Research Papers

Application of the Continuous and Discontinuous Fields of Plastic Deformations to the Evaluation of the Initial Thickness of Bent Tubes

[+] Author and Article Information
Zdzisław Śloderbach

Department of Applications of
Chemistry and Mechanics,
Opole University of Technology,
Opole 45-758, Poland
e-mail: z.sloderbach@po.opole.pl

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received April 27, 2016; final manuscript received September 28, 2016; published online November 7, 2016. Assoc. Editor: Curt Bronkhorst.

J. Eng. Mater. Technol 139(1), 011009 (Nov 07, 2016) (10 pages) Paper No: MATS-16-1126; doi: 10.1115/1.4034944 History: Received April 27, 2016; Revised September 28, 2016

The paper presents the derived equations for calculations of the initial wall thickness g0 of a tube bent to elbow. The expressions for calculating g0 are presented in a suitable measure of the “great active actual radius Rj” in the bending zone for an exact-generalized solution (continuous fields) and for three formal simplifications (discontinuous fields) of the first-, second-, and third-orders. The expressions to calculate the components of deformation for a generalized solution (continuous fields) are obtained on the basis of kinematically admissible fields of plastic deformations. In any case, a value of initial tube thickness depends on the radius and on the angle of bending αb on the external diameter of the tube, on the displacement of the neutral axis, and on the allowable (required) elbow thickness according to European, American, or other national technical standard or regulations. The initial thickness also depends on the coordinates of the point where the allowable thickness was determined and on the technological–material coefficient of the bending zone range k (defined during the tests). The obtained calculation results are presented in the form of graphs and in table.

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Fig. 1

Scheme of a tube bending machine with a rotating template and a mandrel, (1) mandrel, (2) clamp bolt, (3) rotating template, (4) sliding slat, (5) planisher, (6) bed of sliding slat, (7) bending pipe, and (8) mandrels rod

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Fig. 2

Scheme of the connection of a elbow and a straight section of a pipe when their internal and external diameters are equal, then (g1min  < gt), where gt is the thickness of the tube, rtm is the radius of the template, and m is the straight section of the elbow

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Fig. 3

(a) and (b) The schemes of connections of elbows with straight sections of pipelines: (a) for the case when centering isdone on the external pipe side (both dext are equal), (b) forthe case when centering is done based on the internal pipe side (both dint are equal). In both cases, it should be, that (g0 > g1min > gt). Remaining parameters as on Fig. 2.

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Fig. 4

Geometrical and dimensional quantities pertaining to tube-bending processes

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Fig. 5

Schematic picture of the elbow cross section and its characteristic parameters

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Fig. 6

Variation of the initial thickness versus R for four calculation methods, for α=β=0 deg, kαb=180 deg and (g0=g0ext max,g0′=g0ext max′,g0″=g0ext max″,g0‴=g0ext max‴)

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Fig. 7

Variation of the maximum initial thickness (g0ext′=g0ext max′,g0int′=g0int max′) of the wall of a tube subjected to bending, calculated according to a simplification of the first order and in measures of logarithmic strains, versus the bending radius R, when the required minimum wall thickness of the bent elbow is g1 = 4.5 mm and (α=β=0 deg, kαb=180 deg)

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Fig. 8

Exemplary calculations of longitudinal deformations φ1, depending on the neutral axis displacement y0, where (dzdext)

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Fig. 9

Exemplary calculations of the wall thickness distribution g1, depending on the neutral axis displacement y0, where (dzdext)



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