Research Papers

Free Vibration of Curvilinearly Stiffened Shallow Shells

[+] Author and Article Information
Peng Shi

School of Aerospace Engineering,
Beijing Institute of Technology,
Beijing 100081, China
e-mail: shipeng010321@163.com

Rakesh K. Kapania

Mitchell Professor
Department of Aerospace
and Ocean Engineering,
Virginia Polytechnic Institute and State University,
Blacksburg, VA 24060
e-mail: rkapania@vt.edu

C. Y. Dong

School of Aerospace Engineering,
Beijing Institute of Technology,
Beijing 100081, China
e-mail: cydong@bit.edu.cn

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 30, 2013; final manuscript received December 2, 2014; published online January 27, 2015. Assoc. Editor: Marco Amabili.

J. Vib. Acoust 137(3), 031006 (Jun 01, 2015) (15 pages) Paper No: VIB-13-1337; doi: 10.1115/1.4029360 History: Received September 30, 2013; Revised December 02, 2014; Online January 27, 2015

The free vibration of curvilinearly stiffened shallow shells is investigated by the Ritz method. Based on the first-order shear deformation shell theory and three-dimensional (3D) curved beam theory, the strain and kinetic energies of the stiffened shells are introduced. The stiffener can be placed anywhere within the shell, without the need for having the stiffener and shell element nodes coincide. Numerical results with different geometrical shells and boundary conditions and different stiffener locations and curvatures are analyzed to verify the feasibility of the presented Ritz method for solving the problems. The results show good agreement with those using other methods, e.g., using a converged set of results obtained by Nastran.

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Grahic Jump Location
Fig. 1

Geometry and coordinate system of a doubly curved shallow shell

Grahic Jump Location
Fig. 2

A space curve on a shell

Grahic Jump Location
Fig. 3

A plate with a curvilinear eccentric stiffener: E = 73 GPa, ρ = 2837 kg/m3, μ = 0.33, hs = 6.25 mm, hf = 22.4 mm, and bf = 4.06 mm

Grahic Jump Location
Fig. 4

A spherical shell with a curvilinear stiffener: E = 73 GPa, ρ = 2837 kg/m3, μ = 0.33, a = 1.5698 m, b = 1.5698 m, Rx = Ry = 2.54 m, hs = 12.5 mm, hf = 44.8 mm, and bf = 8.12 mm

Grahic Jump Location
Fig. 5

A spherical shell with concentric orthogonal stiffeners: E = 68.9 MPa, ρ = 7728.97 kg/m3, μ = 0.3, a = 1.5698 m, b = 1.5698 m, Rx = Ry = 2.54 m, hs = 99.451 mm, hf = 201.76 mm, and bf = 101.06 mm

Grahic Jump Location
Fig. 6

A cylindrical shell with curvilinear stiffeners: E = 73 GPa, ρ = 2837 kg/m3, μ = 0.33, a = 0.8 m, b = 0.8 m, Rx = 1 m, hs = 6.25 mm, hf = 89.6 mm, bf = 16.24 mm.

Grahic Jump Location
Fig. 7

A cylindrical shell with four curvilinear stiffeners: E = 73 GPa, ρ = 2837 kg/m3, μ = 0.33, a = 0.8 m, b = 1.2 m, Rx = 2.5 m, hf = 67.2 mm, and bf = 12.18 mm

Grahic Jump Location
Fig. 8

Effects of the stiffener size on the first four natural frequencies of a simply supported curvilinearly stiffened cylindrical shallow shell

Grahic Jump Location
Fig. 9

Effects of the stiffener size on the first four natural frequencies of a clamped curvilinearly stiffened cylindrical shallow shell




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