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Research Papers: Design and Analysis

Shakedown Limit Load Determination for a Kinematically Hardening 90 deg Pipe Bend Subjected to Steady Internal Pressures and Cyclic Bending Moments

[+] Author and Article Information
Hany F. Abdalla

Department of Mechanical Engineering, The American University in Cairo, New Cairo 11835, Egypthany_f@aucegypt.edu

Maher Y. A. Younan

School of Sciences and Engineering, The American University in Cairo, New Cairo 11835, Egyptmyounan@aucegypt.edu

Mohammad M. Megahed

Department of Mechanical Design and Production, Faculty of Engineering, Cairo University, Giza 12613, Egyptmmegahed47@yahoo.com

J. Pressure Vessel Technol 133(5), 051212 (Jul 20, 2011) (10 pages) doi:10.1115/1.4003474 History: Received December 08, 2010; Revised January 10, 2011; Published July 20, 2011; Online July 20, 2011

A simplified technique for determining the lower bound shakedown limit load of a structure, employing an elastic–perfectly plastic (EPP) material model, was previously developed and successfully applied to a long radius 90 deg pipe bend (Abdalla, 2006, “Determination of Shakedown Limit Load for a 90 Degree Pipe Bend Using a Simplified Technique,” ASME J. Pressure Vessel Technol., 128, pp. 618–624). The pipe bend is subjected to steady internal pressure magnitudes and cyclic bending moments. The cyclic bending includes three different loading patterns, namely, in-plane closing, in-plane opening, and out-of-plane bending moment loadings. The simplified technique utilizes the finite element (FE) method and employs a small displacement formulation to determine the shakedown limit load without performing lengthy time consuming full elastic-plastic (ELPL) cyclic loading FE simulations or conventional iterative elastic techniques. In the present research, the simplified technique is further modified to handle structures employing an elastic-linear strain hardening material model following Ziegler’s linear kinematic hardening (KH) rule. The shakedown limit load is determined through the calculation of residual stresses developed within the pipe bend structure accounting for the back stresses, determined from the KH shift tensor, responsible for the rigid translation of the yield surface. The outcomes of the simplified technique showed an excellent correlation with the results of full ELPL cyclic loading FE simulations. The shakedown limit moments output by the simplified technique are utilized to generate shakedown diagrams (Bree diagrams) of the pipe bend for a spectrum of steady internal pressure magnitudes. The generated Bree diagrams are compared with the ones previously generated employing the EPP material model. These indicated relatively conservative shakedown limit moments compared with the ones employing the KH rule.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Stress-strain curve of an elastic-linear strain hardening material

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Figure 2

Cyclic moment loading pattern employed in the full ELPL cyclic loading FE simulations

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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 (KH and EPP), and elastic moments of the pipe bend subjected to IPC bending loading

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Figure 5

Normalized limit, shakedown (KH and EPP), and elastic moments of the pipe bend subjected to IPO bending loading

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Figure 6

Normalized limit, shakedown (KH and EPP), and elastic moments of the pipe bend subjected to OP bending loading

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Figure 7

Shakedown behavior of the output critical point of the 0.3PY case under cyclic IPO bending loading using the shakedown limit moment output by the simplified technique

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Figure 8

Loading path of the output critical point for the 0.3PY case showing translation of the yield surface preserving its initial size (cyclic IPO bending loading)

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Figure 9

Loading path inscribed within the initial yield surface of the output critical point for the 0.3PY case employing an elastic–perfectly plastic material model (cyclic IPO bending loading)

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Figure 10

Reversed plasticity behavior of the output critical point for the 0.3PY case upon exceeding the shakedown limit moment (cyclic IPO bending loading)

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Figure 11

Shakedown behavior of the output critical point of the 0.7PY case under cyclic IPO bending loading using the shakedown limit moment output by the simplified technique

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Figure 12

Ratcheting behavior of the output critical point for the 0.7PY case upon exceeding the shakedown limit moment (cyclic IPO bending loading)

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Figure 13

Zoomed view of the narrow PEEQ spectrum shown in Fig. 1 illustrating ratcheting of the output critical point of the 0.7PY case (cyclic IPO bending loading)

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