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 (final) 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 plotted 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 experiments, on a pyramid type three-roller plate-bending machine. Uncertainty analysis of the empirical correlation is repo rted using the McClintock’s method.Keywords: Roller bending,Springback,Analytical study,Empirical modeling, Uncertainty analysis1. IntroductionLarge and medium size tubes and tubular 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 sequentially; 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 multiple 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 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 asingle pass, for which the plate roller forming machine should 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 repeatability 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 elasto-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. 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) discussed 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 the 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 et 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 includes 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 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 desired radius of curvature,for multiple pass three-roller forming of cylinders, considering real material behavior and change of Young’smodulus of elasticity (E) under deformation and shows that the springback is larger than the springback calculated with constant E.Literature review reveals that 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 programming. 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 analysis 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 versus 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 producing 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. Desired curvature in this case is the functionof 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 () and bottom-roller radius () (Raval, 2002). The capacity of the plate bender is defined 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 analysisBending analysis is based on some of the basic assumptions summarized 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 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 andto 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 circumferential 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 is of the form(1)where is the strain associated with the extension of middle surface, the bending strain and ρ 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 the radius of curvature is more than several times the sheet thickness, the neutral surface approximately coincides with the middle surface. If the general stress–strain curve for the material takes the form(2) Then, for the plastic bending, applied moment per unit width can be of the form (Marciniak and Duncan, 1992)(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 several 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 the stress–strain 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 ().• Bottom roller radius, top roller radius and center distance between bottom 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 radius of curvature (R) and desired radius of curvature () can be given by(4)3. Analytical models of top roller position (U) for desired radius of curvature ()For the desired radius of curvature (), value of loaded radius of curvature (R) can be calculated using the 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 (towards the axis of the central roller). Fig. 1shows 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 interface can be neglected for simplification (i.e. a≈). 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 OY’X, segment Y’X can be assumed to be equal to half the center distance between bottom rollers (i.e. ). So, from triangle OY’X in Fig. 3Simplification of the above equation will result in the form（5）where A= (4/R), B = 8 and C = a2/R.From Eq.(5), top roller position (U) can be obtained for the loaded radius of curvature （R）,calculated from desired radius of curvature().Fig.3 Bend plate in loaded condition without considering contact point shift 3.2. Concept 2Concept 1 discussed above, neglects the contact point shift at the bottom rollers plate interfaces, whereas concept 2 suggests 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 curvature. Then, for the larger loaded radius of curvature (R), length of arc (s) between the points L’H in Fig. 4 is assumed to be equal to L’H’(i.e. ). 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 into total N number of small segments defining the nodal points , , . . ., at each segment intersection as shown in Fig. 4. Each small segment of the arc s, i.e. L H being the arc length d(s)equal to ((a/2)/N) is considered as a straight line at an angle of (θ/N), (2θ/N), . . ., θ, respectively with the horizontal. Incremental x and y co-ordinates at each nodal point are calculated using the relationship (Gandhi and Raval, 2006):(6) where for total N number of segment (i.e. i=1, 2, . . .,N)Then, from the summation of ‘x’ co-ordinates 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, considering the GHO(7) In Fig. 4, considering the HOL’This can be derived to the form(8)The contact point shift between the plate and bottom rollers are obtained by(9) 3.3. 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 radius of curvature (R), center distance between bottom rollers (a) and bottom roller radius () considering actual contact point shift can be obtained as discussed below.Fig. 5 Geometry of three-roller bending process.From the OPQ in Fig. 5where , andExpanding and rearranging, this can be derived to the form(10) Replacing R from Eq. (10) into Eq. (4) and simplifying,(11) whereEq. (11) represents the top roller position (U) as a function of final radius of curvature (). From Eq. (11), it can be observed that top rolle position (U) is the function of• Bottom roller radius ()• Center distance between bottom rollers (a).• Material property parameters (E, v K, and n).• Thickness of plate (t).• Final radius of curvature (Assumption of constant radius of curvature between the roller plate interfaces and plane strain condition has eliminated the effect of top roller radius () 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 distance between bottom rollers (a), radius of the bottom rollers () and radius of the top roller (). Further, loaded radius of curvature (R) can be calculated from the desired final radius of curvature () considering the spring back. To develop the empirical model, data set were generated from the 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 andbottom roller radius. Effect of top roller radius () on top roller position (U) was neglected with the assumption of no contact point shift at the top roller plate interface (i.e. uniform radius of the supported plate length). Fig. 6shows the plot of U versus R for the data set for three different bottom roller radiuses () i.e. 95, 90 and 81.5mm. These data sets were generated with top roller radius () 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 () 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 () equal to 105mm and bottom roller radius () equal to 81.5 mm), a functional relationship of the form given by Eq. (12)can be assumed.(12) Constants (c) and (m) were evaluated using method of least square. For the different center distance between bottom rollers (a) i.e. 375, 390, 405, 425, 440, 455 and 470mm, values of constants (c) and (m) were found to be different. Hence, variation of constant (c) and (m) were plotted against center distance (a) as shown in Figs. 7 and 8. The top roller position (U) is derived with new constant () and ().，(13)Fig. 6 –U vs. R for different bottom roller radius () and centerdistances between bottom rollers (a), = 105mm.Fig. 7 – Constant c for different center distancebetween bottom rollers (a), = 105mm.where constants and were obtained as a function of center distance between bottom rollers (a).Similarly, from the U versus R plots for the other machines with top roller radius equal to 105mmand bottom roller radius equal to 90, 95, 100 and 105mm, theempirical equation for top roller position (U) was derived in the form given by Eq.(13).Where, constants () and () were obtained as a function of center distance between bottom rollers (a), for the different machines and are presented in Table 1.,So, unified empirical equation considering all different machines can be obtained as below(14) where P, Q and S are constants, which depend on bottom roller radius (). From the P versus , Q versus and S versus plots, constants P, Q and S were obtained by applying the generalized method of least square and method of differential corrections (Devis, 1962) to the generated dataset, as a function of bottom roller radius () given by Eqs. (15)–(17)(15)(16)(17) Replacing P, Q and S from Eqs. (15)–(17) into Eq. (14)(18)Fig. 8 – Constant m for different center distancebetween bottom rollers (a), = 105mm.Assuming the unit width of the plate and plane strain condition, from Eqs. (3) and (4)(19)Replacing R from Eq. (19) into Eq. (18)(20) Eq. (20)is the empirical equation for top roller position (U) considering contact point shift, where 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 ().Eq. (20)is the generalized equation of top roller position (U) as its derivation is based on the trend equations and it is applicable to any range of the parameters under consideration. Further from Table 1, for the range of bottom roller radius () from 81.5 to 105mm, range of variation of P, Q and S was observed to be 0.0636–0.0593, 2.0673–2.0651 and 0.9631–0.952, respectively. By averaging the P, Q and S, top roller position (U) can be derived to the form given by Eq. (21), which neglects the effect of bottom roller radius () and is applicable to the machine with the range of the bottom roller radius from 81.5 to 105mm.（21）Analytical model developed under present work is based on the assumption of constant radius of curvature between the bottom roller plate interfaces. However, in actual practice, plate has been observed with the varying radius of curvature between the roller supports due to nonsymmetrical moment distribution around the top roller axis. Research reported by Hensen and Jannerup (1979) has described the curvature functions for finding the varying curvature between the roller supports in loaded condition. However, in the derivation, as none of the essential variables can be expressed as explicit functions of input quantities describing geometry and material characteristics, calculations were possible only with multi-loop iterative procedure. Hence, to avoid the multi-loop iterative procedure, empirical model as described above is useful to predict/obtain top roller position (U). With the help of the experimental data on the curvature distribution for plate in loaded condition, empirical model for the top roller position(U) can be developed as per the procedure discussed in foregoing sections. This will include the effect of top rollerplate contact point shift and will lead to the more accurate prediction of top roller position for desired radius of curvature.5． Uncertainty analysisThe uncertai nty analysis is carried out in accordance with the McClintock’s method with the following assumed uncertainties in the various parameters:• Uncertainty in strain hardening exponent (n) =±10%.• Uncertainty in strength coefficient (K, N/mm2) =±15%.•Uncertainty in thickness of plate =±0.29mm (5mm≤t < 8 mm), ±0.32mm (8mm≤t<10mm), ±0.35mm (10mm≤t<12mm) and ±0.39mm (12mm≤t<15mm).• Uncertainty in center distance between bottom rollers (a) =±1mm.• Uncertainty in loaded radius (R, mm)=±1%.Uncertainty in strain hardening exponent (n) is assumed based on its variation with percentage elongation at 2 and 8-in. gauge length where as, uncertainty in strength coefficient(K) is assumed based on its variation over the range of the tensile strength for different grades of the carbon manganese steel as per ASME Section 2 (ASME, 2001a). Uncertainty in the loaded radius (R) is assumed based on the rules for the construction of pressure vessel as per ASME Section 8 (ASME, 2001b). Uncertainty in thickness is assumed based on the thickness tolerances of hot rolled steel plates for5–20mm thickness as per DIN 1016 (DIN, 1987). As the center distance between bottom rollers was set with the help of the scale having least count of 1mm, its uncertainty is assumed to be equal to±1mm. The resultant uncertainties in the top roller positions are found to be in the range of .6． ConclusionDeveloped analytical and empirical models were verified with the experiments on three-roller cylindrical bending. Following important conclusions were derived out of the reported work:(1) Analytical model based on concept 3, simplifies the calculation procedure for the machine-setting parameters as it expresses the top roller position as an explicit function of desired radius of curvature.(2) Agreement of empirical results with that of the experiments and analytical results based on concept 3 proves the correctness of the procedure.(3) For the small to medium scale fabricators, where the volume of production does not permit the acquisition of automated close loop control systems, developed models can be proved to be simple tool for the first hand estimation of machine setting parameters for required product dimensions.(4) Consideration of effect of initial strain and change of modulus of elasticity during deformation on spring back, in analytical/empirical model will further improve the accuracy of prediction of top roller position.(5) Further, empirical model based on the experimental loaded curvature distribution between roller supports would consider the top roller-plate contact point shift and will lead to more accurate prediction of top roller position. AcknowledgementsThe authors gratefully acknowledge the assistance provided by Dr. E.V. Ramakrishnan, Professor and Head, Department of English, V.N. South Gujarat University, India for English language editing. Authors are also thankful to the reviewers whose learned comments have helped a lot in improving the quality of this work.1.ASME Boiler and Pressure Vessel Code, 2001. Section 2,Part A and D.2.ASME Boiler and Pressure Vessel Code, 2001. Section 8, Division 1.3.Devis, D.S., 1962. Nomography and Empirical Equations, 2nd ed. Reinhold Publishing Corporation, NY, USA.4.DIN 1016, 1987. Steel Flat Products; Hot Rolled Sheet and Strip; Limit Deviations, Form and Mass Tolerances, Revision 87.5.Ditter, G.E., 1979. Mechanical Behavior of Materials under Tension Mechanical Metallurgy, 2nd ed. Mc-Graw Hill, NY, USA, pp. 329–348.6.Gandhi, A.H., Raval, H.K., 2005. Stress strain curve for multiple pass loading of ductile material. In: Proceedings of the International Conference on Recent Advances in Mechanical & Materials Engineering, Kuala Lumpur, Malaysia, May 30–31, pp. 175–180.7.Gandhi, A.H., Raval, H.K., 2006. Analytical modeling of top roller position for multiple pass (3-roller) cylindrical forming of plates, in: Proceedings of International Mechanical Engineering Congress and Exposition, Chicago, IL, USA, November 5–10, Paper no. IMECE2006-14279.8.Hardt, D.E., Roberts, M.A., Stelson, K.A., 1982. Closed-loop shape control of a roll-bending process. J. Dynam. Syst. Meas. Control 104, 317–321.9.Hardt, D.E., Constantine, E., Wright, A., 1992. A model of sequential bending process for manufacturing simulation. J. Eng. Ind. Trans. ASME 114, 181–187.10.Hensen, N.E., Jannerup, O., 1979. Modeling of elastic-plastic bending of beams using a roller bending machine. J. Eng. Ind., Trans. ASME 101, 304–310.11.Hosford, W.F., Caddell, R.M., 1993. Metal Forming Mechanics and Metallurgy. PTR Prentice Hall, NJ, USA12.Hua, M., Sansome, D.H., Baines, K., 1995. Mathematical modeling of the internal bending moment at the top roll contact in multi-pass four-roll thin-plate bending. J. Mater. Process. Technol. 52, 425–459.13.Hua, M., Baines, K., Cole, I.M., 1999. Continuous four-roll plate bending: a production process for the manufacture of single seamed tubes of large and medium diameters. Int. J. Mach. Tools Manuf. 36, 905–935.14.Kajrup, G., Flamholz, A., 2003. Bending the wind-roll bending challenges in fabricating conical cylinder wind towers, The Fabricator, August 14, ().15.Marciniak, Z., Duncan, J.L., 1992. The Mechanics of Sheet Metal Forming. Edward Arnold Ltd., London.16.Raval, H.K., 2002. Experimental & theoretical investigation of bending process, (vee & three roller bending), PhD Thesis, South Gujarat University, India.17.Shin, J.G., Park, T.J., Yim, H., 2001. Kinematics based determination of the rolling region in roll bending for smoothly curved plates. J. Manuf. Sci. Eng. 123, 284–290. Sidebottom, O.M., Gebhardt, C.F., 1979. Elastic springback in plates and beams formed by bending, Exp. Mech., 371–376.18.Yang, M., Shima, S., 1988. Simulation of pyramid type three roller bending process. Int. J. Mech. Sci. 30 (12), 877–886.19.Yang, M., Shima, S., Watanabe, T., 1990. Model base control for three roller bending process of channel bar. J. Eng. Ind. Trans. ASME 112, 346–351.。