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Research Papers: Design and Analysis

Plastic Instability Pressure of Toroidal Shells

[+] Author and Article Information
Vu Truong Vu

Department of Mechanical Engineering, University of Liverpool, Liverpool L69 3GH, UKvutruongvu@gmail.com

J. Blachut

Department of Mechanical Engineering, University of Liverpool, Liverpool L69 3GH, UKem20@liverpool.ac.uk

J. Pressure Vessel Technol 131(5), 051203 (Jul 28, 2009) (10 pages) doi:10.1115/1.3148824 History: Received February 07, 2009; Revised May 08, 2009; Published July 28, 2009

Abstract

This paper considers the determination of plastic instability pressure in toroidal shells under internal uniform pressure. Analytical and numerical approaches, as well as verification by experiments, are presented. This work is inspired by Mellor’s treatment (1983, Engineering Plasticity, Ellis Horwood Ltd., Chichester; 1960, “The Ultimate Strength of Thin-Walled Shells and Circular Diaphragms Subjected to Hydrostatic Pressure,” Int. J. Mech. Sci., 1, pp. 216–228; 1962, “Tensile Instability in Thin-Walled Tubes,” J. Mech. Eng. Sci., 4(3), pp. 251–256), which assumed that plastic instability occurs at the maximum load. A closed-form formula of plastic instability condition is derived analytically. This expression for toroidal shells turns out to be the general case of spherical and cylindrical shells given by Mellor. Then the corresponding pressure is obtained by semi-analytical analysis for a material with the strain hardening characteristic, $σ=A(B+ε)n$. For the numerical approach, plastic instability pressure is the maximum pressure at which a small pressure increment causes a very large deformation. This is identified by the slope of pressure—change of volume curve approaching zero. Both approaches predict the onset of instability at the inner equator point. Experimental results of two nominally identical stainless steel toroidal shells correlated well to both approaches in terms of the magnitude of pressure and failure location.

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Figures

Figure 2

Detail of a 90 deg elbow

Figure 3

Shapes of untested, tested virgin coupon (above) and tested coupon with a seam

Figure 7

Point marking on an elbow for wall-thickness measurement

Figure 9

(a) Shape of toroid T1 after burst, (b) burst position of toroid T1, (c) shape of toroid T2 after burst, and (d) burst position of toroid T2

Figure 16

Figure 1

Shape of untested toroid T1

Figure 6

Geometry of toroidal shell

Figure 8

Experimental and numerical pressure versus change of volume curves of toroids T1 and T2

Figure 10

Toroid shapes before (T2, right-hand side) and after test (T1)

Figure 11

View of both toroids T2 (left) and T1 after test. (a) Side view and (b) plan view.

Figure 12

Critical subtangent z to the generalized instability strain hardening curve (adapted from Swift (4))

Figure 13

Distribution of meridional stress σ1 along the meridian for various ratios of R/r

Figure 14

Value of critical subtangent z versus ratio R/r of toroid. Value z=0.557 is for cylindrical shells.

Figure 15

Analytical and experimental true stress-true strain curves

Figure 4

Normal stress-normal strain curves of tensile coupons

Figure 5

True stress-true strain curve of the virgin coupon

Errata

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