1、Analytical and empirical modeling of top roller position for three-roller cylindrical bending of plates and its experimental verificationA.H. Gandhi, H.K. RavalAbstract：Reported work proposes an analytical and empirical model to estimate the top roller position explicitly as a function of desired (f

2、inal) radius of curvature for three-roller cylindrical bending of plates, considering the contact point shift at the bottom roller plate interfaces. Effect of initial strain and change of material properties during deformation is neglected. Top roller positions for loaded radius of curvature are plo

3、tted for a certain set of data for center distance between bottom rollers and bottom roller radius. Applying the method of least square and method of differential correction to the generated data, a unified correlation is developed for the top roller position, which in turn is verified with the expe

4、riments, on a pyramid type three-roller plate-bending machine. Uncertainty analysis of the empirical correlation is reported using the McClintocks method.Keywords: Roller bending, Springback, Analytical study, Empirical modeling, Uncertainty analysis1. IntroductionLarge and medium size tubes and tub

5、ular sections are extensively in use in many engineering applications such as the skeleton of oil and gas rigs, the construction of tunnels and commercial and industrial buildings (Hua et al., 1999). The hull of ships may have single, double or higher order curvatures, which can be fabricated sequen

6、tially; first by roll forming or bending (to get the single curvature), and then line heating (to get the double or higher order curvature). As roller bending is performed at least once in the sequential process, its efficient performance is a prerequisite for the accurate forming of the double or m

7、ultiple curvature surfaces (Shin et al., 2001). In view of the crucial importance of the bending process, it is rather surprising to find that roller-bending process in the field has been performed in a very nonsymmetrical manner. Normal practice of the roller bending still heavily depends upon the

8、experience and skill of the operator. Working with the templates, or by trial and error, remains a common practice in the industry. The most economical and efficient way to produce the cylinders is to roll the plate through the roll in a single pass, for which the plate roller forming machine should

9、 be equipped with certain features and material-handling devices, as well as a CNC that can handle the entire production process (Kajrup and Flamholz, 2003).Many times most of the plate bending manufacturers experience Low productivity due to under utilization of their available equipment. The repea

10、tability and accuracy required to use the one-pass production method has always been a challenging task.Reported research on the forming of cylindrical shells mostly discusses the modeling and analysis of the process. Hensen and Jannerup (1979) reported the geometrical analysis of the single pass el

11、asto-plastic bending of beams on the three-roller pyramid benders by assuming triangular moment distribution between the rollers. Developed model for the bending force and bending moment was based on the contact point shift between the plate and top roll fromthe vertical centerline of the top roll.

12、Hardt et al. (1982) described closed loop shape control of three-roller bending process. The presented scheme accomplishes the shape control by measuring the loaded shape, the loaded moment and effective beam rigidity of the material in real time. Yang and Shima (1988) and Yang et al. (1990) discuss

13、ed the distribution of curvature and bending moment in accordance with the displacement and rotation of the rolls by simulating the deformation of work piece with Ushaped cross-section in a three-roller bending process. They reported the relationship between the bending moment and the curvature of t

14、he work piece by elementary method, which was further used to build up a process model combining the geometries for three-roller bending process. Developed process model was further applied to the real time control system to obtained products with constant and continuously varying curvature. Hardt e

15、t al. (1992) reported a process model for use in simulation of the manufacturing of cylindrical shells from the plates, which require sequential bending, by incorporating the prior bend history. They modeled the process with series of overlapping two-dimensional three point bends, where overlap incl

16、udes the plastic zone from the previous bends. Hua et al. (1995) reported the mathematical model for determining the plate internal bending resistance at the top roll contact for the multi-pass four-roll thin plate bending operations along with the principle mechanisms of bending process for single

17、pass and multi-pass bending. Shin et al. (2001) have reported a kinematics based symmetric approach to determine the region of the plate to be rolled, in order to form smoothly curved plates. Gandhi and Raval (2006) developed the analytical model to estimate the top roller position as a function of

18、desired radius of curvature, for multiple pass three-roller forming of cylinders, considering real material behavior and change of Youngsmodulus of elasticity (E) under deformation and shows that the springback is larger than the springback calculated with constant E. Literature review reveals that

19、only limited studies are available on the continuous three-roller bending of plates. With reported analytical models, it is difficult to find the top roller position explicitly as a function of the desired radius of curvature and hence it requires solving the set of equations by nonlinear programmin

20、g. Use of the close loop shape control or adaptive control or CNC control system can improve the accuracy and the consistency of the process but acquisition and maintenance of such a system is costly and may not be affordable to the small scale to medium scale fabricators. Purpose of the present ana

21、lysis is to develop the model for prediction of the top roller position as a function of the desired radius of curvature explicitly for cylindrical shell bending. Development of the model is based on analytical and empirical approach. Empirical model is developed based on the top roller position ver

22、sus loaded radius of curvature plots, which is obtained geometrically for a set of data of center distance between bottom rollers and bottom roller radius.Fig. 1 Schematic diagram of three-roller bending process.Fig. 1 shows the schematic diagram of three-roller bending process, which aimed at produ

23、cing cylindrical shells. The plate fed by two side rollers and bends to a desired curvature by adjusting the position of center top roller in one or several passes. Distance between bottom rollers can be varied. During deformation, axes of all the three rollers are set parallel to each other. Desire

24、d curvature in this case is the function of plate thickness (t), plate width (w), material properties (E, n, K, and v), center distance between two bottom rollers (a), top-roller position(U), top-roller radius (rt) and bottom-roller radius (r1) (Raval, 2002). The capacity of the plate bender is defi

25、ned by the parameters such as tightest bend radius with the maximum span and designed thickness of the plate and the amount of straight portion retained at the end portions of the plate.Fig2 Deformation in fiber ABO2. Bending analysis Bending analysis is based on some of the basic assumptions summar

26、ized below:The material is homogeneous and has a stable microstructure throughout the deformation process. Deformation occurs under isothermal conditions. Plane strain conditions prevail. The neutral axis lies in the mid-plane of the sheet. Bauschinger effect is neglected. Analysis is based on power

27、 law material model, Pre-strain is neglected. Change of material properties during deformation is neglected. Plate is with the uniform radius of curvature for supported length between bottom rollers.2.1. Geometry of bendingIn thin sheets, normal section may be considered to remain plane on bending a

28、nd to converge on the center of curvature (Marciniak and Duncan, 1992). It is also considered that the principal direction of forces and strain coincide with the radial and circumferential direction so that there is no shear in the radial plane and gradient of stress and strain are zero in circumfer

29、ential direction. The middle surface however may extend. Fibers away from the middle surface are deformed as shown in Fig. 2. Initially the length of the fiber AB0 is assumed as l0 in the flat sheet. Then, under the action of simultaneous bending and stretching the axial strain of the fiber AB0 is o

30、f the form 1=lnlml0+ln(1+y)=a+b (1)where a is the strain associated with the extension of middle surface,b the bending strain and is the radius of curvature of the neutral surface.2.2. Moment per unit width for bending without tensionIn the case of simple bending without applied tension and where th

31、e radius of curvature is more than several times the sheet thickness, the neutral surface approximately coincides with the middle surface. If the general stressstrain curve for the material takes the form f=Keqn (2) Then, for the plastic bending, applied moment per unit width can be of the form (Mar

32、ciniak and Duncan, 1992) M=2K(1nn+22n+2)tn+2 (3)2.3 Elastic spring back in plates formed by bendingIn practice, plates are often cold formed. Due to spring back, the radius through which the plate is actually bent must be smaller than the required radius. The amount of spring back depends up on seve

33、ral variables as follows (Raval, 2002;Sidebottom and Gebhardt, 1979): Ratio of the radius of curvature to thickness of plates, i.e. bend ratio. Modulus of elasticity of the material. Shape of true stress versus true strain diagram of the material for loading under tension and compression. Shape of t

34、he stressstrain diagram for unloading and reloading under tension and compression, i.e. the influence of the Bouschinger effect. Magnitude of residual stresses and their distribution in the plate before loading. Yield stress (y). Bottom roller radius, top roller radius and center distance between bo

35、ttom rollers. Bending history (single pass or multiple pass bending, initial strain due to bending during previous pass).Assuming linear elastic recovery law and plane strain condition (Marciniak and Duncan, 1992; Hosford and Caddell, 1993), for unit width of the plate, relation between loaded radiu

36、s of curvature (R) and desired radius of curvature (Rf) can be given by RRf=1-6K'tn-1Rn-1n+22n1-v2E (4)3. Analytical models of top roller position (U) for desired radius of curvature (Rf)For the desired radius of curvature (Rf), value of loaded radius of curvature (R) can be calculated using the

37、 Eq. (4). From the calculated value of loaded radius of curvature (R), top roller position (U) can be obtained using the concepts described below.3.1. Concept 1Application of load by lowering the top roller will result in the inward shift of contact point at the bottom roller plate interface (toward

38、s the axis of the central roller). Fig. 1 shows that distance between plate and bottom roller contact point reduces to a from a. Raval (2002) reported that for the larger loaded radius of curvature (R), top roller position (U) is very small, and hence, contact point shift at the bottom roller plate

39、interface can be neglected for simplification (i.e. aa'). Fig. 3 shows the bend plate with uniform radius of curvature (R) between roller plate interfaces X and Y, in the loaded condition. As top roller position (U) is small for the larger loaded radius of curvature(R), in triangle OYX, segment

40、YX can be assumed to be equal to half the center distance between bottom rollers (i.e. a2). So, from triangle OYX in Fig. 3a22+R-U2=R2Simplification of the above equation will result in the form AU2-BU+C=0 （5） where A= (4/R), B = 8 and C = a2/R.From Eq.(5), top roller position (U) can be obtained fo

41、r the loaded radius of curvature（R）,calculated from desired radius of curvature(Rf). Fig.3 Bend plate in loaded condition without considering contact point shift3.2. Concept 2Concept 1 discussed above, neglects the contact point shift at the bottom rollers plate interfaces, whereas concept 2 suggest

42、s the method for the approximation of these contact point shift for the particular top roller position (U). It was assumed that the plate spring back after its exit from the exit side bottom roller and hence between the roller plate interfaces, plate is assumed to be with the uniform radius of curva

43、ture. Then, for the larger loaded radius of curvature (R), length of arc (s) between the points LH in Fig. 4 is assumed to be equal to LH(i.e. a2). In order to obtain contact point shift at bottom roller plate interface, portion of the plate in between the bottom rollers plate interfaces is divided

44、into total N number of small segments defining the nodal points s1, s2, . . .,sN-1 at each segment intersection as shown in Fig. 4. Each small segment of the arc s, i.e. L s1, s2, . . .,sN-1H being the arc length d(s)equal to (a/2)/N) is considered as a straight line at an angle of (/N), (2/N), . .

45、., , respectively with the horizontal. Incremental x and y co-ordinates at each nodal point are calculated using the relationship (Gandhi and Raval, 2006):dxi=dscosi=a2Ncosidyi=dssini=a2Nsini (6)where for total N number of segment (i.e. i=1, 2, . . .,N)i=iNThen, from the summation of x co-ordinates

46、and y co-ordinates of all the nodal points, top roller position (U) for the particular value of loaded radius of curvature (R) can be obtained in two different ways as follows.Fig. 4 Bend plate in loaded condition (assuming the platewith constant radius of curvature between the supports).In Fig. 4,

47、considering the GHO OG2=r12-(a2-X)2=r12-a2-i=iNdxi2 U=Y+JG=i=1Ndyi+r1-OG (7)In Fig. 4, considering the HOLX2 + R U JG2= R2This can be derived to the formU2-2(R + JG)U + X2 + R + JG2-R2 = 0 (8)The contact point shift between the plate and bottom rollers are obtained by shift=a2-X=a2-i=1Nd(x)i (9)3.3.

48、 Concept 3Fig. 5 shows the loaded plate geometry assuming constant loaded radius of curvature (R) between the bottom roller plate interfaces with top roller position (U) and center distance between bottom rollers (a). Relationship of top roller position (U)with other operating parameters viz loaded

49、radius of curvature (R), center distance between bottom rollers (a) and bottom roller radius (r1) considering actual contact point shift can be obtained as discussed below.Fig. 5 Geometry of three-roller bending process.From the OPQ in Fig. 5OQ2 - (OQ - U)2 = PQ2where OQ= R + r1, PQ=a/2 and OQ'=

50、OQ=OP+UExpanding and rearranging, this can be derived to the formR=U2+a28U-r1=4U2-8r1U+a28U (10) Replacing R from Eq. (10) into Eq. (4) and simplifying,U=Gnn+2E2Rf4Gn-1n+2E-3K'tn-1Un-11-v222n (11)whereG = 4U2 - 8r1U + a2Eq. (11) represents the top roller position (U) as a function of final radiu

51、s of curvature (Rf). From Eq. (11), it can be observed that top rolle position (U) is the function of Bottom roller radius (r1) Center distance between bottom rollers (a). Material property parameters (E, v K, and n). Thickness of plate (t). Final radius of curvature (Rf).Assumption of constant radi

52、us of curvature between the roller plate interfaces and plane strain condition has eliminated the effect of top roller radius (rt) and width of the plate (b).4. Development of empirical modelAs described earlier, top roller position (U) is the function of loaded radius of curvature (R), center dista

53、nce between bottom rollers (a), radius of the bottom rollers (r1) and radius of the top roller (rt). Further, loaded radius of curvature (R) can be calculated from the desired final radius of curvature (Rf) considering the spring back. To develop the empirical model, data set were generated from the

54、 geometry for the required top roller position (U) in order to obtain the particular value of loaded radius of curvature (R), with a set of values of center distance between bottom rollers and bottom roller radius. Effect of top roller radius (rt) on top roller position (U) was neglected with the as

55、sumption of no contact point shift at the top roller plate interface (i.e. uniform radius of the supported plate length). Fig. 6 shows the plot of U versus R for the data set for three different bottom roller radiuses (r1) i.e. 95, 90 and 81.5mm. These data sets were generated with top roller radius

56、 (rt) as 105mm, for range of loaded radius of curvature (R) from 1400 to 3800mm; center distance between bottom rollers (a) from 375 to 470mm and bottom roller radius (r1) from 81.5 to 105mm. From these data, correlation for top roller position (U) was derived which is described as follows.From the study of the U versus R plots for the particular machine (with top roller radius (rt) equal to 105mm and bottom roller radius (r1) equal to 81.5 mm), a functional relationship of the form given by Eq. (12) can be assumed.U=cRm (12)Constants (c) and (m) were evaluated using