Method for calculating radius of symmetric three-roller rolled plate

Abstract

The invention discloses a radius calculation method for a symmetric three-roller rolled plate, and the method achieves the precise prediction through four-stage modeling: firstly, building a plate elastic-plastic constitutive model, and analyzing a real stress-strain relation; secondly, according to the missile plastic bending theory, stress and bending moment expressions in the loading / unloading stage are established respectively, and the target curvature radius after unloading is obtained through integral calculation; constructing a three-roller contact point coordinate system, analyzing the stress balance of the plate in combination with material mechanics, and associating the bending moment of the contact point of the upper roller with the maximum elastic bending moment to determine a curvature parameter; and finally, through curvature averaging processing and Matlab loop iteration, a coupling model of a geometrical relationship and mechanical balance is established, and the average curvature radius of the plate and the pressing amount of the upper roller are solved. According to the method, the elastic-plastic theory, the bending deformation theory and numerical calculation are innovatively combined, the calculation precision of the bending process parameters is effectively improved through theoretical derivation and cyclic verification, and a theoretical basis is provided for forming control of the three-roller plate bending machine.

Description

Technical Field

[0001] The present invention belongs to the technical field of sheet metal forming, and particularly relates to a method for calculating the radius of a symmetric three-roll bent sheet. Background Art

[0002] With the continuous development of offshore engineering technology, large cylindrical structures are increasingly widely used in the fields of offshore platforms, submarine pipelines, shipbuilding, etc., such as the support cylinders of offshore oil and gas platforms, the tower bases of offshore wind power generation, the main cylindrical components of floating foundations, and the lifting cylinders of installation ships. The manufacturing accuracy of the cylinder is crucial, deeply affecting the strength and safety of the structure and restricting its application and development in offshore engineering. At present, the development of plate rolling machines in China is rapid, and remarkable progress has been made in the fields of design and production manufacturing, and plate rolling machines of various structural forms can be produced to meet the needs of different industries.

[0003] Among them, the symmetric three-roll is the most classic rolling process. As a simple and effective forming equipment for steel plate rolling, the plate rolling machine forms a three-point bending of the steel plate through the downward pressure of the middle roll to generate plastic deformation. As Figure 1 shown, in the rolling forming process, the sheet moves to the right with the rotation of the three rolls and enters the three-point bending loading area, from elastic-plastic loading, and then gradually undergoes elastic-plastic unloading until it is plastically bent into an arc shape with the target radius. For the symmetric three-roll rolling process, the elastic-plastic deformation in the rolling forming process is complex and variable, lacking a theory for accurately controlling the target radius. Therefore, there are problems such as insufficient accuracy, high scrap rate, and long debugging cycle in the rolling process. How to accurately predict the rolling forming radius of the sheet and develop a process optimization plan that can estimate parameters such as the downward pressure can achieve the low-cost construction of large-diameter cylinders with high efficiency and accuracy. Summary of the Invention

[0004] The technical problem to be solved by the present invention is as follows:

[0005] The existing problems in the symmetric three-roll rolling process, such as insufficient accuracy, high scrap rate, and long debugging cycle.

[0006] To solve the above technical problems, through practice and summarization, the inventor has obtained a method for calculating the radius of a symmetric three-roll bent plate, which can simply, conveniently, accurately and reliably predict and calculate the bending radius of the plate. Precise prediction is achieved through four-stage modeling: First, establish the elastoplastic constitutive model of the plate to analyze the true stress-strain relationship; second, deduce the elastoplastic bending theory, establish the stress and bending moment expressions in the loading / unloading stages respectively, and obtain the target curvature radius after unloading through integral calculation; then construct the coordinate system of the three-roll contact points, analyze the force balance of the plate combined with mechanics of materials, and correlate the bending moment at the upper-roll contact point with the maximum elastic bending moment to determine the curvature parameters; finally, through curvature averaging and Matlab cyclic iteration, establish a coupling model of geometric relationship and mechanical balance, and solve the average curvature radius of the plate and the downward pressure of the upper roll. This method innovatively combines elastoplastic theory, bending deformation theory and numerical calculation, and effectively improves the calculation accuracy of the bending process parameters through theoretical derivation and cyclic verification, providing a theoretical basis for the forming control of three-roll plate bending machines.

[0007] The specific steps include:

[0008] Step 1: Establish the elastoplastic constitutive model of the plate

[0009] When the plate is bent and formed, elastoplastic deformation occurs. Analyze the process of elastoplastic deformation change, and establish an elastoplastic constitutive model to fit the relationship between the true stress and true strain of the plate;

[0010] Step 2: Establish the elastoplastic bending theory

[0011] Analyze the stress-strain relationship of the plate in the loading stage and unloading stage, obtain the stress expression of the plate in the loading stage, and according to the integral calculation of mechanics of materials, obtain the bending moment expression of the plate in the loading stage, and obtain the maximum elastic bending moment expression and maximum unloading bending moment expression in the loading stage, obtain the stress expression in the unloading stage, and according to the integral calculation of mechanics of materials, obtain the bending moment expression of the plate in the unloading stage. When the bending moment is zero, obtain the target curvature radius expression after the plate rebounds;

[0012] Step 3: Analyze the forces in the process of rolling the plate under the symmetric three-rolls

[0013] Establish a coordinate system with the center of one of the rolls as the coordinate origin, obtain the position coordinates of the contact points of the three rolls and the plate, analyze the forces on the plate, and obtain the relationship between the bending moment in the loading stage and the bending moment in the unloading stage; and combine the three-point bending equilibrium relationship to obtain the expression of the bending moment at the contact point of the upper roll and the plate and the pressure generated at the contact point of the three rolls and the plate;

[0014] Step 4: Calculate the deformation in the process of plate rolling

[0015] 4.1) Average the curvatures of each section according to the curvature change situation in each stage;

[0016] 4.2) Deduce the relational expressions between relevant parameters, establish a Matlab loop program, and determine the average curvature radius and the downward pressure of the upper roller during the plate rolling process according to the loop.

[0017] In the preferred solution, the equation of the elastoplastic constitutive model is:

[0018]

[0019] σ * = Eε - (E - E s )(ε m - ε e )(2)

[0020] Where: E is Young's modulus; E s is the material strengthening coefficient; ε and σ are the material strain and stress respectively; ε e and σ s are the strain and stress when the material begins to yield respectively; ε b is the strain corresponding to the ultimate strength of the plate; σ * is the stress of the material in the unloading stage; ε m = y / ρ m is the maximum strain value of the material before unloading, y is the distance from any point to the center layer, and ρ m is the minimum curvature radius of the material before unloading.

[0021] In the preferred solution, the specific steps of step 2 include:

[0022] Under the plane section assumption, ε = y / ρ, where ρ is the curvature radius of the neutral layer. Combining with the bilinear hardening model equation, the stress expression of the plate in the loading stage can be obtained as follows:

[0023]

[0024] Through the integral calculation of material mechanics, the bending moment of the plate in the loading stage is obtained as follows:

[0025]

[0026] And the maximum elastic bending moment of the plate is M e = Eε e t 2 / 6;

[0027] Continue to load, then the plate enters the bending moment plastic stage;

[0028] When the bending radius reaches R m start to unload, at this time R m ≤ t / 2ε e , the maximum bending moment is:

[0029]

[0030] After entering the unloading stage, the bending radius of the sheet is denoted as ρ u , ρ u gradually increases, and the stress expression of the sheet in the unloading stage can be obtained as follows:

[0031]

[0032] Through the integral calculation of material mechanics, the bending moment of the sheet in the unloading stage is obtained as follows:

[0033]

[0034] When the bending moment unloading is completed, let M * = 0, and the forming radius R0 of the sheet after springback can be obtained:

[0035]

[0036] That is, the relationship between the curvature radii before and after the springback of the sheet is expressed, and the forming radius R0 of the sheet can be accurately predicted therefrom.

[0037] In the preferred solution, step 3 specifically includes:

[0038] Let the contact points of the lower left roller, the upper roller, and the lower right roller with the sheet be A, B, and C respectively, the corresponding angles at the respective contact points be θ1, θ0, and θ2, the center distance between the two lower rollers be 2a, and the radius be r2; the radius of the upper roller be r1; the thickness of the sheet to be processed be t; and d be the downward pressure of the upper roller;

[0039] Taking the center of the lower left roller as the origin of the coordinate system to establish a plane rectangular coordinate system, the coordinate relationships of A, B, and C are obtained:

[0040]

[0041] where O1 and O2 are the centers of the left and right rollers respectively, and O is the center of the theoretical forming radius of the sheet;

[0042] Pressures F A , F B , F C are generated at the three contact points A, B, and C. When three-point bending occurs, the bending moment of the sheet is approximately linearly distributed, and the bending moments of the AB segment and the BC segment are:

[0043]

[0044] where the bending moment M B at B is the maximum bending moment of the sheet;

[0045] It can be seen from formula (4) that when MB >M e When the bending moment is greater than M, plastic deformation will occur in the sheet metal, causing the sheet metal to be bent and formed;

[0046] Let M B =M max Substituting into Equation (5), the radius of curvature R at B can be obtained; m ;

[0047] Combined with the geometric relationship, from the three-point bending equilibrium relationship, it can be known that F A , F B , F C and the bending moment M at B B are related as follows:

[0048]

[0049] Where: is the average radius of the loading section AC.

[0050] In the preferred solution, step 4.1) specifically includes:

[0051] It can be seen from formulas (4) and (7) that the curvature κ of the sheet metal = 1 / ρ is determined by the bending moment;

[0052] It can be seen from formula (10) that during the elastic AD loading stage, the bending moment changes linearly with x, and its curvature also increases linearly with the length. Then the average curvature in the elastic stage is The average radius of curvature is ρ em =t / 4ε e ;

[0053] In the DB elastoplastic loading stage, the curvature κ of the sheet metal increases with the increase of the bending moment until the maximum bending moment M at point B max generates its maximum curvature κ B =1 / R m , and it can be seen from formula (4) that the curvature in this section changes non-linearly;

[0054] For the convenience of calculation, the average curvature of DB is approximated as

[0055] Combined with formulas (4) and (10), the approximate average curvature of the entire loading stage AB:

[0056]

[0057] Similarly, during the unloading stage BC, the bending moment of the sheet metal decreases linearly until the bending moment at point C is zero. At this time, the sheet metal rebounds to the forming radius R0;

[0058] From formulas (7) and (10), the average curvature of the BC section during the unloading stage is:

[0059]

[0060] The preferred local solution, step 4.2) is as follows: Combining the geometric condition (9), the downward pressure of the upper roll during the three-roll bending process is calculated through Matlab programming.

[0061] The preferred local solution, the calculation process of the downward pressure of the upper roll during the three-roll bending process is as follows:

[0062] And the average curvature radii of the left and right segments of point B are obtained according to formulas (12) and (13) respectively. First, define and And a minimum value e = 0.001 is set. Then calculate R 1m = 2×R1×R2 / (R1 + R2) and R 2m = (R1 + R2) / 2. Enter a while loop. As long as R 2m + R 1m > e, perform the following operations:

[0063] 1.1) Let k = k + 1;

[0064] 1.2) Calculate R mm = (R 1m + R 2m ) / 2;

[0065] 1.3) Calculate k1 = (R mm + R2) / (R1 - R2);

[0066] 1.4) Set the initial φ1 and φ2;

[0067] Enter another while loop. As long as φ2 - φ1 > e, perform the following operations:

[0068] 2.1) Calculate φ = (φ1 + φ2) / 2;

[0069] 2.2) Calculate

[0070] 2.3) Calculate OA and OC according to formula (16);

[0071] 2.4) Adjust φ1 and φ2 according to the relationship between O1O2 and l;

[0072] 2.5) Each time in the loop, kk0 = kk0 + 1, and record and O1O2(kk0) = O1O2;

[0073] After the loop ends, calculate the relevant values according to formula (16);

[0074] Finally, if a1 > l / 2, then let R 2m = Rmm ; Otherwise, let R 1m = R mm kk0;

[0075] Where: is the radius simplified after cyclic operation. The geometric relationships in the above cyclic process are as follows:

[0076]

[0077] Relationships between various angles:

[0078]

[0079] Combined with formulas (14) and (15), the relationships of each side are obtained:

[0080]

[0081] During the plate rolling process, the plate is subjected to the downward pressure of the upper roll and the supporting force of the lower roll, and the downward pressure of the upper roll depends on the downward displacement of the upper roll;

[0082] Combined with the above geometric relationships and relevant cyclic results, when the geometrically simplified radius R and θ0 are obtained, the downward displacement of the upper roll can be calculated through formulas, and thus the relationship between the downward displacement of the upper roll and the forming radius can be obtained;

[0083]

[0084] Compared with the prior art, the present invention has the following beneficial effects:

[0085] For the cold bending forming process of the plate under the three-roll bending, a mechanical model considering the loading and unloading processes of material elastoplasticity is established; based on the accurate mechanical model of the cold bending forming process, the influence laws of the mechanical parameters and geometric parameters in the rolling process on the forming radius are analyzed, and a calculation algorithm for accurately predicting the forming radius of the plate is formed; based on the accurate mechanical model of the cold bending forming process, the prestress distribution of the plate after rolling forming is calculated, which provides reference and guidance for the subsequent processing and assembly processes of the plate. Brief Description of the Drawings

[0086] Figure 1 is a schematic diagram of the plate during symmetric three-roll bending;

[0087] Figure 2 is a flowchart of the radius calculation method of the present invention;

[0088] Figure 3 is a bilinear hardening material model;

[0089] Figure 4 Schematic diagram of the force analysis and bending moment distribution during rolling bending;

[0090] Figure 5 Calculation flow chart of the bending model

[0091] Figure 6 Distribution of normal stress in the steel plate before and after removing the external force

[0092] Figure 7 Relationship diagram between the loading bending moment and the unloading bending moment and the curvature Specific implementation manner

[0093] A method for calculating the radius of a symmetric three-roll bent plate, characterized by comprising:

[0094] Step 1, establish an elastic-plastic constitutive model of the plate

[0095] When the plate is bent and formed, elastic-plastic deformation occurs. Analyze the change process of elastic-plastic deformation, and establish an elastic-plastic constitutive model to fit the relationship between the true stress and the true strain of the plate;

[0096] Step 2, establish an elastic-plastic bending theory

[0097] Analyze the stress-strain relationship of the plate in the loading stage and the unloading stage, obtain the stress expression of the plate in the loading stage, calculate according to the integral of material mechanics, obtain the bending moment expression of the plate in the loading stage, and obtain the maximum elastic bending moment expression and the maximum unloading bending moment expression in the loading stage. Obtain the stress expression in the unloading stage, calculate according to the integral of material mechanics, obtain the bending moment expression of the plate in the unloading stage, and when the bending moment is equal to zero, obtain the expression of the target curvature radius after the plate rebounds;

[0098] Step 3, force analysis of the plate rolling process under symmetric three rolls

[0099] Establish a coordinate system with the center of one of the rolls as the coordinate origin, obtain the position coordinates of the contact points between the three rolls and the plate, conduct a force analysis on the plate, and obtain the relationship between the bending moment in the loading stage and the bending moment in the unloading stage; and combine the three-point bending equilibrium relationship to obtain the expression of the bending moment at the contact point between the upper roll and the plate and the pressure generated at the contact point between the three rolls and the plate;

[0100] Step 4, deformation calculation of the plate bending process

[0101] 4.1) Average the curvatures of each section according to the curvature change situation of each stage;

[0102] 4.2) Deduce the relational expressions between relevant parameters, establish a Matlab loop program, and determine the average curvature radius and the downward pressure of the upper roll during the plate rolling process according to the loop.

[0103] Among them, the equation of the elastic-plastic constitutive model is:

[0104]

[0105] σ * = Eε - (E - E s )(ε m - ε e ) (2)

[0106] Where: E is Young's modulus; E s is the material strengthening coefficient; ε and σ are the material strain and stress respectively; ε e and σ s are the strain and stress when the material begins to yield respectively; ε b is the strain corresponding to the ultimate strength of the sheet; σ * is the stress of the material in the unloading stage; ε m = y / ρ m is the maximum strain value of the material before unloading, y is the distance from any point to the center layer, and ρ m is the minimum curvature radius of the material before unloading. The true stress - true strain relationship diagram of the material is as shown in Figure 3 shown.

[0107] Among them, the specific steps of step 2 include:

[0108] When the steel plate is bent and stressed, its curvature radius gradually changes with the change of the bending force. The greater the bending moment, the greater the degree of bending. The steel plate gradually develops from elastic deformation at the beginning to elastoplastic deformation. The bending process of the steel plate is a process of elastic bending - elastoplastic bending - plastic bending change.

[0109] Under the plane - section assumption, ε = y / ρ, where ρ is the curvature radius of the neutral layer. Combining with the bilinear hardening model equation, the stress expression of the sheet in the loading stage can be obtained as follows:

[0110]

[0111] Through the integral calculation of material mechanics, the bending moment of the sheet in the loading stage is obtained as follows:

[0112]

[0113] And the maximum elastic bending moment of the sheet is obtained as M e = Eε e t 2 / 6;

[0114] Continue to load, then the sheet enters the bending moment plastic stage;

[0115] When the bending radius reaches R m start to unload. At this time, R m ≤ t / 2ε e , the maximum bending moment is:

[0116]

[0117] After entering the unloading stage, the bending radius of the sheet is denoted as ρ u , ρ u gradually increases, and the stress expression of the sheet in the unloading stage can be obtained as follows:

[0118]

[0119] Through the integral calculation of material mechanics, the bending moment of the sheet in the unloading stage is obtained as follows:

[0120]

[0121] When the bending moment unloading is completed, let M * = 0, and the forming radius R0 of the sheet after springback can be obtained:

[0122]

[0123] That is, the relationship between the curvature radii before and after the springback of the sheet is expressed, and the forming radius R0 of the sheet can be accurately predicted therefrom.

[0124] In the preferred solution, step 3 specifically includes:

[0125] As Figure 1 shown, the contact points of the lower left roller, the upper roller and the lower right roller with the sheet are A, B and C respectively, the corresponding angles with the respective contact points are θ1, θ0 and θ2, the center distance between the two lower rollers is 2a, and the radius is r2; the radius of the upper roller is r1; the thickness of the sheet to be processed is t; d is the downward pressure of the upper roller. In the roll bending forming process, the steel plate starts to be loaded from point A, the curvature κ = 1 / ρ gradually increases until the bending moment at point B reaches the maximum bending moment M max (the maximum bending moment M max > M e , in order to produce plastic deformation), then there is a critical point of elastic loading between point A and point B, denoted as D, which is the maximum elastic deformation here. Then the AD section is the elastic loading section, and the DB section is the elastic-plastic loading section until the maximum elastic-plastic deformation is obtained at point B, where both the bending moment and the curvature reach the maximum values. After the steel plate passes through B, it enters the BC section and starts to unload, the bending moment gradually decreases, and the curvature gradually decreases. When it reaches point C, the unloading is completed, the bending moment is equal to 0, but the curvature will not become 0 and there will be residual deformation due to plastic deformation. The steel plate reaches the target forming radius through cold bending.

[0126] The simplified force analysis and deformation shape of the steel plate during the roll bending process are as Figure 4 shown. Taking the center of the lower left roller as the origin of the coordinate system, a plane rectangular coordinate system is established. In Figure 4 the coordinate system, the coordinate relational expressions of A, B and C are obtained through geometric relationships:

[0127]

[0128] Among them, O1 and O2 are the centers of the left and right rollers respectively, and O is the center of the theoretical forming radius of the sheet;

[0129] The force analysis of the steel plate is as Figure 4 shown. Ignoring the friction of the three rollers, pressures F A , F B , and F C are generated at the three contact points A, B, and C. When three-point bending occurs, the bending moment of the sheet shows an approximately linear distribution. The bending moments in the AB segment and the BC segment are:

[0130]

[0131] Among them, the bending moment M B at B is the maximum bending moment of the sheet;

[0132] It can be seen from formula (4) that when M B > M e , the sheet will have plastic deformation, causing the sheet to be bent and formed;

[0133] Let M B = M max and substitute it into equation (5), the radius of curvature R m at B can be obtained;

[0134] Combined with the geometric relationship, it can be known from the three-point bending equilibrium relationship that the relationship between F A , F B , F C and the bending moment M B at B is:

[0135]

[0136] Among them: is the average radius of the loading section AC.

[0137] In a preferred solution, the specific steps of step 4.1) include:

[0138] Ignoring the influence of shear deformation, it can be known from formulas (4) and (7) that the curvature κ = 1 / ρ of the sheet is determined by the bending moment;

[0139] It can be known from formula (10) that during the elastic AD loading stage, the bending moment changes linearly with x, and its curvature also increases linearly with the length. Then the average curvature in the elastic stage is The average radius of curvature is ρ em = t / 4ε e ;

[0140] In the DB elastoplastic loading stage, the curvature κ of the sheet increases with the increase of the bending moment until point B reaches the maximum bending moment M max produces its maximum curvature of κ B = 1 / R m . As can be seen from formula (4), the curvature in this section changes non-linearly;

[0141] For the convenience of calculation, we approximate the average curvature of DB as

[0142] Combining formulas (4) and (10), the approximate average curvature of AB during the entire loading stage is:

[0143]

[0144] Similarly, during the unloading stage BC, the bending moment of the sheet decreases linearly until the bending moment at point C is zero. At this time, the sheet rebounds to the forming radius R0;

[0145] From formulas (7) and (10), the average curvature of the BC section during the unloading stage is:

[0146]

[0147] Among them, step 4.2) is specifically as follows: Combining the geometric condition (9), the downward displacement of the upper roll during the three-roll bending process is calculated through Matlab programming.

[0148] Among them, the calculation process of the downward displacement of the upper roll during the three-roll bending process is as follows:

[0149] From formulas (12 - 13), the approximate curvatures of the steel plate during loading and unloading can be obtained, but the distribution ratio of these two sections is uncertain, and the average curvature radius of the entire AC section of the steel plate cannot be directly obtained, so explicit functions of parameters such as the contact angle and forming radius cannot be obtained. Here, combining the geometric condition (9), the downward displacement of the upper roll during the three-roll bending process is calculated through Matlab programming. Figure 5 It is the flowchart for iterative calculation, and the specific operation is as follows.

[0150] And the average curvature radii of the left and right sections of point B are obtained according to formulas (12) and (13) respectively. First, define and and set a minimum value e = 0.001. Then calculate R 1m = 2×R1×R2 / (R1 + R2) and R 2m = (R1 + R2) / 2, and enter a while loop. As long as R 2m + R 1m > e, the following operations are performed:

[0151] 1.1) Let k = k + 1;

[0152] 1.2) Calculate R mm = (R 1m + R 2m ) / 2;

[0153] 1.3) Calculate k1 = (R mm + R2) / (R1 - R2);

[0154] 1.4) Set the initial φ1 and φ2;

[0155] Enter another while loop and perform the following operations as long as φ2 - φ1 > e:

[0156] 2.1) Calculate φ = (φ1 + φ2) / 2;

[0157] 2.2) Calculate

[0158] 2.3) Calculate OA and OC according to formula (16);

[0159] 2.4) Adjust φ1 and φ2 according to the relationship between O1O2 and l;

[0160] 2.5) In each loop, kk0 = kk0 + 1, and record and O1O2(kk0) = O1O2;

[0161] After the loop ends, calculate the relevant values according to formula (16);

[0162] Finally, if a1 > l / 2, then let R 2m = R mm ; otherwise, let R 1m = R mm kk0;

[0163] Where: is the simplified radius after loop calculation. The geometric relationships in the above loop process are as follows:

[0164]

[0165] Relationships between various angles:

[0166]

[0167] Combined with formulas (14) and (15), the relationships of each side are obtained:

[0168]

[0169] During the plate rolling process, the plate is subjected to the downward pressure of the upper roll and the supporting force of the lower roll, and the downward pressure of the upper roll depends on the downward displacement of the upper roll;

[0170] Combining the above geometric relationships and relevant cyclic results, when the geometric simplified radius R and θ0 are obtained, the reduction of the upper roll can be calculated by a formula, and the relationship between the reduction of the upper roll and the forming radius can be obtained.

[0171] d = r1 + r2 + t - [OO1 cosθ1 - (R - r1)cosθ0]. (17)

[0172] The following combines engineering examples to show the specific implementation steps and actual effects of this method.

[0173] Now analyze the Figure 1 mechanical properties of the steel plate shown during the rolling process of a three-roll symmetric plate rolling mill. The main process parameters involved in the plate rolling process are: the radius of the upper roll r1 = 0.19 m, the lower roll r2 = 0.15 m, 2a = 0.540 m; material parameters: E = 205 GPa, Es = 1.5 GPa, υ = 0.3, ε e = 235 MPa.

[0174] According to the above parameters and formula (3), taking the steel plate with a thickness of t = 20 mm during rolling as an example, the stress distribution before and after removing the external force is obtained. As Figure 5 shown.

[0175] The main stress generated by springback is the normal stress of the steel plate. Figure 5 For the steel plate under the bending moment applied by the supporting roll, the outer surface is subjected to tensile stress and the inner surface is subjected to compressive stress. The tensile stress is set as positive and the compressive stress is set as negative; in the elastic stage of the loading deformation zone, according to formula (4), under the above parameters, when the load is applied and the bending moment M = 13.21 kN·m, the steel plate is in the elastic stage and the bending radius R = 10 m. At this time, the stress distribution is shown as a linear distribution by the blue solid line. Continuing to load, the surface material reaches the elastic limit. At this time, the maximum bending moment Me that can be applied in the elastic stage is 15.667 kN·m, and the minimum radius of the steel plate in the elastic stage is R = 8.72 m. Continuing to load, the steel plate enters the elastoplastic stage. It can be calculated from formula (7) that when the radius of the steel plate in the elastoplastic stage is R = 1.484 m, the formed radius R0 after unloading is 2 m. From formula (6), it can be known that a bending moment of M = 23.777 kN·m is applied. At this time, the stress distribution is shown as the red solid line. After gradually unloading, it enters the unloading stage. When the unloading bending moment M = 1.207 kN·m, the springback radius reaches R = 1.7 m. At this time, the stress distribution is shown as the yellow dotted line; finally, when the bending moment is equal to zero, there will be residual stress inside the steel plate, and its distribution is shown as the purple dotted line. At this time, the steel plate with the target formed radius R0 = 2 m is obtained.

[0176] In the four-point bending test, four loading points act on the steel plate structure. A symmetric model is established, and forces are applied at the upper and lower two nodes at the right end of the steel plate. A pure bending region will be generated between these two internal loading points. By changing the magnitudes of the forces applied at the two nodes, the bending moment magnitude can be changed. According to the above theory, under the above parameters, Figure 6 Figure 2 shows the displacement and deformation diagrams of the steel plate under different bending moments. According to symmetry, fixed constraints are applied, and different bending moments are applied at the rightmost end of the steel plate, and simulation is carried out using ANSYS software.

[0177] According to Formulas (4) and (6), the relationship between the loading bending moment and the unloading bending moment of the steel plate with the same curvature is as Figure 6 shown. The * marks in the figure are the ANSYS simulation values of the deformation amounts of each steel plate. It can be seen from the figure that the finite element values and the theoretical values fit each other, verifying the accuracy of the bending theory results of the steel plate of the present invention.

[0178] The maximum bending moments of Q235 steel plates with different thicknesses under different springback radii are shown in Table 1. It can be obtained that when the steel plate thickness is small, as the springback radius increases, the maximum bending moment corresponding to the elastic-plastic stage of the steel plate slowly decreases. When the steel plate is thicker, the corresponding maximum bending moment rapidly decreases. Therefore, the greater the steel plate thickness, the more attention should be paid to the influence of the bending moment on the steel plate. Table 2 shows the springback radii of steel plates with different thicknesses under the target curvature radius. It can be obtained that when the target springback radius of the steel plate is closer to the critical value of the elastic-plastic stage, the ratio to the radius before springback is greater.

[0179] Table 1: Maximum bending moments of Q235 steel plates with different thicknesses under different hardening coefficients (E S = 1500 MPa)

[0180]

[0181] Table 2: Springback radii of steel plates with different thicknesses under the target curvature radius for Q235 material (E S = 1500 MPa)

[0182]

[0183] According to Formulas (8-16) and the Matlab loop command, Table 3 can be obtained. It can be seen from Table 3 that the forming radius R is related to the downward pressure d of the upper roll and the contact angle θ0 between the center of the upper roll and the steel plate.

[0184] Table 3: Springback radii of a 30-mm-thick steel plate under the target radius for Q235 material (E S = 1500 MPa)

[0185]

[0186] These data show the changes in the springback radius, bending force (F B ), maximum bending moment (M max ), deformation amount (d), and contact included angle (θ0) of the steel plate under different forming radii. As the forming radius increases, the bending force and maximum bending moment gradually decrease, and the deformation amount and initial bending angle also decrease accordingly.

[0187] The present invention has conducted in-depth research on the relevant process theories, automatic control models, etc. of the steel plate during roll bending on a roll bending machine, and its research methods and results have guiding significance for the automatic forming of the steel plate roll bending.

Claims

1. A method for calculating the radius of a symmetric three-roll bent sheet, characterized in that, Including: Step 1: Establish an elastoplastic constitutive model of the sheet When the sheet is bent and formed, elastoplastic deformation occurs. Analyze the change process of elastoplastic deformation, and establish an elastoplastic constitutive model to fit the relationship between the true stress and true strain of the sheet. Step 2: Establish an elastoplastic bending theory Analyze the stress-strain relationship of the sheet in the loading stage and unloading stage, obtain the stress expression of the sheet in the loading stage, and according to the integral calculation of material mechanics, obtain the bending moment expression of the sheet in the loading stage, and obtain the maximum elastic bending moment expression and maximum unloading bending moment expression in the loading stage. Obtain the stress expression in the unloading stage, and according to the integral calculation of material mechanics, obtain the bending moment expression of the sheet in the unloading stage. When the bending moment is zero, obtain the target curvature radius expression of the sheet after springback. Step 3: Analyze the force on the sheet during the rolling process under symmetric three rolls Establish a coordinate system with the center of one of the rolls as the coordinate origin, obtain the position coordinates of the contact points between the three rolls and the sheet, analyze the force on the sheet, and obtain the relationship between the bending moment in the loading stage and the bending moment in the unloading stage; and combine the three-point bending equilibrium relationship to obtain the expression of the bending moment at the contact point between the upper roll and the sheet and the pressure generated at the contact point between the three rolls and the sheet. Step 4: Calculate the deformation of the sheet during the bending process 4.1) Average the curvatures of each section according to the curvature change situation of each stage. 4.2) Deduce the relational expressions between relevant parameters, establish a Matlab loop program, and determine the average curvature radius and the downward displacement of the upper roll during the rolling process of the sheet according to the loop.

2. The radius calculation method for symmetric three-roll bending of a sheet according to claim 1, characterized in that, The equation of the elastoplastic constitutive model is: σ * = Eε - (E - E s )(ε m - ε e )(2) Where: E is Young's modulus; E s is the material strengthening coefficient; ε and σ are the material strain and stress respectively; ε e and σ s are the strain and stress when the material begins to yield respectively; ε b is the strain corresponding to the ultimate strength of the sheet; σ * is the stress of the material in the unloading stage; ε m = y / ρ m is the maximum strain value of the material before unloading, y is the distance from any point to the center layer, and ρ m is the minimum radius of curvature of the material before unloading.

3. The radius calculation method of a symmetric three-roll bending sheet according to claim 1, characterized in that, The specific content of step 2 includes: Under the plane section assumption, ε = y / ρ, where ρ is the curvature radius of the neutral layer. Combining with the bilinear hardening model equation, the stress expression of the sheet in the loading stage can be obtained as follows: Through the integral calculation of material mechanics, the bending moment of the sheet in the loading stage is obtained as follows: and obtain the maximum elastic bending moment of the sheet as M e = Eε e t 2 6; Continue to load, and the sheet enters the bending moment plastic stage. When the bending radius reaches R m unloading begins, at this time R m ≤t2ε e , the maximum bending moment is: After entering the unloading stage, the bending radius of the sheet is denoted as ρ u , ρ u gradually increases, and the stress expression of the sheet in the unloading stage can be obtained as follows: Through the integral calculation of material mechanics, the bending moment of the sheet in the unloading stage is obtained as follows: When the bending moment unloading is completed, let M * = 0, and the forming radius R0 after springback of the sheet can be obtained: That is, the relationship between the curvature radii before and after the springback of the sheet is expressed, and the forming radius R0 of the sheet can be accurately predicted therefrom.

4. A method for calculating the radius of a symmetric three-roll bending sheet according to claim 3, characterized in that The specific content of step 3 includes: Let the contact points of the lower left roll, the upper roll and the lower right roll with the sheet be A, B and C respectively, the corresponding angles at the corresponding contact points be θ1, θ0 and θ2, the center distance between the two lower rolls be 2a, and the radius be r2; the radius of the upper roll be r1; the thickness of the sheet to be processed be t; d be the downward displacement of the upper roll. Establish a plane rectangular coordinate system with the center of the lower left roll as the coordinate origin, and obtain the coordinate relational expressions of A, B and C: Among them, O1 and O2 are the centers of the left and right rolls respectively, and O is the center of the circle of the theoretical forming radius of the sheet. Generate pressure F at three contact points A, B, and C A , F B , F C , during three-point bending, the bending moment of the plate shows an approximately linear distribution, and the bending moments in the AB section and the BC section are as follows: where the bending moment M at B B is the maximum bending moment of the sheet As can be seen from Equation (4), when M B > M e , the sheet metal will have plastic deformation, causing the sheet metal to be bent and formed; Let M B = M max Substituting into equation (5), the radius of curvature R at B can be obtained m ; Combined with the geometric relationship, it can be known from the three-point bending equilibrium relationship that F A , F B , F C and the bending moment M B at B are related as follows: Wherein: is the average radius of the loading section AC.

5. A method for calculating the radius of a symmetric three-roll bending sheet according to claim 4, characterized in that, The specific content of step 4.1) includes: It can be seen from formulas (4) and (7) that the curvature κ = 1 / ρ of the sheet is determined by the bending moment. As can be seen from Equation (10), during the elastic AD loading stage, the bending moment varies linearly with x, and its curvature also increases linearly with the length. Therefore, the average curvature in the elastic stage is The average radius of curvature is ρ em = t / 4ε e ; In the DB elastoplastic loading stage, the curvature κ of the sheet increases with the increase of the bending moment until point B reaches the maximum bending moment M max produces its maximum curvature of κ B = 1 / R m , and it can be seen from formula (4) that the curvature in this section changes nonlinearly; For the convenience of calculation, the average curvature of the DB is approximated as Combining formulas (4) and (10), the approximate average curvature of AB in the entire loading stage: Similarly, the bending moment of the sheet in the BC section of the unloading stage decreases linearly until the bending moment at point C is zero, and at this time the sheet rebounds to the forming radius R0. From formulas (7) and (10), the average curvature of the BC section in the unloading stage is:

6. A method for calculating the radius of a symmetric three-roll bending sheet according to claim 5, characterized in that, Step 4.2) is as follows: Combining geometric condition (9), the downward displacement of the upper roll during the three-roll bending process is calculated through Matlab programming.

7. A method for calculating the radius of a symmetric three-roll bending sheet according to claim 6, characterized in that The calculation process of the downward displacement of the upper roll during the three-roll bending process is as follows: And the average curvature radii of the left and right segments of point B are obtained according to formulas (12) and (13) respectively. First, define and and set a minimum value e = 0.

001. Then calculate R 1m = 2×R1×R2 / (R1 + R2) and R 2m = (R1 + R2) / 2. Enter a while loop. As long as R 2m + R 1m > e, perform the following operations: 1.1) Let k = k + 1; 1.2) Calculate R mm = (R 1m + R 2m ) / 2; 1.3) Calculate k1 = (R mm + R2) / (R1 - R2); 1.4) Set the initial φ1 and φ2; Enter another while loop. As long as φ2 - φ1 > e, perform the following operations: 2.1) Calculate φ = (φ1 + φ2) / 2; 2.2) Calculation 2.3) Calculate OA and OC according to formula (16); 2.4) Adjust φ1 and φ2 according to the relationship between O1O2 and l; 2.5) In each loop, kk0 = kk0 + 1 and record and O1O2(kk0) = O1O2; After the loop ends, calculate the relevant values according to formula (16); Finally, if a1 > l / 2, then let R 2m = R mm ; otherwise, let R 1m = R mm kk0; Wherein: is the radius simplified after cyclic operation, and the geometric relationships in the above cyclic process are as follows: Relationships between various angles: Combined with formulas (14) and (15), the relationships of each side are obtained: During the plate rolling process, the plate is subjected to the downward pressure of the upper roll and the supporting force of the lower roll, and the downward pressure of the upper roll depends on the downward displacement of the upper roll; Combining the above geometric relationships and relevant loop results, when the geometric simplified radius R and θ0 are obtained, the downward displacement of the upper roll can be calculated through the formula, and thus the relationship between the downward displacement of the upper roll and the forming radius can be obtained;

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