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Research Papers

Stresses in Half-Elliptic Curved Beams Subjected to Transverse Tip Forces

[+] Author and Article Information
Eduardo Velazquez

e-mail: evelazqu@ucsd.edu

J. B. Kosmatka

e-mail: jkosmatka@ucsd.edu
Department of Structural Engineering,
University of California,
San Diego, La Jolla,
California, 92093-0085

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 4, 2008; final manuscript received April 22, 2012; accepted manuscript posted June 7, 2012; published online October 29, 2012. Editor: Robert M. McMeeking.

J. Appl. Mech 80(1), 011010 (Oct 29, 2012) (7 pages) Paper No: JAM-08-1284; doi: 10.1115/1.4006934 History: Received September 04, 2008; Revised April 22, 2012; Accepted June 07, 2012

In-plane bending of curved beams produces substantial through-thickness normal and shear stresses that can result in structural failures. A half-elliptic curved beam, having a known prescribed variable radius of curvature, is studied as an extension of the previously published circular arc beam. The equations for the normal, tangential, and shear stresses are developed for a curved beam outlined by two confocal half ellipses loaded by a pair of concentrated perpendicular forces on its ends. Closed-form analytical solutions for the stresses are found using an elasticity approach, where the solution is found by using selected terms of the biharmonic equation in elliptic coordinates. For the case of an elliptic beam with an aspect ratio of very close to unity, the solution closely agrees with published circular beam solutions. For other elliptic beam aspect ratios, the calculated stresses display good correlation to detailed finite element model solutions for thickness to semi-axis ratios < 0.1. A parametric study revealed that the maximum normal stress is located at the midplane for high-aspect ratio (a/b ≥ 1) half-elliptic beams, but shifts toward the load tip for low aspect ratio (a/b < 1) beams due to local curvature effects. Moreover, the peak shear stress location moves toward the midplane and the magnitude greatly increases as the aspect ratio is increased. Thus, there are large normal and shear stress interactions occurring near the midplane for high-aspect ratio half-elliptic beams, which is not observed for circular beams. These stress interactions can produce unique failures in materials having low shear strength and through-thickness strength. The current closed-form solution is an improvement on previously published approximate solutions.

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Figures

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Fig. 1

Curved beam under load

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Fig. 2

Curved beam with a < b

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Fig. 3

(a,b) Normal stress for circular specimen with t/r = 0.1

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Fig. 4

(a) Comparison case geometry. (b) NASTRAN model mesh.

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Fig. 5

(a) Normal stress, t/b = 0.11. (b) Shear stress, t/b = 0.11. (c) Tangential stress, t/b = 0.11.

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Fig. 6

(a), (b), (c): Radius of curvature, moment and shear versus angular location

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Fig. 7

NASTRAN model mesh, t/b = 0.53

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Fig. 8

(a) Normal stress, t/b = 0.53. (b) Tangential stress, t/b = 0.53.

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Fig. 9

Maximum normal stress versus angular location

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Fig. 10

Maximum shear stress versus angular location

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Fig. 11

Location of maximum normal and shear stress versus semi-axis ratio

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