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Archive of Applied Mechanics

, Volume 87, Issue 6, pp 927–947 | Cite as

Thick isotropic curved tubes: three-dimensional stress analysis

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Abstract

While the structural analysis of straight beams is straightforward, the behavior of curved beams is more complex to predict. In the present work, a displacement approach of toroidal elasticity is used to analyze thick isotropic curved tubes subjected to axial load, torque, and bending moment. The governing equations are developed in a toroidal coordinate. The method of successive approximation is used to obtain the general solution. The accuracy of the present methodology is tested comparing the numerical results with those obtained by finite element method (FEM) and stress-based toroidal elasticity (SBTE). The proposed methodology is computationally cost-effective, and its results reveal good agreements with FEM and SBTE results. Finally, several numerical examples of stress distributions in thick isotropic curved tubes under axial load, torque, and bending moment are presented. By using the present methodology, displacements as well as stresses are obtained which are important information for fracture analysis.

Keywords

Thick curved tubes Toroidal elasticity Displacement-based solution Successive approximation Mechanical load 

List of symbols

ab

Inside and outside cross-sectional radius of the curved tube

c

Reference length

\({{a_{n}, b_{n}, c_{n}, d_{n}, e_{n}, f_{n}}}\)

Constants of the nth-order complementary solution

\({{A_{ni}, B_{ni}, C_{ni}}}\)

Constants of the nth-order particular solution

E

Young’s modulus

G

Shear’s modulus

\(F_{z}\), \(T_{z}\), \({M}_{{y}}\)

Axial load, torque and bending moment

R

Mean radius of an isotropic curved tube

\({u}_{\varsigma }\), \({u}_{\phi }\), \({u}_{\theta }\)

Displacement components in toroidal coordinates

uvw

Non-dimensional displacement components

\({{{u}_{k}, v_{k}, w_{k}}}\)

The kth-order non-dimensional displacement components

UVW

The first part of Navier function

\({\bar{U}} ,{\bar{V}} , \bar{W}\)

The second part of Navier function

\({\hat{U}} , {\hat{V}} , \hat{W}\)

The third part of Navier function

\({{U_{k}, V_{k}, W_{k}}}\)

The first part of Navier function of the kth order

\({\bar{U}}_{{k}}\), \({\bar{V}}_{{k}}\), \(\bar{W}_{{k}}\)

The second part of Navier function of the kth order

\({\hat{U}}_{{k}}\), \({\hat{V}}_{{k}}\), \(\hat{W}_{{k}}\)

The third part of Navier function of the kth order

\(\varepsilon \)

a / R

\(\upsilon \)

Poisson’s ratio

\(\varsigma , \phi , \theta \)

Toroidal coordinates

\(\tau _{\varsigma \varsigma } ,\tau _{\phi \phi } ,\tau _{\theta \theta } \)

Non-dimensional normal stress components

\(\tau _{\varsigma \phi } ,\tau _{\phi \theta } ,\tau _{\varsigma \theta } \)

Non-dimensional shear stress components

\(\tau _{\varsigma \varsigma k} ,\tau _{\phi \phi k} ,\tau _{\theta \theta k} \)

The kth-order non-dimensional normal stress components

\(\tau _{\varsigma \phi k} ,\tau _{\phi \theta k} ,\tau _{\varsigma \theta k} \)

The kth-order non-dimensional shear stress components

References

  1. 1.
    Asadi, H., Akbarzadeh, A.H., Chen, Z.T., Aghdam, M.M.: Enhanced thermal stability of functionally graded sandwich cylindrical shells by shape memory alloys. Smart Mater. Struct. 24(4), 045022 (2015)CrossRefGoogle Scholar
  2. 2.
    Akbarzadeh, A., Chen, Z.: Thermo-magneto-electro-elastic responses of rotating hollow cylinders. Mech. Adv. Mater. Struct. 21(1), 67–80 (2014)CrossRefGoogle Scholar
  3. 3.
    Von Karman, T.: Uber die Formanderung Dunnwandiger Rohre, insbesondere federnder Ausgleichsrochre. VDI-Zeitdchriff 55, 1889–1895 (1911)Google Scholar
  4. 4.
    Brazier, L.G.: On the flexure of thin cylindrical shells and other thin sections. Proc. R. Soc. Lond. A 116(773), 104–114 (1927)CrossRefMATHGoogle Scholar
  5. 5.
    Ting, T.C.T.: Pressuring, shearing, torsion and extension of a circular tube or bar of cylindrically anisotropic material. Proc. R. Soc. Lond. A 452, 2397–2421 (1996)CrossRefMATHGoogle Scholar
  6. 6.
    Ting, T.C.T.: New solution to pressuring, shearing, torsion and extension of cylindrically anisotropic elastic circular tube or bar. Proc. R. Soc. Lond. A 455, 3527–3542 (1999)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, T., Chung, C.T., Lin, W.L.: A revisit of a cylindrically anisotropic tube subjected to pressuring, shearing, torsion and extension and a uniform temperature change. Int. J. Solids Struct. 37, 5143–5159 (2000)CrossRefMATHGoogle Scholar
  8. 8.
    Boyle, J.T.: The finite bending of curved pipes. Int. J. Solids Struct. 17, 515–529 (1981)CrossRefMATHGoogle Scholar
  9. 9.
    Reissner, E.: On finite pure bending of curved tubes. Int. J. Solids Struct. 17(9), 839–844 (1981)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Emmerling, F.A.: Flexible Shells. Springer, Berlin (1984)MATHGoogle Scholar
  11. 11.
    Lang, H.A.: The theory of toroidal elasticity. Res Mech. Int. J. Struct. Mech. Mater. Sci. 26(4), 289–351 (1989)Google Scholar
  12. 12.
    Gohner, O.: Schubspannungsverteilung im Querschnitt einer Schraubenfeder. Ing. Arch. 1, 619–664 (1930)CrossRefMATHGoogle Scholar
  13. 13.
    Ancker, C.J., Goodier, J.N.: Pitch and curvature corrections for helical springs. 58-APM, ASME (1958)Google Scholar
  14. 14.
    Kornecki, A.: Stress distribution in a pressurized thick-walled toroidal shell—a three dimensional analysis. College of Aeronautics, Cranfield, England, Note 137 (1963)Google Scholar
  15. 15.
    McGill, D.J., Rapp, I.H.: Axisymmetric stresses and displacements in thick-walled elastic torus. J. Eng. Mech. Div. 99(3), 629–633 (1973)Google Scholar
  16. 16.
    Redekop, D.: A displacement solution in toroidal elasticity. Int. J. Press. Vessels Pip. 51, 1–21 (1992)CrossRefGoogle Scholar
  17. 17.
    Zhu, Y., Redekop, D.: An out-of-plane displacement solution in toroidal elasticity. Int. J. Press. Vessels Pip. 58(3), 309–319 (1994)CrossRefGoogle Scholar
  18. 18.
    Eric, R.: On finite bending of pressurized tubes. Trans. ASME 26, 386–392 (1959)Google Scholar
  19. 19.
    Madureira, L.R., Melo, F.Q.: Hybrid formulation solution for stress analysis of curved pipes with welded bending joints. Eng. Fract. Mech. 77(15), 2992–2999 (2010)CrossRefGoogle Scholar
  20. 20.
    Fonseca, E.M.M., de Melo, F.J.M.Q.: Numerical solution of curved pipes submitted to in-plane loading conditions. Thin Walled Struct. 48(2), 103–109 (2010)CrossRefGoogle Scholar
  21. 21.
    Kolesnikov, A.M.: Large bending deformations of pressurized curved tubes. Arch. Mech. 63(5–6), 507–516 (2011)Google Scholar
  22. 22.
    Christo Michael, T., Veerappan, A.R., Shanmugam, S.: Effect of ovality and variable wall thickness on collapse loads in pipe bends subjected to in-plane bending closing moment. Eng. Fract. Mech. 79, 138–148 (2012)CrossRefGoogle Scholar
  23. 23.
    Zhu, Y., Luo, X.Y., Wang, H.M., Ogden, R.W., Berry, C.: Three-dimensional non-linear buckling of thick-walled elastic tubes under pressure. Int. J. Non-Linear Mech. 48, 1–14 (2013)CrossRefGoogle Scholar
  24. 24.
    Levyakov, S.V.: Evaluation of Reissner’s equations of finite pure bending of curved elastic tubes. J. Appl. Mech. 81(4), 1–9 (2013)CrossRefGoogle Scholar
  25. 25.
    Yudo, H., Yoshikawa, T.: Buckling phenomenon for straight and curved pipe under pure bending. J. Mar. Sci. Technol. 20(1), 94–103 (2014)CrossRefGoogle Scholar
  26. 26.
    Alashti, R.A., Ahmadi, S.A.: Buckling of imperfect thick cylindrical shells and curved panels with different boundary conditions under external pressure. J. Theor. Appl. Mech. 52(1), 25–36 (2014)Google Scholar
  27. 27.
    Hamdaoui, M.E.I., Merodio, J., Ogden, R.W., Rodríguez, J.: Finite elastic deformations of transversely isotropic circular cylindrical tubes. Int. J. Solids Struct. 51(5), 1188–1196 (2014)CrossRefGoogle Scholar
  28. 28.
    Djamaluddin, F., Abdullah, S., Ariffin, A.K., Nopiah, Z.M.: Non-linear finite element analysis of bitubal circular tubes for progressive and bending collapses. Int. J. Mech. Sci. 99, 228–236 (2015)CrossRefGoogle Scholar
  29. 29.
    Rodríguez, J., Merodio, J.: Helical buckling and post buckling of pre-stressed cylindrical tubes under finite torsion. Finite Elem. Anal. Des. 112, 1–10 (2016)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Yazdani Sarvestani, H., Hoa, S.V., Hojjati, M.: Three-dimensional stress analysis of orthotropic curved tubes-part 1: single-layer solution. Eur. J. Mech. A Solids 60, 327–338 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Yazdani Sarvestani, H., Hojjati, M.: Three-dimensional stress analysis of orthotropic curved tubes-part 2: laminate solution. Eur. J. Mech. A Solids 60, 339–358 (2016)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Yazdani Sarvestani, H., Hojjati, M.: Failure analysis of thick composite curved tubes. Compos. Struct. 160, 1027–1041 (2017)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Bioresource EngineeringMcGill UniversityMontrealCanada

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