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Research Papers

# Linear and Nonlinear Dynamic Responses of Various Shaped Laminated Composite Plates

[+] Author and Article Information

Department of Mechanical and Materials Engineering, The University of Western Ontario, London, ON, N6A 5B9, Canada

Anand V. Singh1

Department of Mechanical and Materials Engineering, The University of Western Ontario, London, ON, N6A 5B9, Canadaavsingh@eng.uwo.ca

1

Corresponding author.

J. Comput. Nonlinear Dynam 4(4), 041011 (Sep 02, 2009) (13 pages) doi:10.1115/1.3187177 History: Received June 05, 2008; Revised March 13, 2009; Published September 02, 2009

## Abstract

A unified approach to study the forced linear and geometrically nonlinear elastic vibrations of fiber-reinforced laminated composite plates subjected to uniform load on the entire plate as well as on a localized area is presented in this paper. To accommodate different shapes of the plate, the analytical procedure has two parts. The first part deals with the geometry which is interpolated by relatively low-order polynomials. In the second part, the displacement based $p$-type method is briefly presented where the displacement fields are defined by significantly higher-order polynomials than those used for the geometry. Simply supported square, rhombic, and annular circular sector plates are modeled. The equation of motion is obtained by the Hamilton’s principle and solved by beta-$m$ method along with the Newton–Raphson iterative scheme. Numerical procedure presented herein is validated successfully by comparing present results with the previously published data, convergence study, and fast Fourier transforms of the linear and nonlinear transient responses. The geometric nonlinearity is seen to cause stiffening of the plates and in turn significantly lowers the values of displacements and stresses. Also as expected, the frequencies are increased for the nonlinear cases.

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## Figures

Figure 6

Time histories for simply supported laminated square plate 0 deg/90 deg/90 deg/0 deg] subject to uniform step load on a circular patch of R=0.125 m at (0.25, 0.25). Solid line: t0=∞, dashed line: t0=8 s, dotted line: t0=4 s, and dashed-dotted line: t0=0.5 s. (a) Linear displacement and (b) nonlinear displacement.

Figure 8

Fast Fourier transforms of the nonlinear displacement histories for simply supported laminated square plate [0 deg/90 deg/90 deg/0 deg] subject to uniform step load. (a) Log(amplitude) versus frequency for t0=∞ s and (b) log(amplitude) versus frequency for t0=0.5 s.

Figure 9

Fast Fourier transforms of the linear displacement histories for simply supported laminated square plate [0 deg/90 deg/90 deg/0 deg] subject to uniform step load on a circular patch of R=0.125 m at (0.25, 0.25). (a) Log(amplitude) versus frequency for t0=∞ s and (b) log(amplitude) versus frequency for t0=0.5 s.

Figure 13

Time histories for simply supported laminated rhombic plate [0 deg/90 deg/90 deg/0 deg] with α=30 deg subject to uniform step load on a circular patch. Solid line: position 1, dashed line: position 2, dotted line: position 3, dashed-dotted line: position 4. (a) Linear displacement, (b) nonlinear displacement, (c) linear stress, and (d) nonlinear stress.

Figure 14

Time histories for simply supported laminated rhombic plate [0 deg/90 deg/90 deg/0 deg] with α=30 deg subject to uniform step load on a circular patch with t0=0.5 s located at position 3. Solid line: linear and dotted line: nonlinear. (a) Displacement and (b) stress.

Figure 15

Annular sector plate

Figure 16

Time histories for simply supported laminated annular sector plate [0 deg/90 deg/90 deg/0 deg] subject to uniform step load. Solid line: α=30 deg, dashed line: α=60 deg, dotted line: α=90 deg, dashed-dotted line: α=120 deg. (a) Linear displacement, (b) nonlinear displacement, (c) linear stress, and (d) nonlinear stress.

Figure 17

Time histories for simply supported laminated annular sector plate [0 deg/90 deg/90 deg/0 deg] subject to uniform step load. Solid line: α=30 deg, dashed line:α=60 deg, dotted line: α=90 deg, and dashed-dotted line: α=120 deg. (a) Linear displacement, (b) nonlinear displacement, (c) linear stress, and (d) nonlinear stress.

Figure 1

Figure 2

Figure 3

Skew plate

Figure 4

Nonlinear displacement time history for simply supported square plate [0 deg/90 deg], subjected to step loading. Solid line: present method and +: Ref. 4.

Figure 5

Time histories for simply supported laminated square plate [0 deg/90 deg/90 deg/0 deg] subject to uniform step load. Solid line: t0=∞, dashed line: t0=8 s, dotted line: t0=4 s, and dashed-dotted line: t0=0.5 s. (a) Linear displacement and (b) nonlinear displacement.

Figure 7

Fast Fourier transforms of the linear displacement histories for simply supported laminated square plate [0 deg/90 deg/90 deg/0 deg] subject to uniform step load. (a) Log(amplitude) versus frequency for t0=∞ s and (b) log(amplitude) versus frequency for t0=0.5 s.

Figure 10

Fast Fourier Transforms of the nonlinear displacement histories for simply supported laminated square plate [0 deg/90 deg/90 deg/0 deg] subject to uniform step load on a circular patch of R=0.125 m at (0.25, 0.25). (a) Log(Amplitude) versus frequency for t0=∞ s and (b) log(amplitude) versus frequency for t0=0.5 s.

Figure 11

Time histories for simply supported laminated rhombic plate [0 deg/90 deg/90 deg/0 deg] subject to uniform step load. Solid line: α=0 deg, dashed line: α=30 deg, and dotted line: α=60 deg. (a) Linear displacement, (b) nonlinear displacement, (c) linear stress, and (d) nonlinear stress.

Figure 12

Time histories for simply supported laminated rhombic plate [45 deg/−45 deg/45 deg/−45 deg] subject to uniform step load. Solid line: α=0 deg, dashed line: α=30 deg, dotted line: α=60 deg. (a) Linear displacement, (b) nonlinear displacement, (c) linear stress, and (d) nonlinear stress.

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