JP7644342B2 - METHOD FOR PREDICTION OF METAL PLATE SHAPE AND METAL PLATE MANUFACTURING METHOD - Google Patents

Description

The present invention relates to a method for predicting the shape of a metal sheet after rolling, and a method for manufacturing a metal sheet using the method.

Various methods have been proposed to predict the shape of metal plates, such as thin or thick plates, after they have been rolled.

Patent Document 1 discloses a method for evaluating flatness by multiplying the crown ratio change of a rolled material (thick plate) by a shape change coefficient, thereby reducing the thermal strain difference caused by the temperature difference applied to the steel plate during rolling. In the flatness evaluation formula used here, the crown ratio change defined at a fixed point on the steel plate is multiplied by the shape change coefficient to convert it into the steepness (hereafter referred to as flatness) equivalent to residual stress, which is then used as the flatness.

Patent Document 2 discloses a shape prediction method based on the residual stress of a steel plate. In this method, first, the plate surface temperature distribution of the steel plate is measured, and based on this, the residual stress distribution in the plate longitudinal direction after air cooling is calculated, and the residual stress distribution is then approximated into a rectangle. Next, the residual stress at the buckling critical point is calculated using a predetermined simplified prediction formula from the rectangularly approximated residual stress distribution. The calculated residual stress at the buckling critical point is then compared with the rectangularly approximated residual stress to determine whether or not there is a shape defect.

Non-Patent Document 1 discloses a buckling model formulated with a triangular residual stress distribution.

Patent document 3 discloses a method for predicting the shape of a metal plate, which includes the following steps. That is, this method includes the steps of (a) determining the widthwise distribution of buckling stress using the widthwise distribution of longitudinal residual stress of the metal plate, (b) separating the stress components into those that undergo wave-shape transformation and those that remain in the metal plate after buckling when it is determined that buckling will occur, and (c) predicting the wave shape after buckling using the stress components that undergo wave-shape transformation.

Japanese Patent Application Publication No. 6-15321 Japanese Patent Application Publication No. 8-187505 Patent No. 4262142

Journal of the Japan Society for Technology of Plasticity: Plasticity and Processing, Vol. 28, No. 312 (1987-1), pp. 58-66

However, the method disclosed in Patent Document 1 does not take into consideration the shape dead zone that should exist. In other words, there is no concept of buckling, and all inherent residual stress is converted into a wave shape. Note that the shape dead zone refers to a stress region in which even if residual stress exists in the metal plate, out-of-plane deformation due to buckling does not occur up to the buckling limit.

In addition, the method disclosed in Patent Document 2 takes into account the buckling phenomenon of metal plates, but judges buckling by approximating the residual stress distribution into a rectangle as a simple formula for predicting buckling, and does not provide a model that accurately reflects the stress distribution. The same is true for the method disclosed in Non-Patent Document 1, which approximates the residual stress distribution into a triangle.

In this regard, the method disclosed in Patent Document 3 is a shape prediction method based on residual stress inherent in a metal plate, and enables buckling judgment and shape prediction at the time of buckling by inputting the residual stress distribution distributed in the width direction, thereby improving the accuracy of wave-shaped buckling judgment compared to the methods disclosed in Patent Documents 1 and 2 and Non-Patent Document 1. However, as a result of careful consideration by the inventors, it was found that there is room for improvement in prediction accuracy, as described below.

If the shape after buckling cannot be accurately predicted, it is impossible to determine whether or not to send the metal plate to the straightening line. In other words, if the prediction is not accurate, even if the shape is good, there is a possibility that the plate will be sent to the refinement process, resulting in increased costs. Therefore, it is essential to improve the accuracy of predicting the shape of metal plates.

The present invention has been made in consideration of these points, and aims to provide a method for predicting the shape of a metal plate that is highly accurate and can accurately perform the separation of the metal plate required for the refining process in advance, and a method for manufacturing a metal plate using this method.

To achieve the above objective, the inventors investigated methods for predicting the shape of metal plates, and came to the following findings.

As disclosed in Patent Document 3, it is known that the residual stress (elongation strain) of a metal plate is separated into a component that converts into a wave shape and a component that remains in the metal plate even after buckling. The invention of Patent Document 3 uses this component that converts into a wave shape to predict the shape of the metal plate after buckling.

The present invention is a further development of the invention of Patent Document 3. After intensive research, the inventors found that in the invention of Patent Document 3, an error occurs in the shape profile because it is assumed that the shape profile calculated by the buckling eigenvalue analysis remains the same even after buckling. The inventors then came up with the idea that the shape of the metal plate can be appropriately predicted by recalculating (eigenvalue analysis calculation using a theoretical buckling model) with the elastic strain difference reallocated due to the occurrence of the shape profile that occurs after buckling, rather than calculating with the plastic elongation strain difference of the evaluation target in the buckling eigenvalue analysis. That is, it was previously thought that the disclosed shape profile after buckling was determined by the plastic elongation strain difference used in the initial calculation, but as a result of intensive research by the inventors, it was found that the shape profile is determined by the elastic strain difference that is reallocated to the metal plate after the buckling shape occurs, that is, the elastic strain difference that exists in the plate even after buckling. The present invention is based on the above findings, and the gist of the present invention is as follows.

The present invention is a method for predicting the shape of a metal plate, and is characterized in that when predicting the post-buckling wave shape based on the difference in plastic elongation strain imparted to the metal plate, the wave shape profile is determined by buckling analysis using the difference in elastic strain that remains in the metal plate even after buckling.

The following three algorithms are specific for implementing the metal plate shape prediction method.

The first metal plate shape prediction method (algorithm) includes a first step of setting a plastic elongation-strain difference , which is the difference in the width direction of the strain that elongates in the longitudinal direction of the metal plate during rolling, when the rolled metal plate buckles ; a second step of converting the plastic elongation-strain difference into a standardized elongation-strain difference in the first calculation of the repeated calculation, and using the standardized elongation-strain difference calculated in the tenth step described below in the second and subsequent calculations of the repeated calculation; and a second step of performing a buckling analysis based on the standardized elongation-strain difference, and calculating the buckling inherent strain difference, the wave pitch, and the standardized shape of the metal plate, which are criteria for determining whether the metal plate buckles. a third step of calculating a profile; a fourth step of determining whether the calculation is the first calculation of the repeated calculations or the second or subsequent calculations; a fifth step of comparing the plastic elongation strain difference with the buckling inherent strain difference to determine whether buckling has occurred if the calculation is the first calculation of the repeated calculations in the fourth step; and a fifth step of comparing the plastic elongation strain difference with the buckling inherent strain difference to determine whether buckling has occurred if the calculation is the second or subsequent calculation of the repeated calculations in the fourth step, or if the plastic elongation strain difference is greater than the buckling inherent strain difference and buckling has occurred in the fifth step. a sixth step of separating the difference between the plastic elongation strain difference and the buckling shape transformation elongation strain difference, which is a component that is converted into the shape of the metal plate and is a difference between the difference between the plastic elongation strain difference and the buckling shape transformation elongation strain difference, calculating an elongation strain error, which is the difference between the geometric shape elongation strain difference that appears in the shape of the metal plate after out-of-plane deformation and the buckling shape transformation elongation strain difference, and determining a wave height at which an integrated value of the square of the elongation strain error is a minimum value to determine a wave shape profile; a seventh step of adding up the elastic buckling inherent strain difference and the elongation strain error to calculate an elastic strain difference inherent in the metal plate after buckling; and a seventh step of calculating the elastic strain difference by using the elastic strain difference calculated in the seventh step. a ninth step of calculating an error between the standardized elongation-strain difference set in the second step and the standardized elongation-strain difference calculated in the eighth step and comparing the square of the error with a predetermined threshold; and a tenth step of correcting the standardized elongation-strain difference set in the second step by a value obtained by multiplying the error by a relaxation coefficient if the square of the error in the ninth step is greater than the predetermined threshold, and wherein the second step to the tenth step are repeatedly performed until the square of the error in the ninth step becomes equal to or less than the predetermined threshold.

The second metal plate shape prediction method (algorithm) includes a first step of setting a plastic elongation and strain difference , which is the difference in the width direction of the strain that elongates in the longitudinal direction of the metal plate during rolling when the rolled metal plate buckles; a second step of setting the number of repeated calculations to M ; a third step of converting the plastic elongation and strain difference into a standardized standardized elongation and strain difference in the first calculation of the repeated calculations, and using the standardized elongation and strain difference corrected in the tenth step described below in the second and subsequent calculations of the repeated calculations; and a third step of performing a buckling analysis based on the standardized elongation and strain difference, and calculating a buckling inherent strain difference, which is a criterion for determining whether the metal plate buckles, a wave pitch, and a shape prediction method for the metal plate. a fourth step of calculating a standardized shape profile of the plastic elongation strain difference; a fifth step of determining whether the calculation is the first calculation of the repeated calculations or the second or subsequent calculations; a sixth step of comparing the plastic elongation strain difference with the buckling inherent strain difference to determine whether buckling has occurred when it is determined in the fifth step that the calculation is the first calculation of the repeated calculations; and a sixth step of comparing the plastic elongation strain difference with the buckling inherent strain difference to determine whether buckling has occurred when it is determined in the sixth step that the plastic elongation strain difference is greater than the buckling inherent strain difference and that buckling has occurred, a seventh step of dividing the buckling shape transformation elongation strain difference into M-1 divisions to calculate an M-1 division buckling shape transformation elongation strain difference, and replacing the buckling inherent strain difference of the first iteration of the iterative calculation with an initial buckling inherent strain difference; and if it is determined in the fifth step that the iterative calculation is the second or subsequent iterative calculation, or if it is determined in the sixth step that the plastic elongation strain difference is greater than the buckling inherent strain difference and buckling will occur, correcting the buckling shape transformation elongation strain difference using the M-1 division buckling shape transformation elongation strain difference and the initial buckling inherent strain difference to obtain a geometric shape elongation strain difference that appears in the shape of the metal plate after the out-of-plane deformation and a geometric shape elongation strain difference that appears in the shape of the metal plate after the out-of-plane deformation and a geometric shape elongation strain difference that appears in the shape of the metal plate after the out-of-plane deformation. The method includes an eighth step of calculating an elongation-strain error, which is the difference between the corrected buckling shape transformation elongation-strain difference, and determining a wave height at which the integrated value of the square of the elongation-strain error is the minimum value to determine a wave shape profile; a ninth step of replacing the buckling inherent strain difference with an elastic strain distribution and adding the elongation-strain error to calculate an elastic strain difference inherent in the metal plate after buckling; and a tenth step of calculating a standardized elongation-strain difference corresponding to the elastic strain difference using the elastic strain difference calculated in the ninth step, and is characterized in that the second step to the tenth step are performed the number of times set in the second step.

The third metal plate shape prediction method (algorithm) includes a first step of setting a plastic elongation and strain difference, which is the difference in the width direction of the strain that elongates in the longitudinal direction of the metal plate during rolling when the rolled metal plate buckles , a second step of setting the number of repeated calculations to M , a third step of dividing the plastic elongation and strain difference into M and calculating a plastic elongation and strain difference according to the number of repeated calculations, and a third step of converting the plastic elongation and strain difference into a standardized standardized elongation and strain difference if the plastic elongation and strain difference is equal to or less than the buckling inherent strain difference in the first calculation of the repeated calculations or in the seventh step described below, and a fourth step of using the normalized elongation and strain difference calculated in the eleventh step below when it is determined that buckling will occur; a fifth step of performing a buckling analysis based on the normalized elongation and strain difference to calculate a buckling inherent strain difference, a wave pitch, and a normalized shape profile of the metal plate, which are criteria for determining whether buckling will occur in the metal plate; a sixth step of determining whether buckling has occurred in the seventh step below, which is one step before the repeated calculation; a seventh step of comparing the plastic elongation and strain difference with the buckling inherent strain difference to determine whether buckling has occurred in the seventh step; When it is determined that the difference in elongation and strain is equal to or less than the buckling inherent strain difference and buckling will not occur, the number of repeated calculations is checked, and if the number is not the final number, the process returns to the second step in an eighth step. When it is determined that buckling occurs in the sixth step, or when it is determined that the difference in the plastic elongation and strain is greater than the buckling inherent strain difference and buckling will occur in the seventh step, the plastic elongation and strain difference is separated into the buckling inherent strain difference and a buckling shape transformation elongation and strain difference, which is a component that transforms the shape of the metal plate and is a difference between the plastic elongation and strain difference and the buckling inherent strain difference, and a geometric shape elongation and strain difference that appears in the shape of the metal plate after out-of-plane deformation, The method includes a ninth step of calculating an elongation-strain error, which is the difference with the buckling shape transformation elongation-strain difference, and determining a wave height at which the integrated value of the square of the elongation-strain error is the minimum value to determine a wave shape profile; a tenth step of replacing the buckling inherent strain difference with an elastic strain distribution and adding up the elongation-strain error to calculate an elastic strain difference inherent in the metal plate after buckling; and an eleventh step of using the elastic strain difference calculated in the tenth step to calculate a normalized elongation-strain difference corresponding to the elastic strain difference, and is characterized in that the second step to the eleventh step are performed the number of times set in the second step.

In the method for predicting a shape of a metal plate, the width direction distribution of the longitudinal residual stress of the metal plate calculated by converting the buckling inherent strain difference may be regarded as a thermal stress distribution based on the width direction temperature distribution before and after cooling.

In the method for predicting a shape of a metal plate, the width direction distribution of the longitudinal residual stress of the metal plate calculated by converting the buckling inherent strain difference may be set as the residual stress distribution imparted at least during rolling or straightening.

In the method for predicting the shape of a metal plate, the shape of the metal plate before cooling or the shape of the metal plate after cooling may be measured, and the geometric shape elongation strain difference appearing in the shape of the metal plate after out-of-plane deformation may be superimposed on the plastic elongation strain difference.

According to another aspect of the present invention, a flat plate is manufactured by using the metal plate shape prediction method to determine whether or not to perform shape correction in a finishing process, making the width-direction distribution of the longitudinal residual stress of the metal plate a thermal stress distribution based on the width-direction temperature distribution before and after cooling, and controlling the width-direction temperature distribution so that the metal plate does not buckle, based on the shape of the metal plate predicted by the shape prediction method.

In the manufacturing method of the metal plate, the predicted steepness may be used to determine whether or not to perform shape correction in the refinement process.

According to the present invention, the elastic strain difference that is redistributed due to the shape deformation after buckling is calculated, and recalculation is performed using this elastic strain difference (eigenvalue analysis calculation using a theoretical buckling model), so the shape of the metal plate can be predicted with high accuracy. As a result, the separation of the metal plate required for the refining process can be performed with high accuracy. In addition, by using the results of this metal plate shape prediction, it is also possible to manufacture the metal plate flat.

FIG. 1 is a diagram illustrating a target wave shape. 1 is a flowchart illustrating a conventional shape prediction method. 1 is an explanatory diagram showing a shape prediction result of a metal plate when a conventional shape prediction method and FEM are used; 1A and 1B are explanatory diagrams showing the shape of a metal plate when a conventional shape prediction method and FEM are used; 4 is a flowchart illustrating a feature prediction method according to the first embodiment. FIG. 2 is an explanatory diagram illustrating a feature prediction method according to the first embodiment; FIG. 4 is an explanatory diagram showing a shape prediction result of a metal plate when the shape prediction method according to the first embodiment is used; FIG. 4 is an explanatory diagram showing a shape prediction result of a metal plate when the shape prediction method according to the first embodiment is used; 10 is a flowchart illustrating a feature prediction method according to a second embodiment. FIG. 11 is an explanatory diagram showing a shape prediction result of a metal plate when the shape prediction method according to the second embodiment is used. FIG. 11 is an explanatory diagram showing a shape prediction result of a metal plate when the shape prediction method according to the second embodiment is used. 13 is a flowchart illustrating a feature prediction method according to a third embodiment.

Embodiments of the present invention will be described below with reference to the drawings. In this specification and the drawings, elements having substantially the same functional configuration are designated by the same reference numerals to avoid redundant description.

First, the wave shape to be considered will be explained using FIG. 1. The X-axis is the longitudinal direction in which periodic waves occur, the Y-axis is the plate width direction, and the Z-axis is the plate thickness or wave height direction. The wave shown in FIG. 1 shows a typical ear wave. Note that periodic buckling waves (X direction in the figure) such as medium waves and quarter waves occurring at an intermediate position between ear waves and medium waves are also included. The definition of steepness or flatness used as the size of the wave is the wave piece amplitude height H of the width direction edge of the metal plate divided by the wave pitch P (period P) and multiplied by 100 to express it as a percentage. In addition, this wave shape is called a width direction position shape profile (wave shape profile) cut out from the Y-Z cross section at the position indicated by the wave piece amplitude height H, and the shape profile obtained in the buckling analysis described later is a standardized profile of 0 to 1 without a height dimension. In this embodiment, the wave shape profile distribution h(x, y) = H x W(y) x Sin(2πx/P) is predicted by multiplying the normalized profile by the wave piece amplitude height H.

<Conventional shape prediction method>
As described above, the present invention is a further development of the invention disclosed in Patent Document 3. Therefore, before describing the feature prediction method according to this embodiment, a description will be given of the conventional feature prediction method disclosed in Patent Document 3. Fig. 2 is a flowchart showing the conventional feature prediction method.

(Step X1)
In step X1, a plastic elongation strain difference Δε _pl (y) of the metal plate to be evaluated at an arbitrary position y divided into N parts by a predetermined width in the plate width direction (y direction) is set. The plastic elongation strain difference Δε _pl (y) is the difference in the width direction of the strain (hereinafter referred to as "elongation strain") that elongates in the longitudinal direction of the metal plate during rolling when the rolled metal plate buckles (when out-of-plane deformation occurs in the steel plate). In the following description, the definitions of elongation strain and elongation strain difference are the same.

(Step X2)
In step X2, the plastic elongation strain difference Δε _pl (y) is converted into an elongation strain difference Δε _normal (y) normalized to a value between 0 (zero) and 1, with 1 being the maximum value.

(Step X3)
In step X3, an eigenvalue analysis calculation is performed using a theoretical buckling model. This theoretical buckling model is a model disclosed in Patent Document 3. That is, the theoretical buckling model is a model that performs buckling analysis using a wave-shaped buckling equation created based on a model formulated with a triangular residual stress distribution shown in Non-Patent Document 1. In the theoretical buckling model, if the normalized elongation strain difference Δε _normal (y), the plate thickness t of the metal plate, the plate width B of the metal plate, and the tension Ut acting on the metal plate are input, the buckling inherent strain difference Δε _cr (y) (criterion for buckling occurrence) similar to the plastic elongation strain difference Δε _pl (y), the wave pitch P generated by buckling, and the height profile W (y) normalized to 0 to 1 in the width direction cross section at the time of buckling are output. Note that in the model disclosed in Patent Document 3, the residual stress distribution is input to obtain the buckling stress distribution, but here, elongation strain is used instead of stress.

Here, the longitudinal stress acting on a metal plate is essentially a residual stress component that becomes 0 (zero) when integrated in the plate width direction and a tension (unit tension) component overlap. However, in the buckling model, the tension component is given separately, so here, it is assumed that it becomes 0 when integrated in the plate width direction, that is, the residual stress component becomes 0 when the integrated value in the plate width direction or the average in the plate width direction is taken, and the distribution of the residual stress σ _res (y) and the distribution of the plastic elongation strain difference Δε _pl (y) are converted by the following formula (1).
Δε _pl (y)=-(σ _res (y)/E-Max(σ _res (y))/E) ...(1)

(Step X4)
In step X4, the occurrence of buckling is judged. The plastic elongation strain difference Δε _pl (y) is compared with the buckling inherent strain difference Δε _cr (y). If Δε _pl (y) is greater than Δε _cr (y), it is judged that the metal plate will buckle, and the process proceeds to step X5. Incidentally, the buckling inherent strain difference Δε _cr (y) has a similar relationship to the plastic elongation strain difference Δε _pl (y), so it may be judged at any position in the width direction y, but here the comparison is made at their respective maximum values. On the other hand, if Δε _pl (y) is equal to or less than Δε _cr (y), it is judged that the metal plate will not buckle and is flat, and the process proceeds to step X6.

(Step X5)
In step X5, the shape of the metal sheet is predicted using a shape prediction model. In the shape prediction model, the plastic elongation strain difference Δε _pl (y) is separated into a component that is converted into a shape deformation after buckling (buckling shape transformation elongation strain difference Δε _ts (y)) and a component that remains in the metal sheet even after buckling (buckling inherent strain difference Δε _cr (y)), and then, assuming that the wave pitch P calculated in step X3 based on the buckling shape transformation elongation strain difference Δε _ts (y) does not change, the wave height distribution h(x,y)=H×W(y)×Sin(2πx/P) obtained by multiplying the standardized height profile W(y) by the wave piece amplitude height H is determined so that the error with the geometric elongation strain difference distribution is small, thereby predicting the shape of the metal sheet. This shape prediction model is the same as the model disclosed in Patent Document 3. However, while the model disclosed in Patent Document 3 predicts the shape using stress, here, elongation strain difference is used instead of stress.

As described above, the shape of the metal plate (wave piece amplitude height H) is output in step X5, or the fact that the metal plate is flat is output in step X6, and shape prediction ends.

<Verification of conventional shape prediction methods>
In the conventional method, the shape (wave height width distribution h(y)) of the metal sheet is predicted as described above. The inventors verified the shape predicted by this conventional shape prediction method by comparing it with the analysis result by the finite element method (FEM).

Figures 3 and 4 are explanatory diagrams showing the shape of a metal sheet when using a conventional shape prediction method and FEM under the same metal sheet conditions and the same plastic elongation strain difference. Note that Figure 3 shows the plate thickness direction displacement in the longitudinal direction displacement of the plate width center part, i.e., the wave height direction displacement, and Figure 4 shows an enlarged view of the wave height profile of cross section A in Figure 3.

Referring to Figure 3, the wave pitch after buckling is almost the same when comparing the conventional shape prediction method with FEM. On the other hand, referring to Figure 4, the wave height displacement in the width direction center part is almost the same when comparing the conventional shape prediction method with FEM, but the wave height displacement in the range surrounded by the dotted line in Figure 4 is different, which shows that an error occurs in the shape profile.

The inventors conducted extensive research and found that the cause of the error in the shape profile is that in the conventional shape prediction method, the shape profile calculated in the buckling eigenvalue analysis in step X3 is assumed to remain the same even after buckling. The inventors then came up with the idea that the shape of the metal plate can be appropriately predicted by recalculating (eigenvalue analysis calculation using a theoretical buckling model) the elastic strain difference that exists in the metal plate as residual stress even after buckling, which is redistributed to the metal plate due to the generation of buckling waves after buckling, rather than calculating using the plastic elongation strain difference of the evaluation target in the buckling eigenvalue analysis. In the following, the shape prediction method for the metal plate according to this embodiment will be described based on the above findings.

<Feature Prediction Method According to First Embodiment>
A shape prediction method for a metal plate according to a first embodiment will be described. Fig. 5 is a flowchart showing the shape prediction method according to the first embodiment. Fig. 6 is an explanatory diagram showing a schematic diagram of the shape prediction method according to the first embodiment. In the first embodiment, steps A2 to A10 described later are repeated to perform convergence calculation.

(Step A1)
In step A1, the plastic elongation strain difference Δε _pl (y) of the metal plate to be evaluated is set as shown in Fig. 6(a) This step A1 is similar to the above-mentioned step X1.

(Step A2)
In step A2, in the first calculation of the iterative calculation, the plastic elongation-strain difference Δε _pl (y) is converted into a normalized elongation-strain difference Δε _normal (y) of normalization between 0 and 1. In the second and subsequent calculations of the iterative calculation, the normalized elongation-strain difference Δε _normal (y) corrected in step A10 described later is used as the normalized elongation-strain difference in the next and subsequent steps. This step A2 is the same as the above step X2 when there is no iterative calculation.

(Step A3)
In step A3, an eigenvalue analysis calculation is performed using a theoretical buckling model. This step A3 is the same as step X3, and if the normalized elongation strain difference Δε _normal (y), the thickness t of the metal plate, the width B of the metal plate, and the tension Ut acting on the metal plate are input, the buckling inherent strain difference Δε _cr (y) (criterion for buckling occurrence), which is similar to the plastic elongation strain difference Δε _pl (y), the wave pitch P, and the normalized height profile (shape profile) W (y) of 0 to 1 in the width direction cross section at the time of buckling are output. Note that, in the case of the second or subsequent calculation of the repeated calculation, the buckling inherent strain difference Δε _cr (y) is corrected by the elongation strain error Δε _er (y) in step A7 described later, so it is not similar to the plastic elongation strain difference Δε _pl (y).

(Step A4)
In step A4, it is determined whether the calculation is the first calculation of the repeated calculations or the second or subsequent calculations. If it is the first calculation, the process proceeds to step A5, which will be described later, and if it is the second or subsequent calculations, the process proceeds to step A6, which will be described later. In step A5, it is determined whether buckling has occurred, but in the case of the second or subsequent calculations, since it has been determined that the metal plate will buckle, step A5 can be omitted.

(Step A5)
In step A5, the occurrence of buckling is judged. This step A5 is the same as step X4, and the plastic elongation strain difference Δε _pl (y) is compared with the buckling inherent strain difference Δε _cr (y). If Δε _pl (y) is greater than Δε _cr (y), it is judged that the metal plate buckles, and the process proceeds to step A6. Incidentally, as in the conventional method, the buckling inherent strain difference Δε _cr (y) has a similar relationship with the plastic elongation strain difference Δε _pl (y), so it may be judged at any position in the width direction y, but here, the maximum values of each are compared. On the other hand, if Δε _pl (y) is equal to or less than Δε _cr (y), it is judged that the metal plate does not buckle and is flat, and the process proceeds to step A11.

(Step A11)
In step A11, as in step X6, it is output that the metal plate is flat, and the shape prediction ends. When the metal plate is flat, the plastic elongation strain difference Δε _pl (y) remains in the metal plate as an elastic strain difference Δε _el (y), that is, the elastic strain difference Δε _el (y) existing in the metal plate without buckling is calculated by the following formula (2). Furthermore, the residual stress σ _res (y) existing in the metal plate is also calculated by multiplying the elastic strain difference Δε _el (y) calculated by the following formula (3) by the Young's modulus E. N represents the number of nodes equally divided in the width direction. Hereinafter, N will be referred to as the number of nodes.
Δε _el (y)=-(Δε _pl (y)-(ΣΔε _pl (y))/N) ...(2)
σ _res (y)=E×Δε _el (y) ...(3)

As described above, the longitudinal stress acting on a metal plate essentially overlaps the residual stress component and tension (unit tension) component, which are zero when integrated in the plate width direction. However, since the tension component is given separately in the buckling model, it is assumed here that integration in the plate width direction is zero, that is, the residual stress component is the integral value in the plate width direction or the average in the plate width direction is zero, and the residual stress distribution and the distribution of the plastic elongation strain difference Δε _pl (y) are converted using the above formula (1).

(Step A6)
In step A6, the shape of the metal plate is predicted using a shape prediction model. This step A6 is the same as step X5. First, as shown in FIG. 6B, the plastic elongation strain difference Δε _pl (y) is separated into a component that is converted into a shape deformation after buckling (buckling shape transformation elongation strain difference Δε _ts (y)) and a component that remains in the metal plate even after buckling (buckling inherent strain difference Δε _cr (y)). Then, the buckling inherent strain difference Δε _cr (y) is subtracted from the plastic elongation strain difference Δε _pl (y) to calculate the buckling shape transformation elongation strain difference Δε _ts (y) using the following formula (4).
Δε _ts (y)=Δε _pl (y)−Δε _cr (y) ...(4)

Next, assuming that the wave pitch P calculated in step A3 does not change with respect to the buckling shape transformation elongation strain difference Δε _ts (y) calculated from the above formula (4), the wave height width distribution h (x, y) = H × W (y) × Sin (2πx / P) obtained by multiplying the standardized height profile W (y) by the wave piece amplitude height H is predicted as the shape of the metal sheet after buckling. Here, the steepness λ (y) of the waveform after buckling is expressed by the wave piece amplitude height H, the standardized height profile W (y), and the wave pitch P for each width direction position y, that is, it is expressed by the following formula (5) using the geometric shape elongation strain difference Δε _gs (y) that appears in the shape after out-of-plane deformation. However, in general, when expressing the size of the wave in rolling operation, it is often expressed as (5') as a double amplitude of the maximum amplitude H of the half amplitude wave. In addition, this formula (5) is rearranged by the geometric shape elongation strain difference Δε _gs (y) to obtain formula (6).
λ(y)=2×H×W(y)/P=(2/π)√Δε _gs (y) ...(5)
λ=2×H/P ・・・・・・(5')
Δε _gs (P, H, W(y)) = (π/2×W(y)×2×H/P) ² ...(6)

Since the standardized height profile W(y) is a profile standardized as a value between 0 and 1, the geometric shape elongation strain difference Δε _gs (P, H, W(y)) can be calculated by assuming the maximum amplitude H of the single amplitude wave. If it is considered to be substantially the same as the buckling shape transformation elongation strain difference Δε _ts (y), the elongation strain difference Δε _er (y) (hereinafter referred to as "elongation strain error Δε _er (y)") that is not transformed into a wave shape as shown in FIG. 6(c) is expressed by the following formula (7). Then, for example, by using the least squares method, the wave piece amplitude H of the single amplitude wave at which the integrated value of the square of the elongation strain error Δε _er (y) expressed by the following formula (8) is the minimum value is determined, and h(x, y) = H × W(y) × Sin (2πx/P) is predicted as the shape of the metal plate after buckling.
Δε _er (y)=Δε _gs (P, H, W(y))−Δε _ts (y) ...(7)
Min[Σ(Δε _er (y)) ² ] ...(8)

(Step A7)
In step A7, the residual stress redistributed after the occurrence of the buckling wave, that is, the elastic strain difference Δε _el ^* (y) inherent in the metal plate is calculated. Specifically, as shown in FIG. 6(d), the elastic buckling inherent strain difference (Δε _{el_cr} (y)) and the elastic elongation strain error (Δε _er (y)) are added together to calculate the elastic strain difference Δε _el ^* (y) inherent in the metal plate redistributed by the occurrence of the buckling wave, according to the following formulas (9) and (9'). In this way, in step A7, the elastic strain difference Δε _el ^* (y) redistributed by the shape deformation after buckling is calculated. In addition, according to the following formula (10), it is possible to calculate the redistributed residual stress σ _res ^* (y) inherent in the metal plate by multiplying the elastic strain difference Δε _el ^* (y) by the Young's modulus E.
Δε _{el_cr} (y)=-(Δε _cr (y)-(ΣΔε _cr (y))/N) ...(9')
Δε _el ^* (y) = - {Δε _pl (y) - Δε _gs (P, H, W(y))} = Δε _{el_cr} (y) + Δε _er (y) ... (9)
σ _res ^* (y) = E×Δε _el ^* (y) ... (10)

(Step A8)
In step A8, the normalized elongation-strain difference Δε _{normal *(y) is determined again based on the residual stress distribution of the redistributed metal plate after buckling. Specifically, in the convergence calculation in which steps A2 to A10 are repeatedly performed as described below, the normalized elongation-strain difference Δε normal} ^* (y) having a corresponding value of 0 to 1 is determined using the elastic strain difference Δε _el ^* (y) inherent in the redistributed metal plate according to the following formula (11). Fig. ⁶ (e) shows an example of this normalized elongation-strain difference _{Δε normal *} ⁽ y).
Δε _normal ^* (y)=-(Δε _el ^* (y)-Max(ε _el ^* (y)))/(Max(ε _el ^* (y))-Min(ε _el ^* (y)))...(11)

(Step A9)
In step A9, whether or not a solution has been found in the convergence calculation is judged by whether or not the value of the normalized elongation-strain difference Δε _normal ^* (y) after buckling and redistribution is equal to the value of the normalized elongation-strain difference Δε _normal (y) before redistribution. First, the error Δerror (y) of the normalized elongation-strain difference before and after redistribution is calculated for each width direction position y according to the following formula (12). That is, the error Δerror (y) is the amount of change in the normalized elongation-strain difference Δε _normal (y). Next, as shown in the following formula (13), the square error for each width direction position y is integrated to calculate Δerror ^* , and this error Δerror ^* is compared with a predetermined threshold value. If the error Δerror ^* is greater than a predetermined threshold, it is determined that the reallocated normalized elongation-strain difference Δε _normal ^* (y) has not converged, and the process proceeds to step A10 to perform recalculation (eigenvalue analysis calculation using a theoretical buckling model). On the other hand, if the error Δerror ^* is equal to or smaller than a predetermined threshold, it is determined that the normalized elongation-strain difference Δε _normal ^* (y) has converged, and the process proceeds to step A12.
Δerror(y)=Δε _normal ^* (y)−Δε _normal (y)...(12)
Δerror ^* = {Σ(Δerror(y)) ² }>Threshold value (13)

(Step A10)
In step A10, the normalized elongation-strain difference Δε _normal (y) used in the recalculation is corrected by using Δε _normal ^* (y) and Δerror (y) by returning to step A2. Specifically, the error Δerror (y) is multiplied by the relaxation coefficient α according to the following formula (14), and the result is added to the normalized elongation-strain difference Δε _normal (y) after reallocation, to reset the normalized elongation-strain difference Δε _normal (y) for recalculation. Here, when performing recalculation, if all the errors Δerror (y) are superimposed without multiplying by the relaxation coefficient α, the calculation result will vibrate, and the normalized elongation-strain difference Δε _normal (y) will not converge easily. Therefore, in this embodiment, the normalized elongation-strain difference Δε _normal (y) is made to converge easily by multiplying by the relaxation coefficient α. The relaxation coefficient α is arbitrary, but after extensive investigation, the inventors have confirmed that a value of 0.1 to 0.4 is appropriate.
Δε _normal (y)=Δε _normal (y)+α×Δerror(y)=Δε _normal (y)+α×{Δε _normal ^* (y)−Δε _normal (y)} ...(14)

Next, returning to step A2, the normalized elongation-strain difference Δε _normal (y) corrected in step A10 is used to carry out the subsequent steps A3 to A10. Then, steps A2 to A10 are repeated, and the convergence calculation is performed until it is determined in the convergence determination in step A9 that the normalized elongation-strain difference Δε _normal ^* (y) has converged.

(Step A12)
When it is determined in step A9 that the normalized elongation-strain difference Δε _normal ^* (y) has converged, the process proceeds to step A12, where the shape prediction is terminated. Then, the wave pitch P and the wave height h(x,y)=H×W(y)×Sin(2πx/P) at this time are predicted as the shape of the metal sheet.

When the normalized elongation strain difference Δε _normal ^* (y) converges in this manner, it remains in the metal plate as an elastic strain difference Δε _el ^* (y) inherent in the plate even after buckling, and by multiplying this by the Young's modulus E, the residual stress σ _res inherent in the metal plate can also be calculated using the above-mentioned equations (9) and (10).

As mentioned above, the conventional shape prediction method assumes that the shape profile is maintained even after buckling. However, in reality, the effect of the shape profile changing due to the residual stress distribution that is redistributed after buckling is not taken into account, resulting in an error in the shape profile as shown in Figure 4.

In this regard, the inventors have found that the shape profile changes due to the shape deformation after buckling. According to the shape prediction method of the first embodiment described above, the elastic strain difference Δε _{e l} ^* (y) reallocated due to the shape deformation after buckling is calculated, and recalculation is performed using the elastic strain difference Δε _{e l} ^* (y), so that the shape of the metal plate can be appropriately predicted.

To verify the effect of the first embodiment, the inventors compared the shape prediction results (converged solutions) obtained by performing the first embodiment with the shape prediction results obtained by a conventional shape prediction method and the analysis results obtained by FEM. The verification results are shown in Figs. 7 and 8. Fig. 7 shows the width-wise distribution of the elastic strain difference present in the metal plate after buckling, and Fig. 8 shows the width-wise distribution of the normalized profile.

Referring to Figures 7 and 8, the conventional shape prediction results are slightly different from the analysis results by FEM, whereas the shape prediction results of the first embodiment are almost the same as the analysis results by FEM. Therefore, it was found that the shape of the metal plate can be appropriately predicted according to the first embodiment.

<Feature Prediction Method According to Second Embodiment>
A feature prediction method according to the second embodiment will now be described with reference to a flowchart shown in FIG.

The shape prediction method of the second embodiment shares with the shape prediction method of the first embodiment the technical feature of performing recalculation using the elastic strain difference reallocated due to the occurrence of a shape profile that occurs after buckling. However, whereas the first embodiment performs convergent calculations, the second embodiment repeats steps B2 to B10, which will be described later, to perform sequential calculations.

(Step B1)
In step B1, the plastic elongation strain difference Δε _pl (y) of the metal plate to be evaluated is set. This step B1 is similar to the above step A1.

(Step B2)
In step B2, the number of iterative calculations (sequential calculations) is set, and it is determined which iteration of the calculation currently being performed is. In this embodiment, the number of iterative calculations is M. In the following description, the calculation currently being performed is represented as the i-th calculation.

(Step B3)
In step B3, in the first iterative calculation (i=1), the plastic elongation-strain difference Δε _pl (y) is converted to a normalized elongation-strain difference Δε _normal (y) of normalization between 0 and 1. In the second and subsequent iterative calculations (i=2 and subsequent), the normalized elongation-strain difference Δε _normal ^* (y) corrected in step B10 described later is replaced with Δε _normal (y) and used as the normalized elongation-strain difference in the subsequent steps.

(Step B4)
In step B4, an eigenvalue analysis calculation is performed using a theoretical buckling model to calculate the buckling inherent strain difference Δε _cr (y), the wave pitch P, and the normalized height profile (shape profile) W(y) of 0-1. This step B4 is similar to the above step A3, and by inputting the normalized elongation strain difference Δε _normal (y), the sheet thickness t of the metal sheet, the sheet width B of the metal sheet, and the tension Ut acting on the metal sheet, the buckling inherent strain difference Δε _cr (y) (criterion for the occurrence of buckling) which is similar to the plastic elongation strain difference Δε _pl (y), the wave pitch P, and the normalized height profile (shape profile) W(y) of 0 to 1 in the width direction cross section at the time of buckling are output.

(Step B5)
In step B5, it is determined whether the calculation is the first iteration (i=1) or the second or subsequent iteration (i=2 or subsequent iteration). If it is the first iteration, the process proceeds to step B6, which will be described later, and if it is the second or subsequent iteration, the process proceeds to step B8, which will be described later. In step B6, it is determined whether buckling has occurred, but in the case of the second or subsequent iteration, since it has been determined that the metal plate will buckle, steps B6 and B7 can be omitted.

(Step B6)
In step B6, the occurrence of buckling is judged. This step B6 is the same as step A5, and the plastic elongation strain difference Δε _pl (y) is compared with the buckling inherent strain difference Δε _cr (y). If Δε _pl (y) is greater than Δε _cr (y), it is judged that the metal plate will buckle, and the process proceeds to step B7. On the other hand, if Δε _pl (y) is equal to or less than Δε _cr (y), it is judged that the metal plate will not buckle and is flat, and the process proceeds to step B12.

(Step B12)
In step B12, the fact that the metal plate is flat is output, and the shape prediction ends. The elastic strain difference Δε _el (y) and the residual stress σ _res (y) inherent in the metal plate are also calculated by the above equations (2) and (3). This step B12 is the same as the above step A11.

(Step B7)
In step B7, the shape of the metal plate per calculation of the repeated calculation (sequential calculation) is predicted using a shape prediction model. Specifically, first, the plastic elongation strain difference Δε _pl (y) is separated into a component that is converted into a shape deformation after buckling (buckling shape transformation elongation strain difference Δε _ts (y)) and a component that remains in the metal plate even after buckling (buckling inherent strain difference Δε _cr (y)). Furthermore, the buckling shape transformation elongation strain difference Δε _ts (y) is divided into M-1 using the following formula (15), and the M-1 division buckling shape transformation elongation strain difference Δε _ts '(y) is calculated. Then, as shown in the following formula (16), the first buckling inherent strain difference Δε _cr (y) is read as the initial buckling inherent strain difference Δε _cr '(y).
Δε _ts '(y) = (Δε _pl (y) - Δε _cr (y))/(M-1) ... (15)
Δε _cr '(y)=Δε _cr (y) ...(16)

(Step B8)
In step B8, the shape of the metal plate in each iteration of the repeated calculation (sequential calculation) is predicted using a shape prediction model. Specifically, first, the buckling shape transformation elongation strain difference Δε _ts (y) is calculated by the following formula (17). That is, in this calculation, the buckling shape transformation elongation strain difference Δε _ts (y) is corrected using the M-1 division buckling shape transformation elongation strain difference Δε _ts '(y) and the initial buckling inherent strain difference Δε _{cr '} (y) calculated in step B7.
Δε _ts (y) = (Δε _ts '(y) x (i-1) + Δε _cr '(y)) - Δε _cr (y) ... (17)

The subsequent shape prediction of the metal plate is the same as in step A6. That is, the geometric shape elongation strain difference Δε _gs (y) and the elongation strain error Δε _{er (y) are calculated by the above formulas (6) and (7) using the buckling shape transformation elongation strain difference Δε ts (} y) calculated by the above formula (17), and the maximum amplitude H of the half amplitude wave at which the integrated value of the square of the elongation strain error Δε _er (y) shown in the above formula (8) is the minimum is determined, and h(x, y) = H × W(y) × Sin (2πx/P) is predicted as the shape of the metal plate after buckling.

(Step B9)
In step B9, the elastic strain difference Δε _el ^* (y) and the residual stress σ _res ^* (y) existing in the metal plate are calculated by the above formulas (9) and (10). This step B9 is similar to the above step A7.

(Step B10)
In step B10, the normalized elongation strain difference Δε _normal ^* (y) is determined again based on the residual stress distribution of the redistributed metal plate after buckling. Specifically, the normalized elongation strain difference Δε normal * (y) having a corresponding value of 0 to 1 is determined using the elastic strain difference _{Δε el} ^* (y) inherent in the redistributed metal plate according to the above formula ⁽ 11). This step B10 is the same as the above step A8. Then, this normalized elongation strain difference Δε _normal ^* (y) is returned to step B2 and becomes the normalized elongation strain difference used in the recalculation.

Next, the process returns to step B2, and the second and subsequent iterative calculations are performed. In the second and subsequent iterative calculations, the normalized elongation-strain difference Δε _normal ^* (y) calculated in the previous step B10 is used as Δε _normal (y), and the subsequent steps B4 to B10 are performed.

(Step B11)
The above steps B2 to B10 are repeated M times, and the shape prediction is completed. The wave amplitude height H at this time is predicted as the shape of the metal plate. When the normalized elongation strain difference Δε _normal ^* (y) converges in this way, the elastic strain difference Δε _el ^* (y) remains in the metal plate even after buckling, and the residual stress σ _res in the metal plate is calculated by multiplying it by the Young's modulus E according to the above equations (9) and (10).

The second embodiment described above can also provide the same effects as the first embodiment, that is, the elastic strain difference Δε _{e l} ^* (y) reallocated due to the shape deformation after buckling is calculated, and recalculation is performed using the elastic strain difference Δε _{e l} ^* (y), so that the shape of the metal plate can be appropriately predicted.

To verify the effect of the second embodiment, the inventors compared the shape prediction results obtained by performing the second embodiment with the shape prediction results obtained by performing the first embodiment, the shape prediction results obtained by performing a conventional shape prediction method, and the analysis results using FEM. The verification results are shown in Figs. 10 and 11. Fig. 10 shows the width-wise distribution of the elastic strain difference present in the metal plate after buckling, and Fig. 11 shows the width-wise distribution of the normalized profile.

Referring to Figures 10 and 11, the shape prediction results of the conventional method deviate slightly from the analysis results using FEM, whereas the shape prediction results of the first and second embodiments are almost identical to the analysis results using FEM. Therefore, it was found that the shape of the metal plate can be appropriately predicted according to the second embodiment.

<Feature Prediction Method According to Third Embodiment>
A feature prediction method according to the third embodiment will now be described with reference to a flowchart of FIG.

In the second embodiment described above, the buckling shape transformation elongation strain difference Δε _ts (y) is divided into M−1 parts and sequential calculation is performed in step B7. In the third embodiment, the plastic elongation strain difference Δε _pl (y) set in step C1 is divided into M parts and sequential calculation is performed.

(Steps C1 to C2)
In step C1, the plastic elongation strain difference Δε _pl (y) of the metal plate to be evaluated is set, and in step C2, the number of repeated calculations (sequential calculations) (M times) is set, and it is determined which calculation (i-th calculation) is currently being performed. Note that these steps C1 to C2 are similar to the above steps B1 to B2, respectively.

(Step C3)
In step C3, the plastic elongation strain difference Δε _pl (y) of the metal plate to be evaluated set in step C1 is divided into M parts, and the plastic elongation strain difference when performing the sequential calculation is defined. Specifically, the M-partitioned plastic elongation strain difference Δε _pl '(y) is calculated by the following formula (18), and the plastic elongation strain difference Δε _pl (y) in the i-th iterative calculation is defined (calculated) by the following formula (19).
Δε _pl '(y)=Δε _pl (y)/M...(18)
Δε _pl (y)=Δε _pl '(y)×i (19)

(Step C4)
In step C4, in the first iteration (i=1) of the iterative calculation, the plastic elongation-strain difference Δε _pl (y) is converted to a normalized elongation-strain difference Δε _normal (y) of 0 to 1. In the second and subsequent iterative calculations (i=2 and subsequent), if it is determined that buckling does not occur in the buckling presence/absence determination in step C7 (the buckling presence/absence determination one iteration before the iterative calculation), the plastic elongation-strain difference Δε _pl (y) is converted to a normalized elongation-strain difference Δε _normal (y). On the other hand, if it is determined that buckling occurs in the buckling presence/absence determination in step C7, the normalized elongation-strain difference Δε _normal ^* (y) corrected in step C11 (to be described later) is replaced with Δε _normal (y) and used as the normalized elongation-strain difference in the next and subsequent steps.

(Step C5)
In step C5, an eigenvalue analysis calculation is performed using a theoretical buckling model to calculate the buckling inherent strain difference Δε _cr (y), the wave pitch P, and the normalized height profile (shape profile) W(y) of 0-1. This step C5 is similar to step B4, and by inputting the normalized elongation strain difference Δε _normal (y), the thickness t of the metal plate, the width B of the metal plate, and the tension Ut acting on the metal plate, the buckling inherent strain difference Δε _cr (y) (criterion for buckling occurrence) similar to the plastic elongation strain difference Δε _pl (y), the wave pitch P, and the normalized height profile (shape profile) W(y) of 0 to 1 in the width direction cross section at the time of buckling are output.

(Step C6)
In step C6, the next step to proceed to is determined based on the result of the buckling determination in step C7 described later (the buckling determination in the previous iterative calculation). If it is determined in the buckling determination in step C7 that buckling will not occur, the process proceeds to step C7 described later, and if it is determined that buckling will occur, the process proceeds to step C9 described later. Once it is determined that buckling will occur, the buckling determination in step C7 can be omitted.

(Step C7)
In step C7, the occurrence of buckling is judged. This step C7 is the same as step B6, and the plastic elongation strain difference Δε _pl (y) is compared with the buckling inherent strain difference Δε _cr (y). If Δε _pl (y) is greater than Δε _cr (y), it is judged that the metal plate will buckle, and the process proceeds to step C9. On the other hand, if Δε _pl (y) is equal to or less than Δε _cr (y), it is judged that the metal plate is flat and not buckled, and the process proceeds to step B8.

(Step C8)
In step C8, it is determined whether the calculation is the final iteration (i=M) or the calculation before that (i=M-1). If the calculation is before the final iteration, the process returns to step C2 and performs the subsequent steps C3 and onward. This is to confirm whether the metal plate will buckle in the final iteration, although it was determined in the sequential calculations that it would not buckle. On the other hand, if it is the final Mth calculation, it is determined that the metal plate will not buckle and is flat, and the process proceeds to step C13.

(Step C13)
In step C13, the metal plate is finally output as being flat, and the shape prediction ends. The elastic strain difference Δε _el (y) and the residual stress σ _res (y) inherent in the metal plate are also calculated by the above equations (2) and (3). This step C13 is similar to the above step B12.

(Step C9)
In step C9, the shape of the metal sheet is predicted using a shape prediction model. This step C9 is the same as step A6, and involves calculating the buckling shape transformation elongation strain difference Δε _ts (y) using the above formula (4), calculating the geometric shape elongation strain difference Δε _gs (y) and the elongation strain error Δε _er (y) using the above formulas (6) and (7), and determining the maximum amplitude H of the half amplitude wave at which the integrated value of the square of the elongation strain error Δε _er (y) shown in the above formula (8) is the minimum, thereby predicting h(x,y)=H×W(y)×Sin(2πx/P) as the shape of the metal sheet after buckling.

(Step C10)
In step C10, the elastic strain difference Δε _el ^* (y) and the residual stress σ _res ^* (y) existing in the metal plate are calculated by the above formulas (9) and (10). This step C10 is similar to the above step B9.

(Step C11)
In step C11, the normalized elongation-strain difference Δε _normal ^* (y) is determined again based on the residual stress distribution of the metal plate redistributed after buckling, and the corrected normalized elongation-strain difference Δε _normal *(y) is determined by the above ^formula (11). This step C11 is the same as the above step B10.

Next, the process returns to step C2 to repeat the calculations, in which the subsequent steps B4 to B11 are performed using the normalized elongation strain difference Δε _normal ^* (y) calculated in the previous step C11.

(Step C12)
The above steps C2 to C11 are repeated M times, and the shape prediction is completed. The wave amplitude height H at this time is predicted as the shape of the metal plate. When the normalized elongation strain difference Δε _normal ^* (y) converges in this way, the elastic strain difference Δε _el ^* (y) remains in the metal plate even after buckling, and the residual stress σ _res in the metal plate is calculated by multiplying it by the Young's modulus E according to the above equations (9) and (10).

The third embodiment described above can also provide the same effects as the first and second embodiments. That is, the elastic strain difference Δε _{e l} ^* (y) reallocated due to the shape deformation after buckling is calculated, and recalculation is performed using the elastic strain difference Δε _{e l} ^* (y), so that the shape of the metal plate can be appropriately predicted.

<Other embodiments>
Regarding the buckling inherent strain difference Δε _cr (y) that is separated from the plastic elongation strain difference Δε _pl (y) in the above steps A6, B7, and C9 and remains in the metal plate even after buckling, it was confirmed that the residual stress of the metal plate corresponding to the buckling inherent strain difference Δε _cr (y) can be determined by considering the results of FEM analysis taking into account the transformation during cooling, and that the longitudinal residual stress distribution across the width of the metal plate can be determined as thermal stress based on the width-wise temperature distribution before and after cooling. Therefore, the residual stress is determined based on the temperature distribution before and after cooling. In such a case, it is possible to predict the residual stress distribution and the shape during cooling simply by measuring the temperature.

The residual stress of the metal plate corresponding to the above-mentioned buckling inherent strain difference Δε _cr (y) may be defined by adding at least the residual stress distribution imparted during rolling or straightening to the measured width-direction distribution of the longitudinal residual stress of the metal plate. In such a case, the residual stress distribution and cold shape of a metal plate without controlled cooling after rolling or straightening can be predicted.

In addition, the difference in elongation strain Δε _pl (y) set in steps A1, B1, and C1 may be superimposed on the difference in elongation strain obtained by measuring the shape of the metal sheet before or after cooling and assuming that the shape has become flat. In this case, by superimposing the strain due to the rolling shape and the thermal strain before or after cooling, it becomes possible to predict the residual stress and shape with higher accuracy, and the accuracy of the judgment of whether or not the shape can be corrected is improved.

By using the metal plate shape prediction method according to the first, second or third embodiment described above, it is possible to manufacture flat metal plates.

Specifically, the shape of the metal plate is predicted using the above embodiment, and a decision is made on whether or not to perform shape correction in the refining process based on the prediction result. In this case, highly accurate shape prediction is possible, and a flat plate can be manufactured by controlling the width-wise temperature distribution before and after cooling to prevent buckling.

Here, the decision on whether or not to perform shape correction in the refinement process may be based on the presence or absence of buckling as well as on the flatness level according to the user's needs, for example, 2% or less, 1% or less, 0.5% or less. In such cases, it is more practical to set judgment criteria according to the user's needs rather than judging only on the presence or absence of buckling.

Therefore, the predicted steepness may be used to determine whether or not to perform shape correction in the above-mentioned refinement process. In such a case, the judgment is made based on a predetermined steepness rather than on the presence or absence of buckling, which makes it possible to rank the flatness of the metal plate and determine whether or not to apply a correction process according to the rank and customer needs. This allows for judgment within a more practical range, and allows for optimization of the correction process.

Although the embodiments of the present invention have been described above, the present invention is not limited to these examples. It is clear that a person skilled in the art can come up with various modified or revised examples within the scope of the technical ideas described in the claims, and it is understood that these also naturally fall within the technical scope of the present invention.

Taking the manufacturing process of thick plates as an example, we predicted the flatness (shape prediction) of metal plates and conducted an experiment to determine whether the plates could be passed through the finishing process. In the manufacturing process of thick steel plates as metal plates, the thick steel plates pass through a finishing rolling mill, a hot leveler, are water-cooled to a specified temperature in accelerated cooling equipment, and are air-cooled on a cooling bed. At this time, the thermal strain and thermal stress distribution were obtained using a thermometer on the exit side of the rolling mill and before accelerated cooling, and shape predictions were made using a comparative model (Patent Document 3, Patent No. 4262142) and this model (model of the first embodiment), and the plate shapes measured in the cold state were compared. In addition, the steel plates used were a type of steel that is difficult to predict the shape of when strongly cooled in accelerated cooling equipment.

As a result, the rate of false positives, where the shape was judged to be good even though it was poor and required fine threading, was 1% (2 coils out of 200 coils) in the comparative example, while it was 0% (2 coils out of 200 coils) in the example of this application.

The present invention is useful for predicting and controlling the shape of metal sheets after rolling.

Claims (8)

1. A method for predicting a shape of a metal sheet, comprising:
A first step of setting a plastic elongation strain difference , which is a difference in the width direction of the strain that elongates in the longitudinal direction of the metal plate during rolling when the rolled metal plate buckles ;
In the first calculation of the repeated calculation, the plastic elongation and strain difference is converted into a standardized elongation and strain difference, and in the second and subsequent calculations of the repeated calculation, the standardized elongation and strain difference calculated in the tenth step below is used;
A third step of performing a buckling analysis based on the normalized elongation strain difference to calculate a buckling inherent strain difference, which is a criterion for determining whether or not the metal plate will buckle, a wave pitch, and a normalized shape profile of the metal plate;
a fourth step of determining whether the calculation is the first calculation of the repeated calculations or the second or subsequent calculations;
A fifth step of comparing the plastic elongation strain difference with the buckling inherent strain difference to determine whether or not buckling has occurred when it is determined in the fourth step that this is the first calculation of the repeated calculations;
a sixth step of separating the plastic elongation-strain difference into the buckling inherent strain difference and a buckling shape transformation elongation-strain difference, which is a difference between the plastic elongation-strain difference and the buckling inherent strain difference and is a component that is transformed into the shape of the metal plate, calculating an elongation-strain error, which is a difference between a geometric shape elongation-strain difference that appears in the shape of the metal plate after out-of-plane deformation and the buckling shape transformation elongation-strain difference, and determining a wave height at which an integrated value of the square of the elongation-strain error is a minimum value to determine a wave shape profile;
A seventh step of calculating an elastic strain difference inherent in the metal plate after buckling by adding the elastic buckling inherent strain difference and the elongation strain error;
An eighth step of calculating a normalized elongation strain difference corresponding to the elastic strain difference calculated in the seventh step by using the elastic strain difference calculated in the seventh step;
A ninth step of calculating an error between the standardized elongation-strain difference set in the second step and the standardized elongation-strain difference calculated in the eighth step, and comparing the square of the error with a predetermined threshold value;
and a tenth step of correcting the standardized elongation strain difference set in the second step by a value obtained by multiplying the error by a relaxation coefficient when the square of the error is greater than a predetermined threshold value in the ninth step,
A method for predicting a shape of a metal plate, comprising repeating steps 2 to 10 until the square of the error in step 9 becomes equal to or smaller than a predetermined threshold value. 1. A method for predicting a shape of a metal sheet, comprising:
A first step of setting a plastic elongation strain difference , which is a difference in the width direction of the strain that elongates in the longitudinal direction of the metal plate during rolling when the rolled metal plate buckles ;
A second step of setting the number of iterations to M ;
A third step of converting the plastic elongation-strain difference into a standardized elongation-strain difference in the first calculation of the iterative calculation, and using the standardized elongation-strain difference corrected in the tenth step below in the second and subsequent calculations of the iterative calculation;
A fourth step of performing a buckling analysis based on the normalized elongation strain difference to calculate a buckling inherent strain difference, which is a criterion for determining whether or not the metal plate will buckle, a wave pitch, and a normalized shape profile of the metal plate;
A fifth step of determining whether the calculation is the first calculation of the repeated calculations or the second or subsequent calculations;
A sixth step of comparing the plastic elongation strain difference with the buckling inherent strain difference to determine whether or not buckling has occurred when it is determined in the fifth step that this is the first calculation of the repeated calculations;
a seventh step of dividing the plastic elongation strain difference into the buckling inherent strain difference and a buckling shape transformation elongation strain difference, which is a difference between the plastic elongation strain difference and the buckling inherent strain difference and is a component for transforming the shape of the metal plate, dividing the buckling shape transformation elongation strain difference into M-1, calculating M-1 divided buckling shape transformation elongation strain differences, and replacing the buckling inherent strain difference in the first iterative calculation with an initial buckling inherent strain difference;
an eighth step of correcting the buckling shape transformation elongation strain difference using the M-1 division buckling shape transformation elongation strain difference and the initial buckling shape transformation elongation strain difference when it is determined in the fifth step that the calculation is the second or subsequent iteration of the repeated calculations, or when it is determined in the sixth step that the plastic elongation strain difference is greater than the buckling inherent strain difference and buckling will occur, calculating an elongation strain error, which is the difference between the geometric shape elongation strain difference appearing in the shape of the metal plate after the out-of-plane deformation and the corrected buckling shape transformation elongation strain difference, and determining a wave height at which the integrated value of the square of the elongation strain error is the minimum value to determine a wave shape profile;
A ninth step of calculating an elastic strain difference inherent in the metal plate after buckling by replacing the buckling inherent strain difference with an elastic strain distribution and adding the elongation strain error;
A tenth step of calculating a standardized elongation strain difference corresponding to the elastic strain difference calculated in the ninth step,
A method for predicting a shape of a metal plate, comprising: repeating the second step to the tenth step a number of times set in the second step. 1. A method for predicting a shape of a metal sheet, comprising:
A first step of setting a plastic elongation strain difference , which is a difference in the width direction of the strain that elongates in the longitudinal direction of the metal plate during rolling when the rolled metal plate buckles ;
A second step of setting the number of iterations to M ;
A third step of dividing the plastic elongation and strain difference into M parts and calculating a plastic elongation and strain difference according to the number of repeated calculations;
In the first calculation of the repeated calculation or in the seventh step below, if the plastic elongation-strain difference is equal to or less than the buckling inherent strain difference, the plastic elongation-strain difference is converted into a standardized standardized elongation-strain difference, and in the second or subsequent calculation of the repeated calculation and in the seventh step below, if it is determined that buckling occurs, the standardized elongation-strain difference calculated in the eleventh step below is used in the fourth step;
A fifth step of performing a buckling analysis based on the normalized elongation strain difference to calculate a buckling inherent strain difference, which is a criterion for determining whether or not the metal plate will buckle, a wave pitch, and a normalized shape profile of the metal plate;
A sixth step of determining whether or not buckling has occurred in the seventh step immediately before the iterative calculation;
A seventh step of comparing the plastic elongation strain difference with the buckling inherent strain difference to determine whether or not buckling has occurred when it is determined in the sixth step that buckling has not occurred;
In the seventh step, if it is determined that the difference in plastic elongation strain is equal to or less than the difference in inherent buckling strain and buckling does not occur, the number of times of the repeated calculation is confirmed, and if the number is not the final time, the eighth step returns to the second step;
a ninth step of separating the plastic elongation-strain difference into the buckling inherent strain difference and a buckling shape transformation elongation-strain difference, which is a difference between the plastic elongation-strain difference and the buckling inherent strain difference and is a component that is transformed into the shape of the metal plate, calculating an elongation-strain error, which is the difference between the geometric shape elongation-strain difference that appears in the shape of the metal plate after out-of-plane deformation and the buckling shape transformation elongation-strain difference, and determining a wave height at which an integrated value of the square of the elongation-strain error is a minimum value to determine a wave shape profile;
A tenth step of calculating an elastic strain difference inherent in the metal plate after buckling by replacing the buckling inherent strain difference with an elastic strain distribution and adding the elongation strain error;
An eleventh step of calculating a standardized elongation strain difference corresponding to the elastic strain difference calculated in the tenth step,
A method for predicting a shape of a metal plate, comprising: repeating the second step to the eleventh step a number of times set in the second step. The shape prediction method of a metal plate according to any one of claims 1 to 3, characterized in that the width direction distribution of the longitudinal residual stress of the metal plate, calculated by converting the buckling inherent strain difference, is set to a thermal stress distribution based on the width direction temperature distributions before and after cooling. The method for predicting a shape of a metal plate according to any one of claims 1 to 3 , characterized in that a width direction distribution of a longitudinal residual stress of the metal plate, calculated by converting the buckling inherent strain difference, is set to a residual stress distribution imparted at least during rolling or straightening . The method for predicting a shape of a metal plate according to any one of claims 1 to 3 , characterized in that a shape of the metal plate before cooling or a shape of the metal plate after cooling is measured, and the geometric shape elongation strain difference is superimposed on the plastic elongation strain difference. A method for manufacturing a metal plate, comprising: determining whether or not to perform shape correction in a finishing process by using the metal plate shape prediction method according to any one of claims 1 to 6 ; setting a width direction distribution of longitudinal residual stress in the metal plate as a thermal stress distribution based on width direction temperature distributions before and after cooling; and controlling the width direction temperature distribution so that the metal plate does not buckle based on the shape of the metal plate predicted by the shape prediction method, thereby manufacturing a flat plate. 8. The method for manufacturing a metal plate according to claim 7 , wherein the judgment as to whether or not to perform shape correction in the finishing step is made based on the predicted steepness.

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