The paper discusses nonlinear equations of in-plane bending of curved tubes formulated by E. Reissner in terms of two unknown functions and two unknown parameters. To solve the equations, a numerical method based on the finite-difference approximations and Newton–Raphson iteration technique is proposed. Deformations and stresses in tubes of circular and noncircular cross sections are studied for a wide range of geometrical parameters. The accuracy of the equations is evaluated by comparing the numerical results with predictions obtained by a special shell finite element.
Issue Section:
Research Papers
References
References
1.
Reissner
, E.
1981
, “On Finite Pure Bending of Curved Tubes
,” Int. J. Solids Struct.
, 17
(9
), pp. 839
–844
.10.1016/0020-7683(81)90098-62.
Dubyaga
, K. M.
1909
, “The Bending of Thin-Walled Curved Tubes
,” Izv. S.-Peterburgskogo Politekh. Inst.
, 11
, pp. 663
–675
.3.
Bantlin
, A.
1910
, “Formanderung und Beanspruchung Federnder Ausglichrohren
,” VDI Z. (1857–1968)
, 54
(2
), pp. 43
–49
.4.
Karman
, Th. von
1911
, “Uber die Formanderung Dunnwandiger Rohre Insbesonders Federnder Angleschrohre
,” VDI Z. (1857)–(1968)
, 55
, pp. 1889
–1895
.5.
Lorenz
, H.
1912
, “Die Biegung Krummer Rohre
,” Phy. Z.
, 13
(16
), pp. 768
–774
.6.
Hovgaard
, W.
1926
, “Bending of a Curved Tube of Circular Cross Section
,” Proc. Natl. Acad. Sci. U.S.A.
, 12
(6
), pp. 365
–370
.10.1073/pnas.12.6.3657.
Karl
, H.
1943
, “Biegung Gekrummer, Dunnwandiger Rohre
,” ZAMM
, 23
, pp. 331
–345
.10.1002/zamm.194302306048.
Beskin
, L.
1945
, “Bending of Curved Thin Tubes
,” Trans. ASME
, 67
, pp. A1
–A7
.9.
Reissner
, E.
1949
, “On Bending of Curved Thin-Walled Tubes
,” Proc. Natl. Acad. Sci. U.S.A.
, 35
, pp. 204
–288
.10.1073/pnas.35.4.20410.
Clark
, R. A.
Reissner
, E.
1951
, “Bending of Curved Tubes
,” Adv. Appl. Mech.
, 2
, pp. 93
–122
.10.1016/S0065-2156(08)70299-011.
Kafka
, P. G.
Dunn
, M. B.
1956
, “Stiffness of Curved Circular Tubes With Internal Pressure
,” Trans. ASME. Ser. E: J. Appl. Mech.
, 23
(2
), pp. 247
–254
.12.
Rodabaugh
, E. C.
George
, H. H.
1957
, “Effect of Internal Pressure on Flexibility and Stress Intensification Factors of Curved Pipe or Welding Elbows
,” Trans ASME. Ser. E: J. Appl. Mech.
, 79
(4
), pp. 939
–948
.13.
Crandall
, S. H.
Dahl
N. C.
1957
, “The Influence of Pressure on Bending of Curved Tubes
,” Proc 9th Congr. Appl Mech. Bruxelles
, 6
, pp. 101
–111
.14.
Kostovetskii
, D. L.
1960
, “The Effect of Initial Ellipticity of the Cross Section on Bending of Curved Thin-Walled Tube
,” Energomashinostroenie
, 3
, pp. 23
–27
.15.
Jones
, N.
1967
, “In-Plane Bending of Short-Radius Curved Pipe Bend
,” Trans. ASME, Ser. B
, 89
(2
), pp. 271
–277
.10.1115/1.361003916.
Cheng
, D. H.
Thailer
, H. J.
1968
, “In-Plane Bending of Curved Circular Tubes
,” Trans. ASME. Ser. B
, 90
(4
), pp. 666
–670
.10.1115/1.360470617.
Cheng
, D. H.
Thailer
, H. J.
1970
, “On Bending of Curved Circular Tubes
,” Trans. ASME. Ser. B
, 92
(1
), pp. 62
–66
.10.1115/1.342772018.
Chernykh
, K. F.
1960
, “Saint-Venants’s Problem for Thin-Walled Tubes With a Circular Axis
,” Prikl. Mat. Mekh.
, 24
(3
), pp. 423
–432
.19.
Cherniy
, V.P.
2001
, “Effect of Curved Bar Properties on Bending of Curved Pipes
,” Trans. ASME
, 68
, pp. 650
–655
.10.1115/1.135751820.
Timoshenko
, S.
1923
, “Bending Stresses in Curved Tubes of Rectangular Cross Section
,” Trans. ASME
, 45
(1893
), pp. 135
–140
.21.
De Melo
, F.J.M.Q.
Vaz
, M.A.P.
1992
, “The Linear Elastic In-Plane Bending of Curved Hollow Profiles Having a Thin-Walled Rectangular Section
,” J. Strain Anal.
, 27
(3
), pp. 145
–149
.10.1243/03093247V27314522.
Huber
, M. T.
1948
, “The Bending of the Curved Tubes of Elliptic Section
,” Proc. VIIth Int. Congr. Appl. Mech. London
, 1
, pp. 322
–328
.23.
Olesiak
, Z.
1955
, “Stan Naprezenia i Odksztalcenia Sprezystego Zginanej Rury Cienkosciennej o Przekroju Soczewkowatym
,” Arch. Mech. Stosow.
, 7
(4
), pp. 503
–518
.24.
Whatham
, J. F.
1987
, “Analysis of Pipe Bends With Symmetrical Noncircular Cross Sections
,” Trans. ASME. Ser. E: J. Appl. Mech.
, 54
(3
), pp. 604
–610
.10.1115/1.317307625.
Brazier
, L. G.
1927
, “On the Flexure of Thin Cylindrical Shells and Other ‘Thin’ Sections
,” Proc. R. Soc., Ser. A
, 116
(773
), pp. 104
–114
.10.1098/rspa.1927.012526.
Wood
, J. D.
1958
, “The Flexure of a Uniformly Pressurized Circular, Cylindrical Shell
,” Trans. ASME. Ser. E: J. Appl. Mech.
, 25
(4
), pp. 453
–458
.27.
Chwalla
, E.
1933
, “Reine Biegung Schlanker, Dunnwandiger Rohre mit Gerader Achse
,” ZAMM
, 13
(1
), pp. 48
–53
.10.1002/zamm.1933013010828.
Heck
, O. S.
1937
, “Uber der Stabilitat Orthotroper Elliptischer Zylinderschalen bei Reiner Biegung
,” Luftfahrtforschung
, 14
(3
), pp. 137
–147
.29.
Konovalov
, Yu. V.
Bending of Infinite Cylindrical Shell
,” Prikl. Mat. Mekh.
, 4
(5
), pp. 35
–54
.30.
Reissner
, E.
1961
, “On Finite Pure Bending of Cylindrical Tubes
,” Osterr. Ing.-Arch.
, 15
, pp. 165
–172
.31.
Reissner
, E.
Weinitschke
, H. J.
1963
, “Finite Pure Bending of Circular Tubes
,” Q. Appl. Math.
, 20
(4
), pp. 305
–319
.32.
Reissner
, E.
Weinitschke
, H. J.
1966
, “Corrections to “Finite Pure Bending of Circular Tubes
,” Q. Appl. Math.
, 23
(4
), p. 368
.33.
Perrone
, N.
Kao
, R.
1971
, “A General Nonlinear Relaxation Iteration Technique for Solving Nonlinear Problems in Mechanics
,” Trans. ASME. Ser. E: J. Appl. Mech.
, 38
(2
), pp. 371
–376
.10.1115/1.340878534.
Na
, T. Y.
Turski
, C. E.
1974
, “Solution of the Nonlinear Differential Equations for Finite Bending of Thin-Walled Tubes by Parameter Differentiation
,” Aeronaut. Q.
, 25
(1
), pp. 14
–18
.35.
Thurston
, G. A.
1977
, “The Critical Bending Moment of Circular Cylindrical Tubes
,” Trans. ASME. Ser. E: J. Appl. Mech.
, 44
, pp. 173
–175
.10.1115/1.342399036.
Gellin
, S.
1980
, “The Plastic Buckling of Long Cylindrical Shells Under Pure Bending
,” Int. J. Solids Struct.
, 16
(5
), pp. 397
–407
.10.1016/0020-7683(80)90038-437.
Spence
, J.
Toh
, L. H.
1979
, “Collapse of Thin Orthotropic Elliptical Cylindrical Shells Under Combined Bending and Pressure Loads
,” Trans ASME. Ser. E: J. Appl. Mech.
, 46
, pp. 363
–371
.10.1115/1.342455638.
Reissner
, E.
1959
, “On Finite Bending of Pressurized Tubes
,” J. Appl. Mech.
, 26
, pp. 386
–392
.39.
Kostovetskii
, D. L.
1960
, “Bending of Thin-Walled Curved Pipes in the Regions of Large Elastic Deformations
,” Izv. Akad. Nauk SSSR, OTN Mekh. i Mash.
, 3
, pp. 49
–54
.40.
Axelrad
, E. L.
1960
, “Nonlinear Equations of Shells of Revolution and of Bending of Thin-Walled Beams
,” Izv. Akad. Nauk SSSR, OTN Mekh. i Mash., 4
, pp. 84–92.41.
Axelrad
, E. L.
1985
, “Elastic Tubes-Assumptions, Equations, Edge Conditions
,” Thin-Walled Struct.
, 3
(3
), pp. 193
–215
.10.1016/0263-8231(85)90003-542.
Boyle
, J. T.
1981
, “The Finite Bending of Curved Pipes
,” Int. J. Solids Struct.
, 17
(5
), pp. 515
–529
.10.1016/0020-7683(81)90058-543.
Cook
, R. D.
1989
, “Axisymmetric Finite Element Analysis for Pure Moment Loading of Curved Beams and Pipe Bends
,” Comput. Struct.
, 33
(2
), pp. 483
–487
.10.1016/0045-7949(89)90021-744.
Di Pasquantonio
, F.
Macchi
, A.
Villa
, A.
1977
, “Calcolo col Metodo Degli, Elementi Finiti del Coefficientedi Fessibita dei Tubi Curbi Piani
,” Energ. Elettr.
, 54
(4
), pp. 177
–183
.45.
Boussaa
, Dj.
Dang
, K.
Van
Labbe
, P.
Tang
, H. T.
1996
, “Finite Pure Bending of Curved Pipes
,” Comput. Struct.
, 60
(6
), pp. 1003
–1012
.10.1016/0045-7949(96)00011-946.
Kuznetsov
, V. V.
Levyakov
, S. V.
2001
, “Nonlinear Pure Bending of Toroidal Shells of Arbitrary Cross Section
,” Int. J. Solids Struct.
, 38
, pp. 7343
–7354
.10.1016/S0020-7683(00)00411-X47.
Axelrad
, E. L.
1965
, “Refinement of the Upper Critical Loading of Pipe Bending Taking Account of the Geometrical Nonlinearity
,” Izv. Akad. Nauk SSSR, OTN Mekh. i Mash.
, 4
, pp. 123
–139
.48.
Whatham
, J. F.
1982
, “Analysis of Circular Pipe Bends With Flanged Ends
,” Nucl. Eng. Des.
, 72
(2
), pp. 175
–187
.10.1016/0029-5493(82)90213-849.
Abramowitz
, M.
Stegun
, I. A.
1972
. Handbook of Mathematical Functions With Formulas, Graphs, and Tables
, National Bureau of Standards
, Dover Publications, New York.Copyright © 2014 by ASME
You do not currently have access to this content.
