 | 0 | 0 | -1 | 16.08 | 17.5 |

26 | 0 | 0 | 0 | 0 | 1 | 19.31 | 13.9 |

27 | 0 | 0 | 0 | 0 | 0 | 16.41 | 15.5 |

The results of the tests shown in Table

Regression table for tube closing diameter

Term | ||||||
---|---|---|---|---|---|---|

Regression coefficient | Regression coefficient | |||||

Constant | 16.520 | 79.31 | ≤0.001 | 15.4700 | 66.50 | ≤0.001 |

-1.135 | -8.59 | ≤0.001 | -0.3506 | -2.38 | 0.034 | |

-10.078 | -76.27 | ≤0.001 | 10.5533 | 71.51 | ≤0.001 | |

-0.199 | -1.51 | 0.156 | 0.3767 | 2.55 | 0.024 | |

-0.039 | -0.30 | 0.773 | 0.1117 | 0.76 | 0.463 | |

1.277 | 9.66 | ≤0.001 | -2.2839 | -15.48 | ≤0.001 | |

0.172 | 0.48 | 0.640 | 0.2338 | 0.58 | 0.569 | |

-1.303 | -3.64 | 0.003 | 3.1688 | 7.91 | ≤0.001 | |

0.604 | 1.68 | 0.116 | -0.5112 | -1.28 | 0.224 | |

-0.079 | -0.22 | 0.828 | 0.1938 | 0.48 | 0.637 | |

1.162 | 3.24 | 0.006 | 0.2338 | 0.58 | 0.569 | |

0.007 | 0.05 | 0.963 | 0.4106 | 2.62 | 0.021 | |

-1.107 | -7.90 | ≤0.001 | 1.0444 | 6.67 | ≤0.001 | |

-0.332 | -2.37 | 0.034 | -1.4819 | -9.47 | ≤0.001 | |

From the table, the following can be revealed:

As for the response variable

The effective variables include the linear terms [

All linear terms are inversely proportional to the response

The linear term

As for the response variable

The effective variables include all the linear terms, the quadratic terms [

All linear terms are directly proportional to the response

The linear term

The correlation coefficients

From the normal probability plot shown in Figure

Normal probability plot for the data of (a)

Residual plot for the data of (a)

The relation between a specific response variable and two input variables can be presented in a 3D plot called “response surface” or 2D plots called “contour plots,” while holding the other input variables at fixed values. The response surfaces and contour plots for both responses

Response surfaces for (a)

Contour plots for (a)

In this section, the mathematical model previously developed based on the RSM is employed to perform an optimization analysis. The purpose of an optimization analysis is to determine the optimal input variables that satisfy a specific objective function without violating a predefined set of constraints. There are various optimization techniques that have been employed in many previous researches. For example, the genetic algorithm (GA) and artificial neural network (ANN) were employed by Suresh et al. [

The optimization analysis is performed twice: first, when the objective function is to minimize

Optimal values of the input variables based on the defined constraints and objective functions.

Input variables | Constraint type | Uncoded values | Case (1) | Case (2) |
---|---|---|---|---|

Optimal values | ||||

Constrain to region | [0 : 15] | 15^{o} | 15° | |

Constrain to region | [3 : 9] | 9 | 6.57 | |

Hold value | 50 | 50 | 50 | |

Hold value | 140 | 140 | 140 | |

Constrain to region | [400 : 1200] | 828 | 990 |

To confirm the results of the optimization analysis, a confirmation test is carried out. Based on the optimal input variables, the original model, which is the previously developed FE-model, is used to obtain the tube closing diameter

Comparison between the results obtained from the original FE-model and from the optimization analysis.

Case no. | Input variables | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Conf. test | Optimization result | Error% | Conf. test | Optimization result | Error% | ||||||

1 | 15° | 9 | 50 | 140 | 828 | 4.25 | 4.17 | 1.9% | 29.3 | 30.9 | 5.2% |

2 | 15° | 6.5 | 50 | 140 | 990 | 14.6 | 13.88 | 5.2% | 16.87 | 16.9 | 0.17% |

In this study, the parameters involved in tube end closing process have been investigated. The tube end closing process is a metal forming process that is similar to metal spinning as it depends on deforming the tube by using a roller while the tube rotates. The tube upon which the analysis is carried out is manufactured out of AL6061-T6 and has 31.75 mm diameter and 1.778 mm thickness. Although there are many input parameters involved in the process, only five parameters are selected for this study, which are contact depth, roller inclination angle, roller diameter, mandrel curvature, and tube rotational speed. The principles related to the process have been studied and classified into three categories: geometric, motion, and material parameters. After that, the preprocessing setups required to develop the FE-model, such as meshing, material modeling, and boundary settings, have been all discussed. The results obtained by the FE-model under specific process conditions have been presented. In addition, the tube end closing process has been tested experimentally under the same process conditions. The numerical and experimental results have been compared to each other in terms of the tube closing diameter, and a minor deviation of only 1.87% has been recorded. Moreover, a statistical analysis has been carried out based on the response surface method (RSM). The central composite method (CCD) has been selected for the design of experiment (DOE), and a set of 27 tests has been established after selecting the upper and lower levels of the selected controllable variables. Based on the test results obtained by the FE-model, the regression model has been constructed. The correlation coefficients, the normal probability plots, and the residual plots have all proved the accuracy of the regression model. The contact depth has proved to have the most significant effect in the process responses, while the roller diameter has the least effect.

Finally, an optimization analysis has been carried out based on the desirability method. The analysis is aimed to find the finest conditions of the process that satisfy the objective function under two cases. For the first case, the objective function is to minimize the tube-closing diameter only. The finest process conditions have been found to be as follows: [contact depth: 9 mm, roller inclination angle: 15°, roller diameter: 140 mm, mandrel curvature: 50 mm, and tube rotational speed: 828 rpm]. On the other hand, the objective function in the second case is to minimize both the tube-closing diameter and the resultant load. The finest process conditions have been found to be as follows: [contact depth: 6.5 mm, roller inclination angle: 15°, roller diameter: 140 mm, mandrel curvature: 50 mm, and tube rotational speed: 990 rpm]. A confirmation test has been carried out to check the effectiveness of the optimization process. The results obtained from the original model have been compared to those obtained from the regression model and the difference percentages have been recorded. For case (1), the errors in

The FE-Solver and the Optimizer working files that have been used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.