Processing-Scheme Design for Forming Curved Ship Plate and Analysis of Calculation Cases

Abstract

The forming process of curved ship plate suffers from a low degree of automation, mainly due to the lack of an effective processing-scheme design method. In this paper, based on the proposed concept of the “basic amount of forming plasticity”, which can connect the plastic strain induced by the line heating and the deformation to form the target shape, a database is firstly established to describe the plastic strain provided by the heating coil with specific processing parameters, considering the effect of the plate boundary and adjacent heating lines. Secondly, a finite element method is developed and presented to calculate the plastic strain needed to form the target shape. Finally, a processing-scheme design method for forming the curved ship plate is verified by the case study of three typical types of shape: sail-type plates, saddle-type plates, and curved plates with torsion. The verification result shows the processing-scheme design method can provide helpful guidance for the practical forming process in shipyards.

1. Introduction

The manufacturing of ship plates is an important part of shipbuilding. The automation of the manufacturing process of ship plates, involving lofting, marking, pretreatment, cutting, and welding, has made significant progress. However, the forming of the curved ship plate, which is mainly conducted by the line-heating process in the shipyard, still relies on experienced workers with years of training. The low degree of automation in curved ship-plate forming is becoming a critical restricting factor for shipbuilding, and has attracted the attention of many scholars in recent years.

Yao et al. [1] investigated the deformation generated by a single heating line on a flat plate and considered different factors affecting it, such as the inherent strain, edge effect, and size effect. Adan Vega et al. [2] concluded that the deformation caused by multiple heating lines is not simply the superposition of each single heating line. In order to predict the forming deformation, the effects of overlapping, crossing, and parallelism of multiple heating lines have been studied, and the research shows that residual stress is the main factor affecting the inherent strain of multiple heating lines. Cao et al. [3] used a numerical simulation method to obtain the effects of processing parameters such as the heating path, traveling speed, and cooling conditions, and found that when producing longitudinal bending, an accompanying transverse bending is unavoidable.

Ueda et al. [4] took the plastic strain generated after each heating line as the main consideration of study and discussed the principle of path planning for line heating, and this work laid a foundation for the automatic line heating and forming of curved ship plates. Park et al. [5] of the Samsung Research Institute, Korea, categorized the thermal processing for ship-plate forming into two types, linear heating and angular heating, used the generated angular deformation and deflection as a criterion to measure the processing effect, and determined the processing parameters by comparison with the target shape; however, this method is only applicable for the forming of sail-type plates and lacks general applicability for other types of ship plates. Liu et al. [6] proposed a spiral-heating method, which can effectively solve the problems of insufficient heating and over-concentration of heating points in the forming process. Zhao et al. [7] proposed a novel forming method integrated with both cold pressing and line heating, and developed a prototype machine using in-plane shrinkage, which can be calculated from the total strain in the forming process. This is used as the criterion to measure the forming effect, then a simplified algorithm for process-parameter design and path planning is presented and its effectiveness verified by saddle-type plates.

In recent years many scholars have also undertaken much research on the selection of heat sources for line heating. The most widely used heat source in shipyards is the oxy-propane flame, which has the advantages of easy operation and low price. Nonetheless, it is difficult to control the heat output and has low energy efficiency. The laser-heating heat source has a high concentration and the heat-affected area is small, which has become a hot area of research in recent years. Yu [8] et al. studied the flexible manufacturing technology of laser forming, the effect of line heating on material properties, and the mechanism of laser-heating forming has also been taken into consideration. Tsirkas et al. [9] explored the effect of laser-heating parameters on the temperature field of thermoforming using numerical simulation, and proposed that the speed at which the laser moves and the energy magnitude are the most critical factors. Wang et al. [10] established a more completed finite element model for laser heating, but through the study it was found that due to the over-concentration of the heat source of laser heating, it is easy to cause adverse effects such as the wrinkling of the plate. On the other hand, the cost and maintenance of laser-processing equipment is high, which is not suitable for the current needs of a shipyard. Electromagnetic induction heating becomes the most suitable heat source for the time being, which has the advantages of high heating efficiency, low heat loss, and ease of control. Luo et al. [6] established a finite element model of high frequency induction heating to calculate the temperature distribution of the induction coil. Zhang et al. [11] systematically described the characteristics and strain of the temperature field under the induction heat source through both finite element calculation and experiments.

As far as the processing-scheme design for curved ship-plate forming through line heating is concerned, there are three issues that need to be taken into account, as shown in Figure 1. The first question is how much plastic strain can be provided by induction coils with specific size, traveling speed, heating power, and other parameters. Although the majority of scholars choose macroscopic deformations, such as deflection, bending angle, or in-plane shrinkage, as factors for influencing the forming-scheme design, the plastic strain induced by line heating is the fundamental cause of macroscopic deformations, so this paper chooses plastic strain as the primary object for study and discussion.

The second question is how to get the quantity and distribution of plastic strain required by a target shape formed from an initial shape plate. Dong et al. [12] calculated the total strain from the initial shape to the target shape based on the mathematical relationship between strain and radius of curvature, and then categorized the calculated strain distribution into in-plane strain and bending strain, plotted them on the strain-distribution map in terms of principal strain, and expressed them figuratively with arrow vectors, but this method still has the disadvantages of low calculation efficiency and unsatisfying accuracy. Cai et al. [13] presented a new method to calculate the strain by applying the displacement field on the initial plate with boundary constraints from an imaginary mold, which can significantly improve the calculation efficiency. However, this method can only calculate the total strain, not the plastic strain, which limits the effect of heating path planning.

When the total strain or plastic strain distribution needed for a target shape forming is obtained, and it is clear how much such strain can be generated by the available coils, the third issue is to select a suitable induction coil with certain processing parameters and determine the position where the heating lines are to be plotted, i.e., the path-planning problem. Hu et.al. [14] calculated the total strain quantity on the path from the strain distribution diagram, and the process-parameter group in which the strain is closest to the required total strain is selected as the process parameters of this path plan. Nonetheless, not many calculation cases are shown.

Since it is difficult to obtain both the distribution and the history of the temperature and plastic strain induced by the line-heating process by either theoretical analysis or experimental methods, the finite element method becomes a realistic option for research into induction line heating. Dong et al. [15] proposed a method of electromagnetic simulation in COMSOL and heat-transfer simulation and stress–strain simulation in ABAQUS for the line-heating process, and verified the accuracy of the simulation results by comparison with the experimental results; the same method is adopted in this paper.

This article is structured as follows: in response to the current problems, the influence on the generated plastic stain of the processing parameters, such as specific induction coil size, traveling speed, heating power, and other parameters, is investigated by means of theoretical analysis and finite element simulation. A database of the relationship between process parameters and plastic strain is established. The coupling effect on the plastic strain from both the plate boundary and multiple heating lines is discussed. Furthermore, an algorithm for calculating the plastic strain distribution required to form a target curved plate considering material and geometric nonlinearities is developed. Finally, a matching method which helps to select the induction coils with specific heating parameters from the plastic strain database, and to give a processing path, is presented and its effectiveness is verified by case studies of three types of plate: sail-type plates, saddle-type plates, and curved plates with torsion.

2. Calculation Method of the “Basic Amount of Forming Plasticity” and Establishment of a Plastic Strain Database

During the process of line heating, the plate undergoes elastic deformation, elastic-plastic deformation, and even complete plastic deformation. The residual plastic strain after cooling is the main cause of the permanent deformation of the plate, and it can also be regarded as the result of a load that the heating coil can apply on the plate. Curved ship plates, in a plastic forming process, are obtained essentially through the process of applying plastic strain to the initial shape plate to form the target shape, so studying the plastic strain generated by the heating process can provide guidance for curved ship-plate formation.

As previously mentioned, three questions need to be taken into account, and this chapter solves the first question, i.e., how much plastic strain can be provided by induction coils with specific size, traveling speed, heating power, and other parameters.

2.1. The Concept of “Basic Amount of Forming Plasticity”

The deformation effect that line heating brings to the plate can be regarded as the result caused by the equivalent force F and equivalent moment M. F can also be denoted by the macro shrinkage $\Delta l_p$ in the heating line direction of the plastic region:

$$\frac{F}{EA}={\epsilon }_x^{in}=\Delta l_p$$

(1)

where A is the area of the cross section, E is the modulus of elasticity, EA is the compressive (tensile) stiffness of the plastic region, ${\epsilon }_x^{in}$ is the in-plane plastic strain in the x-direction, and $\Delta l_p$ is the shrinkage of the heating area.

Similarly, the equivalent moment M can also be denoted by the macro bending angle $\theta$:

$$\frac{M}{EI}=\frac{({\epsilon }_t^{out}-{\epsilon }_b^{out})}{H}=\theta \prime$$

(2)

where A is the area of the cross section, E is the modulus of elasticity, EI is the bending stiffness of the plastic region, ${\epsilon }_t^{out}$ is the plastic strain at the upper surface, ${\epsilon }_b^{out}$ is the plastic strain at the lower surface, $\theta \prime$ is the derivative of the bending angle, and l_p is the length of the plastic region.

Based on the previous theoretical derivation, the plastic strain is the critical governing factor between macro deformation and line heating, but since it is distributed in all layers and directions, the calculation is complicated, so this paper proposes the concept of the “basic amount of forming plasticity”, which can effectively connect the macro deformation and plastic strain caused by line heating, and has the advantages of convenient calculation and easy analysis, as shown in Figure 2a. The basic amount of forming plasticity includes the size of stable plastic zone, longitudinal shrinkage per unit length in the stable plastic zone, longitudinal bending per unit length in the stable plastic zone, transverse shrinkage of the stable plastic zone, and transverse bending of the stable plastic zone.

As shown in Figure 2b, the plastic zone is the area where the plastic strain is generated by line heating in the plate, and the stable plastic zone is the more stable area in the middle part of the plastic zone. According to the basic theory of line heating, the plastic strain is unstable at both ends of the heating line due to the edge effect during the heating process; in the middle part of the plastic zone, the quantity and distribution of the plastic strain in the cross section perpendicular to the heating line are basically the same. If the heating parameters are the same, but the length of the heating line is different, only the length of the intermediate stable plastic zone is different, and the quantity and distribution of plastic strain always remains the same.

Since the stable plastic zone occupies a large part of the plastic zone, the size of the stable zone is mainly taken for calculating the basic amount of forming plasticity, and the non-stable zone can be ignored, the mesh distribution of the stable plastic zone is shown in Figure 3.

(1)

Longitudinal shrinkage per unit length in the stable plastic zone

Since the length of the heating line may change, the longitudinal shrinkage can be expressed by the per unit length $S_{-\mathrm{long}}$ of the plastic zone and the longitudinal shrinkage multiplied by the heating length. Thus, $S_{-\mathrm{long}}$ can be calculated as:

$$S_{-\mathrm{long}}=\frac{{\displaystyle \sum _n{\epsilon }_{xx}^P·l}}{n·l}$$

(3)

where ${\epsilon }_{xx}^P$ is the mid-plane plastic strain in the longitudinal direction (x-direction), n is the number of elements in the plastic zone in the transverse direction (y-direction), and l is the lengthwise dimension of the element.

(2)

Longitudinal bending per unit length in the stable plastic zone

Based on a similar analysis, the equation for calculating the longitudinal bending per unit length in the plastic zone is:

$$B_v=\frac{S_{-\mathrm{long}}^{\mathrm{top}}-S_{-\mathrm{long}}^{\mathrm{bottom}}}{d}$$

(4)

where $S_{-\mathrm{long}}^{\mathrm{top}}=\frac{{\displaystyle \sum _n{\epsilon }_{xx}^{P-\mathrm{top}}·l}}{n·l}$ is the longitudinal shrinkage per unit length of the upper surface (near the coil), $S_{-\mathrm{long}}^{\mathrm{bottom}}=\frac{{\displaystyle \sum _n{\epsilon }_{xx}^{P-\mathrm{bottom}}·l}}{n·l}$ is the longitudinal shrinkage per unit length of the lower surface (away from the coil), ${\epsilon }_{xx}^{P-\mathrm{top}}$ is the longitudinal (x-directional) plastic strain on the upper surface (near the coil), ${\epsilon }_{xx}^{P-\mathrm{bottom}}$ is the longitudinal (x-directional) plastic strain on the lower surface (away from the coil), and d is the plate thickness.

(3)

Transverse shrinkage of the stable plastic zone

The transverse shrinkage in the plastic zone is the amount of transverse shrinkage $S_{-\mathrm{tran}}$ that can be generated by the equivalent resultant force $F_{-\mathrm{tran}}$ provided by the transverse plastic strain in the stable plastic zone, calculated by the equation

$$S_{-\mathrm{tran}}=\frac{{\displaystyle \sum _n{\epsilon }_{yy}^P·b_c}}{\mathrm{B}}$$

(5)

where ${\epsilon }_{yy}^P$ is the plastic strain in the transverse direction (y-direction), n is the number of elements in the plastic zone in the width direction (y-direction), B is the width of the stable plastic zone, and b_c is the size of the element in the transverse direction.

(4)

Transverse bending of stable plastic zone

The equation for calculating the transverse bending of stable plastic zone is:

$$B_v=\frac{S_{-\mathrm{tran}}^{\mathrm{top}}-S_{-\mathrm{tran}}^{\mathrm{low}}}{d_p}$$

(6)

where $S_{-\mathrm{tran}}^{\mathrm{top}}=\frac{{\displaystyle \sum _n{\epsilon }_{yy}^{P-\mathrm{top}}·b_c}}{\mathrm{B}}$ is the transverse shrinkage on the upper surface (near the coil), $S_{-\mathrm{tran}}^{\mathrm{low}}=\frac{{\displaystyle \sum _n{\epsilon }_{yy}^{P-\mathrm{low}}·b_c}}{\mathrm{B}}$ is the transverse shrinkage on the lower surface (away from the coil), ${\epsilon }_{yy}^{P-\mathrm{top}}$ is the transverse (y-direction) plastic strain on the upper surface (near the coil), is the transverse (y-direction) plastic strain on the lower surface (away from the coil), and d_p is the plate thickness.

2.2. Establishment and Analysis of Plastic Strain Database of a Single Heating Line

Using the previous theoretical approach, the basic amount of forming plasticity provided by a single heating line under different process parameters is calculated through the thermal elastic-plastic FEM. The finite element model, boundary conditions, and material parameters are introduced in Appendix A and Appendix B. The corresponding process parameters are shown in

Table 1. Process parameters.

Plate Thicknesses (m)	Coil Radius (m)	Line Energies (J/m)	Traveling Speeds(m/s)
0.008	0.04	2,500,000	0.005
0.01	0.05	3,000,000	0.01
0.015	0.06	3,500,000	0.015
0.02		4,000,000	0.02
		4,500,000	0.025

, and the line energy is the input energy per unit length in line heating. A database is established and the influence of each processing parameter on the basic amount of forming plasticity is also analyzed to provide a basis for the better selection of each parameter.

2.2.1. Establishment of Plastic Strain Database

The basic amount of forming plasticity with different induction coil radii, different moving speeds of the heating source, and under different heating power have been stored in the database. Due to the length limitations of this article, only the calculation results in the study case of a 0.01-m-thick plate are presented here for discussion, as shown in Figure 4, Figure 5 and Figure 6; the rest of the calculation results for different thicknesses can be found in Appendix C.

(1)

Basic amount of forming plasticity when the radius of the coil is 0.04 m.

(2)

Basic amount of forming plasticity when the radius of the coil is 0.05 m.

(3)

Basic amount of forming plasticity when the radius of the coil is 0.06 m.

The database as mentioned above has been established for different plate thicknesses, coil radii, line energies, and moving speeds of the basic amount of forming plasticity under single heating conditions.

2.2.2. Influence of Different Parameters on the “Basic Amount of Forming Plasticity”

The selection of heating parameters remains a complex problem for plate forming, since one value of the basic amount of forming plasticity may correspond to several processing parameters. A group of commonly used process parameters, 0.01 m plate thickness, 0.05 m coil radius, 0.01 m/s traveling speed, 3500 kJ/m line energy, which corresponds to −0.0312 m longitudinal shrinkage per unit length of stable plastic zone, −0.0312 m transverse shrinkage of stable plastic zone, −0.0066 rad longitudinal bending of stable plastic zone per unit length, −0.2374 rad transverse bending of stable plastic zone, were selected for analysis and discussion. The influence of each parameter on the basic amount of forming plasticity can be analyzed for a better selection of the processing parameters.

(1)

Influence of plate thickness on the basic amount of forming plasticity.

Take the traveling speed of 0.01 m/s and the line energy of 2.0 × 10⁶ J/m as an example, as shown in Figure 7. When the plate thickness increases, the length of the stable plastic zone tends to stabilize, and the width of the stable plastic zone, the longitudinal shrinkage per unit length, the transverse shrinkage and the transverse bending all gradually decrease. The longitudinal bending per unit length gradually increases as the plate thickness increases.

(2)

Influence of coil radius on the basic amount of forming plasticity.

As shown in Figure 8, the length of the stable plastic zone, longitudinal shrinkage per unit length, transverse shrinkage, and transverse bending gradually decrease as the radius of the coil increases, while the width of the stable plastic zone and longitudinal bending per unit length increase as the coil radius increases.

(3)

Influence of line energy on the basic amount of forming plasticity.

As shown in Figure 9, with the increase in line energy, the length of the stable plastic zone firstly shows a slight decrease and then keeps increasing, as the width of the stable plastic zone and transverse shrinkage increase monotonically. Longitudinal shrinkage per unit length and transverse bending continuously increase to reach their maximum values when the line energy is 1.5 × 10⁶ J/m, as longitudinal bending per unit length decreases and reaches the minimum value at the same line energy.

(4)

Influence of coil moving speed on the basic amount of forming plasticity.

As shown in Figure 10, when the moving speed of the heating coil increases, the length and width of the stable plastic zone tend to be stable, and the other parts of the basic amount of forming plasticity, such as longitudinal shrinkage per unit length, transverse shrinkage, and transverse bending, gradually increase, while longitudinal bending per unit length decreases and eventually reaches a stable state.

According to the above analysis, the variations in the relationship between the basic amount of forming plasticity and the process parameters is summarized in

Table 2. The variation relationship between the basic amount of forming plasticity and the process parameters.

Process Parameters	Trends	Length of Stable Plastic Zone	Width of Stable Plastic Zone	Longitudinal Shrinkage per Unit Length	Transverse Shrinkage	Longitudinal Bending per Unit Length	Transverse Bending
Plate thickness
Coil radius
Line energy
Moving speed

, which is helpful for selecting processing parameters.

2.3. Influence of Plate Boundary on the “Basic Amount of Forming Plasticity”

The basic amount of forming plasticity that a single heating line can provide is discussed in Section 2.3, but the forming process is a multi-step process such that a target shape cannot be achieved in a single heating line, and the heating position may be in any place on the plate. In this section, for a heat source with fixed parameters, where the coil radius is 0.05 m, the travel speed of the coil is 0.1 m/s, and the size of the processed plate is 2 m × 1 m × 0.01 m, the boundary effect on the basic amount of forming plasticity is studied.

The coil travels from x = −0.8 m to x = 0.8 m(the coordinate system is shown in Figure 11), and the relationship between the basic amount of forming plasticity and the distance from the heating line to the edge of the center line, d_c, is observed, where d_c is set to vary in the range of 0~0.48 m. The arrangement of the heating line is also shown in Figure 11.

Figure 12 shows that when d_c is less than 0.3 m, the boundary does not affect the heating line, and the basic amount of forming plasticity follows a stable trend and remains unchanged. When the distance d_c exceeds 0.3 m, the transverse shrinkage and transverse bending change firstly, and both become smaller and smaller, but the decreasing trend is irregular. When the distance d_c exceeds 0.35 m, the transverse shrinkage and the transverse bending decrease as the distance d_c increases, and the longitudinal shrinkage per unit length and the longitudinal bending per unit length begin to increase, following an irregular trend.

If the basic amount of forming plasticity is applied to the region near the plate edge, it is necessary to consider the boundary effect on the basic amount of forming plasticity. That means the corresponding quantity of processing parameters should be changed with a relevant multiplier, which can be selected in the diagram according to the position of the heating line.

2.4. Influence of Adjacent Heating Lines on the “Basic Amount of Forming Plasticity”

As suggested in Section 2.4, the adjacent heating lines have an effect on the basic amount of forming plasticity caused by the current heating line; this subsection discusses the effects of adjacent parallel heating lines, and processing parameters are the same as Section 2.4. The basic amount of forming plasticity caused by a single heating line, before and after the neighboring line heating is operated, is compared and discussed. Considering the symmetry, two identical neighboring heating lines are plotted parallelly on two sides of the current heating line.

Three analysis processes are calculated, including differences in heating power Q of adjacent heating lines, distance d_a between adjacent heating lines, and length L_a between adjacent heating lines. The relevant parameters are shown in

Table 3. Processing parameters under each analysis.

	Distance da between Adjacent Heating Lines	Heating Power Q of Adjacent Heating Lines	Length Difference La between Adjacent Heating Lines
Analysis 1	0.1 m, 0.13 m, 0.15 m, 0.2 m, 0.25 m, 0.28 m, 0.3 m, 0.33 m, 0.35 m	35,000 w	0 m
Analysis 2	0.2 m	20,000 w, 25,000 w, 30,000 w, 35,000 w, 40,000 w, 45,000 w	0 m
Analysis 3	0.2 m	35,000 w	−0.2 m, −0.1 m, 0 m, 0.1 m, 0.2 m, 0.3 m, 0.4 m

.

• Analysis 1: Different distance d_a between adjacent heating lines.

The heating line arrangement of Analysis 1 is shown in Figure 13. The red dashed lines (top and bottom) in Figure 14a denote the length and width of the stable plastic zone of a single heating line before the neighboring line heating is operated. While the distance d_a is small, the adjacent heating line has a significant effect on the length and width of the stable plastic zone of the current heating line. When the distance exceeds 0.3 m, the length and width of the stable plastic zone is less influenced by the adjacent heating lines.

In Figure 14b, the red dashed lines are the longitudinal shrinkage per unit length and the transverse shrinkage of a single heating line before the neighboring line heating is operated; at this point, the two lines coincide precisely. The adjacent heating lines can apparently affect the longitudinal shrinkage per unit length and transverse shrinkage, and the longitudinal shrinkage per unit length has a sharper changing trend than the transverse shrinkage. The longitudinal shrinkage per unit length and the transverse shrinkage decrease gradually as the distance d_a increases, and converge to the value of the single heating line.

In Figure 14c, the red dashed lines (top and bottom) denote the longitudinal bending per unit length and the transverse bending of a single heating line before the neighboring line is heated. The longitudinal bending per unit length almost does not vary with differences in distance, d_a. The transverse bending is much greater than the single heating line when the distance d_a is small, decreases gradually as the distance d_a increases, and converges to the value of the single heating line.

• Analysis 2: Different heating power Q of adjacent heating lines.

The heating line arrangement of Analysis 2 is shown in Figure 15. As shown in Figure 16a, it can be observed that the heating-power variation only affects the length of the stable plastic zone, and has no influence on the width of the stable plastic zone. The length of the stable plastic zone increases as the heating power increases.

Figure 16b shows that the adjacent heating lines of different heating powers cause an increase in both the longitudinal shrinkage per unit length and the transverse shrinkage, but with different trends. When the heating powers increase, the transverse shrinkage decreases, while the longitudinal shrinkage per unit length gradually increases, and the changing speed when Q is between 20,000 w and 30,000 w is much greater than that for the other range.

Figure 16c shows that as the heating power increases, the longitudinal bending per unit length gradually increases and the transverse bending finally converges to the value of the single heating line.

• Analysis 3: Different lengths L_a between adjacent heating lines.

The heating line arrangement of Analysis 1 is shown in Figure 17. In Figure 18a, while the length L_a decreases, the length of the stable plastic zone decreases and the width of the stable plastic zone remains unchanged.

Figure 18b shows that both the longitudinal shrinkage per unit length and the transverse shrinkage decrease as the length L_a decreases; the transverse shrinkage remains stable when the length L_a exceeds 0.3 m, but is greater than the value of the single heating line.

Figure 18c shows that when the length L_a decreases, the longitudinal bending per unit length gradually increases, and is always greater than that of a single heating line. While the length L_a decreases, the transverse bending gradually decreases, from an initial value greater than that of a single heating line to a final value less than a single heating line.

Having discussed the above, some conclusions can be drawn: (1) For single line heating, the basic amount of forming plasticity can be significantly increased when there are heating lines around. (2) The distance of adjacent parallel lines has d = 0.3 m as the critical point; at more than 0.3 m, the coupling effect of two times can be ignored. (3) The longer the length of the adjacent parallel heating lines, the higher the basic amount of forming plasticity. (4) When the power increases, the longitudinal shrinkage and longitudinal bending both increase and the transverse shrinkage and transverse bending both decrease. (5) When determining the parameters of the heating lines, a suitable variation multiple should be selected in the diagram to calculate the basic amount of forming plasticity based on the relationship between adjacent heating lines. These conclusions are applied in Section 4.2, for a process-scheme design for curved plate forming.

3. The Plastic Strain Distribution Required for Different Target-Shape-Forming Processes

As previously mentioned, there are three questions that need to be taken into account, and the second question is solved in this chapter, i.e., the quantity and distribution of plastic strain required by a target shape formed from an initial plate.

3.1. Calculation Method

It is complicated to mathematically obtain the plastic strain distribution for the plate to be formed from an initial shape to the target shape. The numerical calculation method is described as follows: Firstly, an analysis model with the same initial shape and elastic-plastic properties as the actual material is built in the FEM software ABAQUS. Secondly, the vertical displacements, h, are applied on each node of the initial plate to make them move to the corresponding positions of the target plate; thus, a plastic strain field is produced with the deformation, and at the same time, a residual stress field also appears. Thirdly, remove the vertical displacements applied in the last step, the plate will spring back to an intermediate shape under the action of the residual stress. The intermediate shape is the deformation to which the initial plate can be deformed by the current plastic strain field. Generally, the intermediate shape is not consistent with the target shape, and an iteration is necessary.

In the calculation iteration, the displacement difference $\Delta h$ between the intermediate shape and the target shape is added to the vertical displacement h, which is to be applied to the initial plate and then the loop of the second and third step mentioned above, until the displacement difference $\Delta h$ is smaller than a given tolerance, which in this paper is one-thousandth of h. The calculation flow is shown in the Figure 19.

3.2. Plastic Strain Distribution Required for Forming Different Types of Curved Plates

According to the previously proposed calculation method, the initial shape of the plate is set as a flat plate and the target shape as three types: the sail-type plate, saddle-type plate, and curved plates with torsion, with a geometry size of 2 m × 1 m × 0.02 m, modulus of elasticity of the material $E=2.1\times {10}^{11}\mathrm{Pa}$, Poisson’s ratio $\mu =0.3$. The stress–strain relationship of the material is an elastic-plastic model with a yield stress of ${\sigma }_y=2.35\times {10}^8\mathrm{Pa}$ and modulus of rigidity $E_t=10\mathrm{GPa}$. Since the ratio of plate thickness d_p to plate width b_p is d_p/b_p = 0.02/1 = 1/50, which is in accordance with the assumption of a thin plate, the shell element is selected for the analysis, in which the plate-thickness direction is divided into five layers with a mesh size of 0.025 m × 0.025 m.

(1)

The quantity and distribution of plastic strain in the saddle-type plate.

Here, to calculate the plastic strain distribution for the plate according to the previous procedure, the initial shape is flat and the target shape is $z=0.1x^2-0.1y^2$. The result is shown in Figure 20.

(2)

The quantity and distribution of plastic strain in the sail-type plate

The initial shape is flat, and the target shape is $z=0.1x^2+0.1y^2$. The displacement and plastic strain are shown in Figure 21.

(3)

The quantity and distribution of plastic strain in the curved plate with torsion.

The initial shape is flat, and the target shape is $z=0.1x^2+0.1y^2+0.1xy$. The displacement and plastic strain are shown in Figure 22.

The distribution and quantity of plastic strain in the foregoing can guide the generation of path planning and be used as a choice of processing parameters.

4. Process Planning Method for Curved Ship-Plate Forming

4.1. Proposing a Matching Method for Curved Ship-Plate Forming

In the actual forming process for curved ship plate, the curvature in one direction is usually formed by a rolling machine, which has the advantage of high efficiency, but can only form a single curvature. After that, the curvature in the other direction is formed by line heating applied to the pre-rolled plate. This paper is not concerned with the parameters and path design for the rolling. To answer the third issue mentioned in Chapter 1, we give the following method (illustrated in Figure 23):

• Step 1: To choose a main governing component from the plastic strain. After the calculation of the quantity and distribution needed to form a pre-rolled plate into the target shape by line heating, using the method proposed in Chapter 3, the main governing component for the heating path planning is chosen from the plastic strain as follows:

  (1)

  The main governing component is chosen from the in-plate plastic strain of the mid-plane, which has three components PE11, PE22, and PE12 (the indexes 1 and 2 indicate the longitudinal and transverse direction), since the curvature in one direction (here assumed as the second direction), has been formed by the roller in advance. Commonly, in the three components, there is only one dominant component of which the value is considerable (for example PE11); the other two components have much smaller values and can be negligible. However, for a target shape with conspicuous torsional curvature, the two components have a remarkable value in a certain area of the mid-plane, and must be taken into consideration when arranging heating lines. The analysis for the torsional curved plated is described later in Step 2-1.

  (2)

  The dominant component of the in-plane plastic strain in the mid-plane, which plays an important role in forming the curvature, is selected as the governing factor for planning the heating path.

• Step 2: To determine the heating area and arrange the heating path for the other curvature that has not yet been formed by the roller.

  (1)

  Determination of the heating area.

Since the line heating only produces compressive plastic strain, the plate region where the main governing component is compressive is considered the heating area. The heating lines will be arranged in this area.

• (2)

  (Determination of the orientation of the heating lines.

The orientation of the heating lines is set in the same direction as the governing component. For example, when the governing component is PE11, then the heating lines are along the first direction, i.e., the longitudinal direction.

• (3)

  Determination of the separating distance between the neighboring heating lines.

The separating distance of the neighboring heating lines is set as the width of the plastic zone induced by a single heating line, which can be obtained from the database established in Chapter 2.

• (4)

  Determination of heating lines. n_l is set as:

  $$n_l=\left[\frac{\mathrm{the}\mathrm{width}\mathrm{of}\mathrm{heating}\mathrm{area}}{\mathrm{the}\mathrm{width}\mathrm{of}\mathrm{the}\mathrm{plastic}\mathrm{zone}}\right]$$

  (7)

  where [] is the operator of the Gaussian function.

• Step 2–1 (if necessary): To determine the heating area and arrange the heating path for the torsional curved plated.

  (1)

  Determination of the heating area.

An additional heating area, where the other two in-plane plastic strain components with remarkable values appear, is chosen as the heating area determined in Step 2. The heating lines for the torsional curvature will be arranged in this area.

• (2)

  Determination of the orientation of the heating lines.

In order to avoid interference with the curvature already formed, the orientation of the heating lines is perpendicular to the direction determined in Step 2.

• (3)

  Determination of the separating distance of the neighboring heating lines.

The separating distance of the neighboring heating lines is set to be equal to the width of the plastic zone induced by a single heating line, which can be obtained from the database established in Chapter 2.

• Step 3: Determination of the corresponding processing parameters for the arranged heating line.

  (1)

  Calculation of the basic amount of forming plasticity. The basic amount of forming plasticity of the plastic zone induced by each heating line is calculated through the method in Chapter 2.

  (2)

  A search is carried out in the database established in Chapter 2 to seek an appropriate combination of processing parameters, upon which the basic amount of forming plasticity provided is closest to that of (1).

• Step 4: Adjustment of the processing parameters according to the prediction of the forming effect.

  (1)

  The processing parameters are adjusted according to the expected curvature after the forming process. If the curvature of the heating line in the plastic area is less than the target curvature, the processing parameters are adjusted by priority, the heating power first and then the coil movement speed.

  (2)

  The processing parameters are adjusted according to the expected deflection after processing. If the deflection of the point in the plastic area is less than the target deflection, the processing parameters are adjusted by priority, the heating power first and then the coil movement speed.

  (3)

  Steps (1)–(2) are repeated until the expected curvature and expected deflection meet the forming accuracy requirements.

• Step 5: Finish the determination of processing parameters and path planning.

In this chapter, three types of typical shape plates, the sail-type plate, saddle-type plate, and curved plate with torsion, with a size of 2 m × 1 m × 0.02 m, are taken as examples for the design of the processing scheme.

4.2. Process Planning and Simulation Verification of Sail-Type Plate

The scheme of coil heating and forming of sail-type plates is introduced in this subsection.

4.2.1. General Scheme

The coordinate system is specified in the previous sections, with the long-edge direction (x-direction, hereafter direction 1 for plastic strain) being longitudinal and the short-edge direction (y-direction, hereafter direction 2 for plastic strain) being transverse, as shown in Figure 24, with a transverse curvature of $c_h$ and a longitudinal curvature of $c_v$. The process for forming is designed according to the method proposed in Section 4.1 as follows:

(1)

For the purpose of reducing processing time and improving efficiency, the transverse curvature is formed in advance by a roller device, i.e., initially a cylinder $z=b{y}^2$, when the target shape is $z=a{x}^2+b{y}^2$.

(2)

The processing parameters and path of the heating line are determined according to Step 1 to Step 5 proposed in Section 4.1, and then described with the following example.

4.2.2. Example 1. Target Shape $z=0.05x^2+0.1y^2$

Step 1: Calculate the quantity and distribution of plastic strain from the initial shape $z=0.1y^2$ to target shape $z=0.05x^2+0.1y^2$ using the method in Chapter 3.

Since transversal curvature $c_h$ is processed by a rolling machine, and according to plastic strain in Figure 25, the value of plastic strain in PE11 is significantly greater than PE22 and PE12, so the longitudinal (PE11), i.e., Figure 25a, is chosen as the main governing component.

Step 2: Since the line heating only produces compressive plastic strain, PE11 is used as the main governing component to determine the heating area based on the distribution of compressive strain. As shown in Figure 25a, the heating area is near the edge of the long side of the curved plate, and multiple heating areas should be carried out at x = 0.3 m/−0.3 m. Finally, as shown in Figure 26a, for the heating area is determined at the edge of the long side in the curved plate, the width is exactly the width of the plastic zone that a single heating line can provide, so only one heating line needs to be arranged, and multiple heating lines should be carried out at x = 0.3 m/−0.3 m, as shown in Figure 26b.

Step 3: The processing parameters are matched with the basic amount of forming plasticity, which is calculated for heating lines 5 and 6, i.e., longitudinal shrinkage per unit length—0.0312 m, longitudinal bending per unit length—0.0452 rad, of the database in Chapter 2. According to the “Effect of different heating positions on plastic strain generation” in Section 2.4, when the heating line is located at the edge of the long side of the plate, the processing parameters are selected according to the basic amount of forming plasticity, and the basic amount of forming plasticity should be increased by about 10% compared to the original value, so the longitudinal shrinkage per unit length is −0.03432 m, and the longitudinal bending per unit length is −0.04972 rad, and the parameters are shown in

Table 4. Heating parameters table.

Heating Lines	Initial Location (m)	End Location (m)	Moving Speed (m/s)	Heating Power (w)
1	(−0.45, 0.4)	(−0.25, 0.4)	0.01	35,000
2	(0.25, 0.4)	(0.45, 0.4)	0.01	35,000
3	(−0.45, -0.4)	(−0.25, −0.4)	0.01	35,000
4	(0.25, -0.4)	(0.45, −0.4)	0.01	35,000
5	(−0.85, 0.45)	(0.85, 0.45)	0.01	45,000
6	(−0.85, −0.45)	(0.85, −0.45)	0.01	45,000

.

According to the parameters of Figure 26b and Table 4, the simulated processing is carried out in ABAQUS.

Step 4: To verify the forming effect.

After the forming process with the heating parameters in Table 4, the curvature of the main frame line in both the processing plate and target shape is shown in

Table 5. Comparison of curvature of main frame line in both processing plate and target shape.

Curvature of Main Frame Line	Left Edge Line	Right Edge Line	Upper Edge Line	Lower Edge Line	Center Horizontal Line	Center Vertical Line
Processing plate	0.205	0.203	0.106	0.098	0.095	0.202
Target plate	0.2	0.2	0.1	0.1	0.1	0.2

; the curvature of the frame lines is hardly different to this. The deflection difference between the processing plate and the target shape at each point is calculated in CloudCompare, and the results are represented in Figure 27. The maximum deflection difference between the two shapes is 0.0034 m, which corresponds to 3.4 mm and is smaller than 4 mm, meeting the actual forming requirements.

Step 5: The processing parameters and path planning of the sail-type plate is determined.

In addition to this, since the longitudinal shrinkage per unit length and the longitudinal bending per unit length of the heating area determine the forming effect, the longitudinal shrinkage per unit length and the longitudinal bending per unit length is chosen for comparative analysis. The basic amount of forming plasticity before and after heating in line 5 is listed in

Table 6. The basic amount of forming plasticity of simulation results.

	Length of the Stable Plastic Zone	Width of the Stable Plastic Zone	Longitudinal Shrinkage per Unit Length	Longitudinal Bending per Unit Length
Processing plate	1.8	0.12	−0.0351	−0.0478
Target plate	1.8	0.125	−0.0322	−0.0462

.

According to Table 6, the difference between the size of the plastic zone, the longitudinal shrinkage, and longitudinal bending of the target shape and the processing shape is quite small. In summary, the successful processing of the plate of $z=0.05x^2+0.1y^2$.

4.3. Process Planning and Simulation Verification of Saddle-Type Plate

The scheme of coil heating and forming of saddle-type plates is introduced in this subsection.

4.3.1. General Scheme

The coordinate system is accordant as specified in the previous sections, with the long-edge direction (x-direction, hereafter direction 1 for plastic strain) being longitudinal and the short-edge direction (y-direction, hereafter direction 2 for plastic strain) being transverse, as shown in Figure 28, with a transverse curvature of $c_h$ and a longitudinal curvature of $c_v$. The procedure for forming is designed according to the proposed method in Section 4.1 as follows:

(1)

For the purpose of reducing processing time and improving efficiency, the transverse curvature is formed in advance by a roller device, i.e., initially a cylinder $z=b{y}^2$, when the target shape is $z=a{x}^2-b{y}^2$.

(2)

The processing parameters and path of the heating line is determined according to Step 1 to Step 5 proposed above, and then described with the following example.

4.3.2. Example 1. Target Shape $z=0.06x^2-0.1y^2$

Step 1: Calculate the quantity and distribution of plastic strain from the initial shape $z=-0.1y^2$ to target shape $z=0.06x^2-0.1y^2$ using the method in Chapter 3.

Since transversal curvature $c_h$ is processed by a rolling machine, and according to the plastic strain in Figure 29, the value of plastic strain in PE11 is significantly greater than PE22 and PE12, so the longitudinal component of plastic strain (PE11), i.e., Figure 25, is chosen as the main governing component.

Step 2: Since the line heating only produces compressive plastic strain, PE11 is used as the main governing component to determine the heating area based on the distribution of compressive strain. As shown in Figure 29a, the heating area is in the middle region of the plate, as shown in Figure 30a. The width of the heating area is 0.76 m, and it is calculated that six heating lines need to be arranged along the longitudinal direction, as shown in Figure 30b.

Step 3: The processing parameters are matched with the basic amount of forming plasticity, which is calculated for the heating lines 1, 2, 3, 4, 5, and 6, i.e., longitudinal shrinkage per unit length—0.0325 m, longitudinal bending per unit length—0.0378 rad; see the database in Chapter 2. According to “Influence of adjacent parallel heating lines” in Section 2.4, when a heating line exists around the heating line and the distance is less than 0.12 m, the processing parameters are selected according to the basic amount of forming plasticity, and the basic amount of forming plasticity should be decreased by about 40% compared to the original value. When the heating line exists around the heating line and the lengths of the two lines are the same, the processing parameters are selected according to the basic amount of forming plasticity, and the basic amount of forming plasticity should be decreased by about 30% compared to the original value. At this point, the reduction in the basic amount of forming plasticity is not a superposition of 40% and 50%, and it can be inferred from the relationship diagram in Section 2.4 that the reduction is approximately 70% of the original value. Thus, the longitudinal shrinkage per unit length is −0.02275 m, and the longitudinal bending per unit length is −0.02646 rad; the parameters are shown in

Table 7. Heating parameters table for the heating lines.

Heating Lines	Initial Location (m)	End Location (m)	Moving Speed (m/s)	Heating Power (w)
1	(−1.0, 0.06)	(1.0, 0.06)	0.01	40,000
2	(−1.0, −0.06)	(1.0, −0.06)	0.01	30,000
3	(1.0, 0.18)	(−1.0, 0.18)	0.01	30,000
4	(1.0, −0.18)	(−1.0, −0.18)	0.01	30,000
5	(−1.0, 0.3)	(1.0, 0.3)	0.01	30,000
6	(−1.0, −0.3)	(1.0, −0.3)	0.01	30,000

.

The simulated processing is carried out in ABAQUS according to the parameters of Figure 30b and Table 7.

Step 4: Verifying the forming effect.

After forming processing, the parameters against the curvature and deflection are adjusted until the requirements are met. The curvature of the main frame line in both the processing plate and target shape is shown in

Table 8. Comparison of curvature of main frame line from processing shape and target shape.

Curvature of Main Frame Line	Left Edge Line	Right Edge Line	Upper Edge Line	Lower Edge Line	Center Horizontal Line	Center Vertical Line
Processing plate	0.204	0.204	0.117	0.115	0.118	0.208
Target plate	0.2	0.2	0.12	0.12	0.12	0.2

. The curvature of the lines has very little difference. The deflection difference between the processing plate and the target plate at each point is calculated in CloudCompare, and the results are represented in Figure 31b. The maximum deflection difference between the two is 0.0039 m, meeting the actual forming requirements of the plate.

Step 5: The processing parameters and path planning of the saddle-type plate are determined.

In addition to this, since the longitudinal shrinkage per unit length and the longitudinal bending per unit length of the heating area determine the molding effect, the longitudinal shrinkage per unit length and the longitudinal bending per unit length are chosen for comparative analysis. The basic amount of forming plasticity before and after heating in the heating area is listed in

Table 9. The basic amount of forming plasticity of simulation results.

	Length of the Stable Plastic Zone	Width of the Stable Plastic Zone	Longitudinal Shrinkage per Unit Length	Longitudinal Bending per unit Length
Target plate	1.75	0.75	−0.0412	−0.040
Processing plate	2.0	0.75	−0.0356	−0.0378

.

According to the results in Table 9, the difference between the size of the plastic zone, the longitudinal shrinkage, longitudinal bending of the target shape and the processing shape is small. In summary, a plate of $z=0.06x^2-0.1y^2$ is successfully processed.

4.4. Process Planning and Simulation Verification of Curved Plate with Torsion

A scheme for coil heating and forming a curved plate with torsion is introduced in this subsection.

4.4.1. General Scheme

The coordinate system is as specified in the previous sections, with the long-edge direction (x-direction, hereafter direction 1 for plastic strain) being longitudinal and the short-edge direction (y-direction, hereafter direction 2 for plastic strain) being transverse, as shown in Figure 32, with a transverse curvature of $c_h$ and a longitudinal curvature of $c_v$. The procedure for forming is designed according to the method proposed in Section 4.1 as follows:

(1)

For the purpose of reducing processing time and improving efficiency, the transverse curvature is formed in advance by a roller device, i.e., the initial shape is a cylinder $z=b{y}^2$,when the target shape is $z=a{x}^2+b{y}^2+cxy$.

(2)

The processing parameters and path planning of the heating line is determined according to Step 1 to Step 5 and Step 2-1 proposed above, and described in the following.

4.4.2. Example 1. $z=0.05x^2+0.1y^2+0.05xy$

Step 1: Calculating the basic amount of forming plasticity required to from $z=0.1y^2$ to $z=0.05x^2+0.1y^2+0.05xy$ using the method in Chapter 3.

Since the transverse direction is processed by a rolling machine in its initial shape, and according to Figure 33, the value of plastic strain in PE11 is bigger than PE22, the PE12 has a remarkable value in a certain region which is shown in Figure 33, and must be taken into consideration, so the longitudinal (PE11), i.e., Figure 33a, is chosen as the main governing component. The distribution of plastic in PE12 must also be considered.

Step 2: Since the line heating only produces compressive plastic strain, PE11 is used as the main governing component to determine the heating area based on the distribution of compressive strain. As shown in Figure 33a, the heating area is near the edge of the long side of the plate, and multiple heating areas should be carried out at x = 0.3–0.9 m/−0.3–0.9 m. Finally, the heating area is shown in Figure 34a. For the heating area at the edge of the long side in the curved plate, the width is exactly the width of the plastic zone that a single heating line can provide, so only one heating line needs to be arranged, and multiple heating lines should be carried out at x = 0.3–0.9 m/–0.3 m–0.9 m, as shown in Figure 34b.

Step2-1: Since the heating plate is a curved plate with torsion, according to Figure 33c, two additional heating lines need to be arranged in the heating area, and the length is 0.4 m, as shown in Figure 35b.

Step 3: The processing parameters are matched with basic amount of forming plasticity, which is calculated for the heating lines 1, 2, 3, 4, 5, and 6 of the database in Chapter 2. Attention must be paid to the influence of the neighboring heating lines; the parameters are shown in

Table 10. Heating parameters table.

Heating Lines	Initial Location (m)	End Location (m)	Moving Speed (m/s)	Heating Power (w)
1-1	(−0.45, 0.4)	(−0.25, 0.4)	0.01	35,000
1-2	(0.25, 0.4)	(0.45, 0.4)	0.01	35,000
1-3	(−0.85, 0.45)	(0.4, 0.45)	0.01	40,000
1-4	(0.3, 0.35)	(0.85, 0.35)	0.01	40,000
1-5	(0.8, 0.1)	(0.8, 0.5)	0.01	40,000
1-6	(0.9, 0.1)	(0.9, 0.5)	0.01	40,000
2-1	(−0.45, 0.4)	(−0.25, 0.4)	0.01	35,000
2-2	(0.25, 0.4)	(0.45, 0.4)	0.01	35,000
2-3	(−0.85, 0.45)	(0.4, 0.45)	0.01	40,000
2-4	(0.3, 0.35)	(0.85, 0.35)	0.01	40,000
2-5	(0.8, 0.1)	(0.8, 0.5)	0.01	40,000
2-6	(0.9, 0.1)	(0.9, 0.5)	0.01	40,000

.

According to the parameters of Figure 35b and Table 10, the simulated processing is carried out in ABAQUS.

Step 4: Verifying the forming effect.

After form processing, adjust the parameters against the curvature and deflection until the requirements are met; the curvature of the main frame line in both the processing plate and target plate is shown in

Table 11. Comparison of curvature of main frame line from processing shape and target shape.

Curvature of Main Frame Line	Left Edge Line	Right Edge Line	Upper Edge Line	Lower Edge Line	Center Horizontal Line	Center Vertical Line
Processing plate	0.206	0.208	0.095	0.096	0.098	0.211
Target plate	0.2	0.2	0.1	0.1	0.1	0.2

, and the curvature of the lines has very little difference. The deflection difference between the processing plate and the target shape at each point is calculated in CloudCompare, and the results are represented in Figure 36. The maximum deflection difference between the two is 0.0035 m, meeting the actual forming requirements of the plate.

Based on the database established in Chapter 2 and the calculation method in Chapter 3, this chapter establishes a machining scheme for typical curvature plates of ships, and verifies the accuracy of this method by using sail-type plates, $z=0.05x^2+0.1y^2$; saddle-type plates, $z=0.06x^2-0.1y^2$; and curved plates with torsion, $z=0.05x^2+0.1y^2+0.05xy$, as examples.

5. Conclusions

In this paper, aiming at the processing-scheme design problem for the curved ship-plate forming, some study work has been carried out, and the following conclusions can be drawn:

• Taking the plastic strain as the critical governing factor of the curved ship-plate forming, the concept of the basic amount of forming plasticity is presented to connect the plastic strain and the deformation from the initial shape to the target shape, and a method for calculating the basic amount of forming plasticity is also proposed.
• For the heating coil under different processing parameters, a database of the plastic strain provided by a single heating line is established. The effect from the plate boundary and adjacent heating lines is analyzed, which can provide a helpful guide for the line-heating-forming parameter determination.
• Based on the finite element method, an algorithm for calculating the plastic strain distribution required to form the target curved plate considering the material and geometric nonlinearities is developed.
• Combining the plastic strain required from the initial shape to the target shape with the induction coil of specific heating parameters from the plastic strain database, a processing-scheme design method for curved ship-plate forming is established, and its effectiveness is verified by the case studies of three typical types of shape: sail-type plates, saddle-type plates, and curved plates with torsion. The verification results show its effectiveness for the practical forming process of different types of curved ship plates.
• Compared with other calculation methods, the processing scheme proposed in this paper has a broader range of applications, and can meet the processing of three different types of plates. The calculation efficiency is faster, and it only takes about 5 min to calculate the required plastic strain. The feasibility of the scheme is higher, and the accuracy of the scheme has been verified on laboratory apparatus.