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Research Papers: Design Automation

Structural Efficiency Measures for Sections Under Asymmetric Bending

[+] Author and Article Information
Neil Buckney

ACCIS,
Department of Aerospace Engineering,
University of Bristol,
Bristol, BS8 1TR, UK
e-mail: neil.buckney@bristol.ac.uk

Alberto Pirrera

ACCIS,
Department of Aerospace Engineering,
University of Bristol,
Bristol, BS8 1TR, UK
e-mail: alberto.pirrera@bristol.ac.uk

Paul M. Weaver

ACCIS,
Department of Aerospace Engineering,
University of Bristol,
Bristol, BS8 1TR, UK
e-mail: paul.weaver@bristol.ac.uk

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received March 28, 2014; final manuscript received September 3, 2014; published online November 14, 2014. Assoc. Editor: Shinji Nishiwaki.

J. Mech. Des 137(1), 011405 (Jan 01, 2015) (15 pages) Paper No: MD-14-1200; doi: 10.1115/1.4028590 History: Received March 28, 2014; Revised September 03, 2014; Online November 14, 2014

Shape factors evaluate the efficiency of material usage in a structure. Previously, they have been developed for simple bending but, in practice, beams often have a more complicated bending response. Therefore, shape factors that account for asymmetric bending are introduced. The shape factors are applied to six example beam sections to demonstrate the effect of shape and load on structural efficiency. The shape factors are also enhanced for inclusion in a more general measure of structural efficiency, the performance index, comprising elements of both geometry and material. Next, a study is performed to show how the asymmetry of a beam section affects structural efficiency. The shape factors can quantitatively evaluate the structural efficiency of beam sections, demonstrating the effect of asymmetric bending on the structural response. Therefore, these shape factors can be used for concept selection and to provide insight into optimal structural design.

Copyright © 2015 by ASME
Topics: Stress , Shapes , Stiffness
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Figures

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Fig. 1

Deformed shape of beams when a downwards deflection is applied to the tip. (a) Simple bending and (b) Asymmetric bending.

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Fig. 2

Bending convention

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Fig. 3

Moments expressed as the resultant Mres and angle α

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Fig. 4

Misalignment between an applied force F and the resultant deflection δ. (a) General case. (b) Zero z-deflection. (c) Zero coupling.

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Fig. 5

Standard sections that are used to demonstrate application of the shape factors. The coordinate system and α, the angle of the RM, are indicated. (a) Tube, (b) Box, (c) I-beam, (d) C-beam,(e) Z-beam, and (f) L-beam.

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Fig. 6

Shape factor for stiffness in the y-direction. (a) Tube, (b) Box, (c) I-beam, (d) C-beam,(e) Z-beam, and (f) L-beam.

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Fig. 7

Shape factor for stiffness in the z-direction. (a) Tube, (b) Box, (c) I-beam, (d) C-beam,(e) Z-beam, and (f) L-beam.

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Fig. 8

Shape factor for general stiffness. The angular coordinate is α, and the radial coordinate is φs,g. (a) Tube, (b) Box, (c) I-beam, (d) C-beam, (e) Z-beam, and (f) L-beam.

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Fig. 9

Stress shape factor. The angular coordinate is α, and the radial coordinate is φf. (a) Tube, (b) Box, (c) I-beam, (d) C-beam,(e) Z-beam, and (f) L-beam.

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Fig. 10

Percentage difference between shape factor formulations for 0 deg < α < 90 deg

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Fig. 11

Schematic of components that describe structural efficiency. The structural index represents the fixed parameters on the left, and the free design parameters are grouped together on the right.

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Fig. 12

The effect of the material distribution in a beam cross section. (a) Hybrid beam cross sections. (b) Normalized flexural rigidity.

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Fig. 13

Mohr's circle for second moment of area with an example section: (a) Principle axes; (b) general rotation; and (c) maximum product second moment of area.

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Fig. 14

Variation of normalized second moment of area with γ and section rotation θ

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Fig. 15

Rectangles with increasing γ. The rectangles were set to have the second moment of areas corresponding to those in Table 5.

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Fig. 16

Mohr's circle for second moment of area. The effect of increasing asymmetry with a constant center is shown.

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Fig. 17

Effect of asymmetry on beam sections. The radial coordinate is gamma, angular coordinate is alpha, and the contour value is described in the subcaption. (a) General stiffness shape factor and (b) β = α − θNA. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

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Fig. 18

Effect of load misalignment on general stiffness shape factor for standard beam sections [28-30]. (a) α = 0 deg, (b) α = 2 deg, and (c) α = 4 deg.

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