Bending Instabilities Of Elastic Tubes

International Journal of Solids and Structures 39 (2002) 2059–2085 www.elsevier.com/locate/ijsolstr

Bending instabilities of elastic tubes
Spyros A. Karamanos *
Department of Mechanical and Industrial Engineering, University of Thessaly, 38334 Volos, Greece Received 25 November 2001; received in revised form 8 December 2001

Abstract The present paper examines instabilities of long thinelastic tubes. Both initially straight and initially bent tubes are analyzed under in-plane bending. Tube response, a combination of ovalization instability and bifurcation instability (buckling), is investigated using a nonlinear finite element (FE) technique, which employs polynomial functions in the longitudinal tube direction and trigonometric functions to describe cross-sectional deformation. Itis demonstrated that the interaction between the two instability modes depends on the value and the sign of the initial tube curvature. The ovalization of initially bent tubes is examined in detail and, in particular, the case of opening moments. Furthermore, the paper emphasizes on bifurcation instability. It is shown that buckling may occur prior to or beyond the ovalization limit point,depending on the value of the initial curvature. Using the nonlinear FE formulation, the location of bifurcation on the primary path is detected, post-buckling equilibrium paths are traced, and the corresponding wavelengths of the buckled configurations are calculated. Moreover, results over a wide range of initial curvature values are presented, extending the findings of previous works. Finally, severalanalytical approaches, introduced in previous research works, are also employed to estimate the moments causing ovalization and bifurcation instability. These approaches are based on nonlinear flexible shell theory or simplified ring analysis. The efficiency and accuracy of those analytical methods with respect to the nonlinear FE formulation are examined. Ó 2002 Elsevier Science Ltd. All rightsreserved.
Keywords: Tube; Pipe bend; Stability; Ovalization; Buckling; Post-buckling; Bifurcation

1. Introduction and literature review The main characteristic of a tube under bending is the distortion (ovalization) of its cross-section (Fig. 1a) because of the inward stress components rv (Fig. 1b). The ovalizing mechanism results in loss of stiffness in the form of limit point instability, referredto as ‘‘ovalization instability’’. Furthermore, the increased axial stress at the compression side due to ovalization may cause bifurcation instability (buckling) in a form of longitudinal wavy-type ‘‘wrinkles’’ usually before a limit moment is reached (Fig. 1a). In the present paper, ovalization and bifurcation instabilities of thin elastic tubes are examined (Fig. 2). The tubes consideredherein are thin, with cross-sectional radius-to-thickness ratio (r=t) greater than 100,

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Tel.: +30-4210-74086; fax: +30-4210-74009. E-mail address: skara@mie.uth.gr (S.A. Karamanos).

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S.A. Karamanos / International Journal of Solids and Structures 39(2002) 2059–2085

Fig. 1. Schematic representation of (a) ovalization vs. bifurcation instability and (b) ovalization mechanism, because of inward stress components rv .

Fig. 2. (a) Tube geometry and (b) loading conditions in initially bent tubes.

and exhibit elastic response (plasticity effects are not considered). The tubes may be initially straight or initially bent with a longitudinalradius of curvature R significantly larger than the cross-sectional radius r ðR=r > 80). The following parameter is employed to describe tube geometry: k¼ r2 Rt pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À l2 ð1Þ

where t is the tube thickness and is the Poisson’s ratio (Fig. 2). Tubes may be subjected to closing moments (‘‘positive’’ bending) or opening moments (‘‘negative’’ bending) as shown in Fig. 2. The present work...