大学本科生毕业设计外文翻译1Analytical and empirical modeling of top roller position for three-roller cylindrical bending of plates and its experimental verificationA.H. Gandhia, H.K. Raval bAbstract：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 reported using the McClintocks method.Keywords: Roller bending Spring back Analytical study Empirical modelingUncertainty 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). 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 大学本科生毕业设计外文翻译2the 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 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。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 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 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.大学本科生毕业设计外文翻译3Fig. 1 Schematic diagram of Fig2 Deformation in fiber AB0three-roller bending process. 2. Bending analysis Bending 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 bending大学本科生毕业设计外文翻译4In thin sheets, normal section may be considered to remain plane on bending and 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 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 AB0 is of the form(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 the ra