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Research Papers

A Nonlinear Finite Element Framework for Viscoelastic Beams Based on the High-Order Reddy Beam Theory

[+] Author and Article Information
G. S. Payette

Graduate Research Assistant

J. N. Reddy

Professor
Life Fellow ASME
e-mail: jnreddy@tamu.edu
Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77843-3123

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the Journal of Engineering Materials and Technology. Manuscript received July 10, 2011; final manuscript received November 18, 2012; published online January 23, 2013. Assoc. Editor: Georges Cailletaud.

J. Eng. Mater. Technol 135(1), 011005 (Jan 23, 2013) (11 pages) Paper No: MATS-11-1147; doi: 10.1115/1.4023185 History: Received July 10, 2011; Revised November 18, 2012

A weak form Galerkin finite element model for the nonlinear quasi-static and fully transient analysis of initially straight viscoelastic beams is developed using the kinematic assumptions of the third-order Reddy beam theory. The formulation assumes linear viscoelastic material properties and is applicable to problems involving small strains and moderate rotations. The viscoelastic constitutive equations are efficiently discretized using the trapezoidal rule in conjunction with a two-point recurrence formula. Locking is avoided through the use of standard low-order reduced integration elements as well through the employment of a family of elements constructed using high-polynomial order Lagrange and Hermite interpolation functions.

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Figures

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Fig. 1

Deformation of a beam structure according to the third-order Reddy beam theory: (a) undeformed configuration and (b) deformed configuration

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Fig. 2

Interpolation functions for a high-order RBT finite element, where n = 6 and i = 1,…, n: (a) Lagrange interpolation functions ψi(1), (b) Hermite interpolation functions ψ∧2i-1(2), and (c) Hermite interpolation functions ψ∧2i(2)

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Fig. 3

Maximum vertical deflection w0(L/2,t) of various viscoelastic beams, each subjected to a uniform vertically distributed load q0. 2 RBT-6 elements employed in each finite element discretization: (a) hinged–hinged beam configuration and (b) pinned–pinned and clamped–clamped beam configurations.

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Fig. 4

Maximum vertical deflection w0(L/2,t) of a hinged–hinged viscoelastic beam subjected to a time-dependent transverse load q(t)

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Fig. 5

A comparison of the time-dependent vertical response w0(L/2,t) (with units of mm) of hinged–hinged beams due to a suddenly applied transverse load q(t). Results are for both viscoelastic as well as elastic beams.

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Fig. 6

A comparison of the time-dependent vertical response w0(L/2,t) (with units of mm) of hinged–hinged viscoelastic and elastic beams due to a periodic concentrated load F(t) (where η=2.744×108Ns/m2)

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