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Research Papers: Design and Analysis

# Lower Bound Limit Load Estimation Using a Linear Elastic Analysis

[+] Author and Article Information
C. Hari Manoj Simha

AMEC NSS, 393 University Avenue, 4th Floor, Toronto, ON, M5G 1E6, Canadareza.adibiasl@amec.com

J. Pressure Vessel Technol 134(2), 021207 (Jan 19, 2012) (10 pages) doi:10.1115/1.4005057 History: Received April 19, 2011; Revised August 22, 2011; Published January 19, 2012; Online January 19, 2012

## Abstract

We present a scheme that utilizes one elastic stress field (no iterations) to compute lower bound limit load multipliers of structures that collapse through gross (or localized) plasticity. A criterion to distinguish between these collapse modes is presented. For structures that collapse through gross plasticity, we demonstrate that the m′ multiplier proposed by Mura (1965, Extended Theorems of Limit Analysis,” Q. Appl. Math., 23 (2), pp. 171–179) is a lower bound in the context of deformation theory. For structures that undergo plastic localization at collapse, we present a criterion that identifies (approximately) the subvolumes of the structure that participate in the collapse. Multiplier m′ is computed over the selected subvolumes, denoted as $m'S$, and demonstrated to be a lower bound multiplier in the context of deformation theory. We consider numerical examples of structures that collapse by localized or gross plasticity and show that our proposed multiplier is lower than the corresponding multiplier obtained through elastic–plastic analysis and the proposed multiplier is not overly conservative.

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## Figures

Figure 1

Left: Entire volume participates in the collapse and right: only part of the volume participates in the collapse

Figure 2

Drawings of the structures analyzed. Material properties are listed. (a) Thick cylinder, (b) torispherical head, (c) nozzle on spherical vessel, and (d ) pressure vessel skirt.

Figure 3

Contour plots of effective stress from elastic–plastic analyses: (a) Thick cylinder, (b) torispherical head, (c) nozzle on spherical vessel, and (d) pressure vessel skirt. Regions in which effective stress is the yield strength are indicated.

Figure 4

Drawings of the structures analyzed. Material properties are listed. (a) Plate with hole, (b) center-cracked plate, (c) plate with multiple cracks, (d) single edge notch bend specimen and (e) statically indeterminate beam.

Figure 5

Finite element results for plate with hole under uni-axial loading. Left: Results from a linear elastic analysis. Right: Elastic-perfectly plastic.

Figure 6

Finite element results for plate with crack under uni-axial loading. Left: Results from a linear elastic analysis. Right: Elastic-perfectly plastic. Notice that, due to the collapse, there is significant displacement of the crack face on the right.

Figure 7

Finite element results for plate with multiple cracks under uni-axial loading. Left: Results from a linear elastic analysis. Right: Elastic-perfectly plastic.

Figure 8

Finite element computations of single-edge notched beam. Top: elastic, bottom: elastic–plastic

Figure 9

Finite element computations of statically indeterminate beam. Top: elastic, bottom: elastic–plastic.

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