**64**, 3, pp. 311-350, 2016

### Analysis of the Nonstability States During Bending Processes of Metallic Tubes at Bending Machines Part I. Derivation of the basic expressions and relationships; Part II. Examples of calculations and analysis of some unstability states

ABSTRACT PART II. In this part of the paper, applying epressions derived in Part I, the exemplary calculations of some selected states of loss of stability during cold bending of thin-walled metal tubes at bending machines are presented. The conditions of the dispersed and localised loss of stability together with formation of the plane state of deformation (PSD) under the plane stress state (PSS) and the cracking criterion based on the technological index A5 as the criteria of unstability were assumed. The majority of calculations were performed for a generalised model of strain and simplification of the 3rd type. It appears that they are two extreme cases: one provides the greatest strains and greatest bending angles and the other provides the lowest strains and lowest bending angles [1]. For simplifications of types 1 and 2, values of the bendng angles are included between these extreme values. The calculations were performeded for the top points of the elbow where strains are the greatest and the wall thickness is the smallest. The calculation results were presented as suitable graphs being useful nomograms.

**Keywords**: allowable strains and stresses; bending angles; neutral layer; wall thickness of elbows

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PART II

Śloderbach Z., Rechul Z., Effect of strain hardening and normal anisotropy on allowable values of strain and stress in pipe-bending processes, Journal of Theoretical and Applied Mechanics, 4(38): 843–859, 2000.

Śloderbach Z., The Conditions of instabilitiy in the plastic deformations analysis [in Polish], Publishing House of the Opole University of Technology, ISBN 978-83-60691-85-4, Opole, pp. 1–188, 2010.

El-Sebaie M. G., Mellor P. B., Plastic instability conditions in the deep-drawing of a circular blank of sheet metal, International Journal of Mechanical Science, 14: 535–556, 1972.

Marciniak Z., Limit deformations in sheet metal stamping [in Polish], WNT, Warszawa, 1971.

Marciniak Z., Kołodziejski J., Fundamentals of metal-working processes [in Polish], Warsaw University of Technology, Warszawa, 1983.

Swift H. W., Plastic instability under plane stress, 1952, Journal of the Mechanics and Physics of Solids, 1(1): 176–187, 1952.

Śloderbach Z., Some problems of mechanics in pipeline bending processes [in Polish], Publishing House of the Wrocław University of Technology, ISBN 83-7085-665-9, Wrocław, pp. 1–218, 2002.

Śloderbach Z., Derivations of relations and analysis of tube bending processes using discontinuous fields of plastic strains. Part. I. Derivation of geometric-analytic relationships, Part. II. Discussion and analysis of the obtained results, International Journal of Applied Mechanics and Engineering, 20(2): 417–426, 427–435, 2015.

Śloderbach Z., Genaralized model of strains during bending of metal tubes in bending machines, Journal of Theoretical and Applied Mechanics, 52(4): 1093–1106, 2014.

Hill R., Mathematical Theory of plasticity, Oxford, Clarendon Press, London, 1985.

Olszak W., Perzyna P., Sawczuk A., Theory of plasticity, PWN, Warszawa, 1965.

Szczepiński W., Theory of plastic working of metals [in Polish], PWN, WN Warszawa, 1993.

Życzkowski M., Combined loading in the theory of plasticity, PWN-Nijhoff, Warszawa–Alpen aan den Rijn, 1981.

Gabryszewski Z., Gronostajski J., Fundamentals of metal-working processes [in Polish], WN PWN, Warszawa, 1991.

Moore G. G., Wallace J. F., The effect of anisotropy on instability in sheet- metal forming, Journal of the Institute of Metals, 93(2): 33–38, 1964/1965.

Śloderbach Z, Pajak J: Determination of the thicknesses of pipe elbows on the basis of expressions resulting from EU-Directive, Archive of Applied Mechanics, 85(5): 629–640, 2015.

Franz W. D., Das Kalt-Biegen von Rohren, Springer-Verlag, Berlin, 1961.

Śloderbach Z., Sawicki T., Evaluation of the Spherical Cap Height in a Test of Hydraulic Forming for Certain Instability Conditions [in Polish], Institute of Fundamental Technological Research Reports (IFTR-Reports), No 4/84, Warszawa, 1984.

Śloderbach Z., A model of deformation geometry in pipe bending processes, Engineering Transactions, 47(1): 3–20, 1999.

Erbel S., Kuczyński K., Marciniak Z., Plastic metal-working processes [in Polish], PWN, Warszawa, 1986.

Tang N. C., Plastic-deformation analysis in tube bending, International Journal of Pressure Vessels and Piping, 77(12): 751-759, 2000.

Dobosiewicz J., Wojczyk K., Life prediction for steam pipeline bends [in Polish], Energetyka, No 3/88, 88–90, 1988.

Dzidowski E. S., Influence of the manufacturing and repair technology criteria on the materiale properties applied in power industry [in Polish], Materials of First Conference PIRE ’98, ISBN 83-909539-5-1, Kudowa Zdrój, pp. 51–56, 1998.

Dzidowski E. S., The mezomechanical aspects of reliability of the pipelines elements manufactured by the cold bending processes [in Polish], Materials Four Conference PIRE 2001, ISBN 83-909539-3-5, Lądek Zdrój, pp. 103–110, 2001.

Dzidowski E. S., Strauchold Sz., Effect of technological factors in pipe bending on damage characteristics and reliability of power pipelines [in Polish], Scientific Booklets of the Opole University of Technology, Elektrical Series 6, Opole, 119–125, 1998.

Seyna F., Ginalski J., Diagnosis methods for safe service of master steam pipelines [in Polish], Dozór Techniczny, No 2/99, 31–33, 1999.

Seyna F., Ginalski J., Numerical methods for long-term evaluating residua life of steam pipelines [in Polish], Dozór Techniczny, No 3 /89, 6–13, 1989.

Seyna F., Ginalski J., Conditions for long-term safe Sservice of steam pipelines [in Polish], Energetyka, No. 6, 214–217, 1987.

Marciniak Z. (ed. ), Plastic metal-working processes. Which methods of forming? On cold, hot or half-hot? [in Polish], Mechanik, SIMP, No 2, Warszawa, 1987.

Zdankiewicz M., European Direktive concern a pressure installations. Design recommendations for pressure vessels. Manufacturing [in Polish], Dozór Techniczny, 2/98, 25–33 and 48, 1998.

EN 13445-4 Unfired Pressure Vessels – Part 4: Fabrication (p. 9. 2. 4 Tube Bents, expression 9. 2-4), 2009 or earlier: Draft Standart EN UFPV Unfired Pressure Vessels – Part 4. Manufacture, CEN/TC54/267 JWGB N277, rev. 5, May, 1996.

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Polish Academy of Sciences, Warsaw, Poland