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RESEARCH PAPERS

Shakedown Limits of a 90-Degree Pipe Bend Using Small and Large Displacement Formulations

[+] Author and Article Information
Hany F. Abdalla

Department of Mechanical Design and Production, Faculty of Engineering, Cairo University, Cairo, Egypthany̱f@aucegypt.edu

Mohammad M. Megahed

Department of Mechanical Design and Production, Faculty of Engineering, Cairo University, Cairo, Egyptmmegahed47@yahoo.com

Maher Y. Younan

Chair, Mechanical Engineering Department,  The American University in Cairo, Cairo 11511, Egyptmyounan@aucegypt.edu

J. Pressure Vessel Technol 129(2), 287-295 (Sep 17, 2006) (9 pages) doi:10.1115/1.2716433 History: Received January 30, 2006; Revised September 17, 2006

In this paper the shakedown limit load is determined for a long radius 90-deg pipe bend using two different techniques. The first technique is a simplified technique which utilizes small displacement formulation and elastic–perfectly plastic material model. The second technique is an iterative based technique which uses the same elastic–perfectly plastic material model, but incorporates large displacement effects accounting for geometric nonlinearity. Both techniques use the finite element method for analysis. The pipe bend is subjected to constant internal pressure magnitudes and cyclic bending moments. The cyclic bending loading includes three different loading patterns, namely, in-plane closing, in-plane opening, and out-of-plane bending. The simplified technique determines the shakedown limit load (moment) without the need to perform full cyclic loading simulations or conventional iterative elastic techniques. Instead, the shakedown limit moment is determined by performing two analyses, namely, an elastic analysis and an elastic–plastic analysis. By extracting the results of the two analyses, the shakedown limit moment is determined through the calculation of the residual stresses developed in the pipe bend. The iterative large displacement technique determines the shakedown limit moment in an iterative manner by performing a series of full elastic–plastic cyclic loading simulations. The shakedown limit moment output by the simplified technique (small displacement) is used by the iterative large displacement technique as an initial iterative value. The iterations proceed until an applied moment guarantees a structure developed residual stress, at load removal, equal to or slightly less than the material yield strength. The shakedown limit moments output by both techniques are used to generate shakedown diagrams of the pipe bend for a spectrum of constant internal pressure magnitudes for the three loading patterns stated earlier. The maximum moment carrying capacity (limit moment) the pipe bend can withstand and the elastic limit are also determined and imposed on the shakedown diagram of the pipe bend. Comparison between the shakedown diagrams generated by the two techniques, for the three loading patterns, is presented.

Copyright © 2007 by American Society of Mechanical Engineers
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References

Figures

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Figure 16

Pipe bend cross-sectional deformation about major axes for (a) in-plane bending and (b) out-of-plane bending loadings

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Figure 17

Pipe bend cross-sectional deformation about major axis for in-plane opening bending loading

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Figure 9

Shakedown diagrams for the out-of-plane, in-plane opening, and in-plane closing bending loadings (small disp.)

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Figure 10

Normalized limit, shakedown, and elastic moments of the in-plane closing bending using the iterative large displacement technique

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Figure 11

Normalized limit, shakedown, and elastic moments of the in-plane opening bending using the iterative large displacement technique

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Figure 12

Normalized limit, shakedown, and elastic moments of the out-of-plane bending using the iterative large displacement technique

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Figure 13

Comparison between normalized shakedown diagrams of small and large displacements for in-plane closing bending loading of the pipe bend

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Figure 14

Comparison between normalized shakedown diagrams of small and large displacements for in-plane opening bending loading of the pipe bend

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Figure 15

Comparison between normalized shakedown diagrams of small and large displacements for out-of-plane bending loading of the pipe bend

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Figure 1

Cyclic loading pattern

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Figure 2

Varying peak cyclic loading pattern used in the iterative large displacement technique

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Figure 3

Schematic diagram of the connected pipe bend for: (a) full geometric model; and (b) half geometric model

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Figure 4

Normalized limit, shakedown, and elastic moments of the pipe bend subjected to in-plane closing bending loading

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Figure 5

Normalized limit, shakedown, and elastic moments of the pipe bend subjected to in-plane opening bending loading

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Figure 6

Normalized limit, shakedown, and elastic moments of the pipe bend subjected to out-of-plane bending loading

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Figure 7

Normalized principal stresses for the 0.3PY case inscribed within the normalized initial yield surface (full elastic–plastic cyclic loading—in-plane opening bending)

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Figure 8

Normalized principal stresses for the 0.4PY case inscribed within the normalized initial yield surface (full elastic–plastic cyclic loading—in-plane opening bending)

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