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Research Papers

Modeling of Fillets in Thin-Walled Beams Using Shell/Plate and Beam Finite Elements

[+] Author and Article Information
K. He

Department of Mechanical Engineering, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250

W. D. Zhu1

Department of Mechanical Engineering, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250wzhu@umbc.edu

1

Corresponding author.

J. Vib. Acoust 131(5), 051002 (Sep 09, 2009) (16 pages) doi:10.1115/1.3142879 History: Received October 17, 2007; Revised April 24, 2009; Published September 09, 2009

Fillets are commonly found in thin-walled beams. Ignoring the presence of a fillet in a finite element (FE) model of a thin-walled beam can significantly change the natural frequencies and mode shapes of the structure. A large number of solid elements are required to accurately represent the shape and the stiffness of a fillet in a FE model, which makes the size of the FE model unnecessarily large for global dynamic and static analyses. In this work the equivalent stiffness effects of a fillet in a thin-walled beam are decomposed into in-plane and out-of-plane effects. The in-plane effects of a fillet are analyzed using the wide-beam and curved-beam theories, and the out-of-plane effects of the fillet are analyzed by modeling the whole fillet section as a slender bar with an irregular cross section. A simple shell/plate and beam element model is developed to capture the in-plane and out-of-plane effects of a fillet on a thin-walled beam. The natural frequencies and mode shapes of a thin-walled L-shaped beam specimen calculated using the new methodology are compared with its experimental results for 28 modes. The maximum error between the calculated and measured natural frequencies for all the modes is less than 2%, and the associated modal assurance criterion values are all over 95%. The methodology is also applied to other thin-walled beams, and excellent agreement is achieved between the natural frequencies from the shell/plate and beam element models and those from the solid element models. While the shell/plate and beam element models provide the same level of accuracy as the intensive solid element models, the degrees of freedom of the shell/plate and beam element models of the thin-walled beams are only about 10% or less of those of the solid element models.

Copyright © 2009 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

The first elastic modes of two L-shaped beams (a) with and (b) without a fillet between the flanges from the solid element models. The natural frequency of the beam in (a) is 3026.7 Hz and that in (b) is 2439.2 Hz. The locations that are used for calculating the associated MAC value are shown as dots in (c).

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Figure 2

The cross section of an L-shaped beam with the fillet between the flanges modeled by two series of shell/plate elements with variable thickness

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Figure 3

(a) Using solid elements to model the fillet and shell/plate elements to model the flanges of a thin-walled L-shaped beam; and (b) using shell/plate elements that are perpendicular to the flanges to model the fillet

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Figure 4

The bridge model of a fillet used in Ref. 3

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Figure 5

The first torsional modes of two L-shaped beams (a) with and (b) without a fillet between the flanges from the solid element models. The radius of the fillet between the flanges is 0.005 m, the thickness of the flanges is 0.005 m, rout=0.0025 m, b=0.03 m, and l=0.15 m. The first torsional natural frequency of the beam in (a) is 543 Hz and that in (b) is 520 Hz. The locations at the outer ends of the two flanges, which are used for calculating the associated MAC value, are shown as dots in (c).

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Figure 7

(a) Double fillets and (b) the straight wide-beam model of the upper fillet region of the double fillets in Ref. 3

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Figure 8

An L-shaped beam with a fillet between the flanges: (a) the whole fillet section from the L-shaped beam and (b) a fillet section with unit length

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Figure 9

The single-fillet region with the applied moments and forces on the tangent sections

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Figure 10

The displacements of the centroid B of the tangent section and the polar coordinates of the curved-beam fillet model

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Figure 11

A curved beam with constant curvature and uniform cross section

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Figure 12

(a) The ith section of the curved beam in Fig. 1 and (b) the free body diagram for calculating its internal moment and forces

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Figure 13

Comparisons of (a) the three coefficients (Cmθ, Cnθ, and Cpθ) and (b) of the two coefficients (Cnθ and Cpθ) as a function of r0/t

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Figure 14

The FE model for calculating the equivalent in-plane stiffness of the fillet

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Figure 15

(a) The whole fillet section in bending and (b) its cross section with the orientations of the principal axes shown in dashed-dotted lines

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Figure 16

(a) The FE model of a filleted L-shaped beam, (b) the front view of the FE model of the fillet in (a), and (c) the free body diagram of the beam element with the rigid links

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Figure 17

A double-fillet region (a) can be modeled as a straight beam (b) with nonuniform cross section and a fixed boundary

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Figure 18

The distributions of the rotational displacements within a double-fillet region for two cases: (a) the fixed boundary is at the lower section and the loads are applied on the top tangent section, and (b) the fixed boundary is at the top tangent section and the loads are applied on the two side tangent sections

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Figure 19

The distribution of the rotational displacements within a single-fillet region shown in Fig. 6. The fixed boundary is at the lower section with the top tangent section subjected to a moment M=10,000 N m and a distributed shear force p=10,000 N/m2. The dimensions of the fillet region are shown in the figure.

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Figure 20

(a) The shell/plate and beam element model of the double fillets with parts of connecting walls, and (b) the front view of the FE model of the double fillets in (a)

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Figure 21

(a) The plane strain FE model of a half single fillet subjected to a distributed shear force on the tangent section, and (b) the deformation of the half fillet in (a)

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Figure 22

(a) I/ARh versus α/π for various r0/t, and (b) h/R versus α/π for various r0/t

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Figure 23

The 17th elastic mode shape of the L-shaped beam calculated from (a) the shell/plate and beam element model and (b) the solid element model

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Figure 24

(a) An aluminum L-shaped beam specimen resting on two soft foams at the two ends, with the excitation and measurement points marked, and (b) its dimensions

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Figure 25

Circles fit to the edges of the fillet between the flanges of the L-shaped beam (left) and the fillet at the outer end of a flange (right) in the cross-sectional plane, marked in black, of the aluminum L-shaped beam, to measure the radii of the fillets.

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Figure 26

(a) The shell/plate and beam element model of the aluminum L-shaped beam, and (b) an enlarged view of the FE model

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Figure 27

The (a) measured and (b) calculated mode shapes of the 24th elastic mode

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Figure 28

The measured and calculated mode shapes of the (a) 22nd and (b) 29th elastic modes

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Figure 29

The mode shapes of the (a) 12th and (b) 18th elastic modes calculated from the models that consider (left) and ignore (right) the fillet between the flanges of the L-shaped beam

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Figure 30

(a) The intensive solid element model of a thin-walled box beam, and (b) its cross-sectional dimensions

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Figure 31

The mode shape of the 18th elastic mode of the box beam calculated from (a) the shell/plate and beam element model and (b) the solid element model

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Figure 32

(a) The intensive solid element model of an I-shaped beam, and (b) its cross-sectional dimensions

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Figure 33

The mode shape of the 14th elastic mode of the I-shaped beam calculated from (a) the shell/plate and beam element model and (b) the solid element model

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