Application of a Weak Form Quadrature Element Method to Nonlinear Free Vibrations of Thin Rectangular Plates

2016 ◽  
Vol 16 (01) ◽  
pp. 1640001 ◽  
Author(s):  
Minmao Liao ◽  
Hongzhi Zhong
Keyword(s):  
Plate Theory ◽  
Weak Form ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Integration Scheme ◽  
Rectangular Plates ◽  
Algebraic Equations ◽  
Quadrature Element Method ◽  
Numerical Integration Scheme ◽  
Element Method

Geometrically nonlinear free vibrations of thin rectangular plates are studied using the recently developed weak form quadrature element method (QEM). The nonlinear von Karman plate theory is employed to express the strain-displacement relations. The weak form description of the plate is formulated on the basis of the variational principle. The integrals involved in the variational description are evaluated by an efficient numerical integration scheme, and the partial derivatives at the integration points are approximated by differential quadrature analogs. A system of algebraic equations is eventually derived, and the nonlinear frequencies and mode shapes are extracted from solving the equations. The efficiency of the method is demonstrated by a convergence study. The accuracy of the method is illustrated by comparing the computed nonlinear to linear frequency ratios with those available in the literature. The influences of the nonlinearity on higher order frequencies and mode shapes are exhibited as well.

A weak form quadrature element method for nonlinear free vibrations of Timoshenko beams

2016 ◽  
Vol 33 (1) ◽  
pp. 274-287 ◽  
Author(s):  
Minmao Liao ◽  
Hongzhi Zhong
Keyword(s):  
Slenderness Ratio ◽  
Weak Form ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Radius Of Gyration ◽  
Element Formulation ◽  
Timoshenko Beams ◽  
Content Type ◽  
Quadrature Element Method ◽  
Element Method

Purpose – The purpose of this paper is to highlight the implementation of a recently developed weak form quadrature element method for nonlinear free vibrations of Timoshenko beams subjected to three different boundary conditions. Design/methodology/approach – The design of the paper is based on considering the geometrically nonlinear effects of axial strain, bending curvature, and shear strain. Then the quadrature element formulation of the beam is introduced. Findings – The efficiency of the method is demonstrated by a convergence study. Ratios of the nonlinear fundamental frequencies to the corresponding linear frequencies are extracted. Their variations with the ratio of amplitude to radius of gyration and the slenderness ratio are examined. The effects of the nonlinearity on higher order frequencies and mode shapes are also investigated. Originality/value – The computed results show fast convergence and compare well with available literature results.

scholarly journals Static and Vibrational Analysis of Partially Composite Beams Using the Weak-Form Quadrature Element Method

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Zhiqiang Shen ◽  
Hongzhi Zhong
Keyword(s):  
Degrees Of Freedom ◽  
Vibrational Analysis ◽  
Weak Form ◽  
Composite Beams ◽  
Free Vibrations ◽  
Computational Accuracy ◽  
Direct Evaluation ◽  
Quadrature Element Method ◽  
Partially Composite ◽  
Element Method

Deformation of partially composite beams under distributed loading and free vibrations of partially composite beams under various boundary conditions are examined in this paper. The weak-form quadrature element method, which is characterized by direct evaluation of the integrals involved in the variational description of a problem, is used. One quadrature element is normally sufficient for a partially composite beam regardless of the magnitude of the shear connection stiffness. The number of integration points in a quadrature element is adjustable in accordance with convergence requirement. Results are compared with those of various finite element formulations. It is shown that the weak form quadrature element solution for partially composite beams is free of slip locking, and high computational accuracy is achieved with smaller number of degrees of freedom. Besides, it is found that longitudinal inertia of motion cannot be simply neglected in assessment of dynamic behavior of partially composite beams.

ANALYTICAL APPROXIMATE SOLUTIONS TO LARGE-AMPLITUDE FREE VIBRATIONS OF UNIFORM BEAMS ON PASTERNAK FOUNDATION

2014 ◽  
Vol 06 (06) ◽  
pp. 1450075 ◽  
Author(s):  
YONGPING YU ◽  
BAISHENG WU
Keyword(s):  
Large Amplitude ◽  
Approximate Solutions ◽  
Free Vibrations ◽  
Integration Scheme ◽  
Nonlinear Frequency ◽  
Pasternak Foundation ◽  
Analytical Approximations ◽  
Numerical Integration Scheme ◽  
Amplitude Vibration ◽  
Large Amplitude Vibration

This paper is concerned with the large-amplitude vibration behavior of simply supported and clamped uniform beams, with axially immovable ends, on Pasternak foundation. The combination of Newton's method and harmonic balance one is used to deal with these vibrations. Explicit and brief analytical approximations to nonlinear frequency and periodic solution of the beams for various values of the two stiffness parameters of the Pasternak foundation, small as well as large amplitudes of oscillation are presented. The analytical approximate results show excellent agreement with those from numerical integration scheme. Due to brevity of expressions, the present analytical approximate solutions are convenient to investigate effects of various parameters on the large-amplitude vibration response of the beams.

VIBRATION AND BUCKLING OF SS-F-SS-F RECTANGULAR PLATES LOADED BY IN-PLANE MOMENTS

2001 ◽  
Vol 01 (04) ◽  
pp. 527-543 ◽  
Author(s):  
JAE-HOON KANG ◽  
ARTHUR W. LEISSA
Keyword(s):  
Beam Theory ◽  
Free Edge ◽  
Free Vibrations ◽  
Variable Coefficients ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
One Dimensional ◽  
Simply Supported ◽  
Free Vibration Frequency ◽  
Edge Conditions

This paper presents exact solutions for the free vibrations and buckling of rectangular plates having two opposite, simply supported edges subjected to linearly varying normal stresses causing pure in-plane moments, the other two edges being free. Assuming displacement functions which are sinusoidal in the direction of loading (x), the simply supported edge conditions are satisfied exactly. With this the differential equation of motion for the plate is reduced to an ordinary one having variable coefficients (in y). This equation is solved exactly by assuming power series in y and obtaining its proper coefficients (the method of Frobenius). Applying the free edge boundary conditions at y=0, b yields a fourth order characteristic determinant for the critical buckling moments and vibration frequencies. Convergence of the series is studied carefully. Numerical results are obtained for the critical buckling moments and some of their associated mode shapes. Comparisons are made with known results from less accurate one-dimensional beam theory. Free vibration frequency and mode shape results are also presented. Because the buckling and frequency parameters depend upon the Poisson's ratio (ν), results are shown for 0≤ν≤0.5, valid for isotropic materials.

Frequency Response of a Three-Node Finite Element for Composite Thin and Thick Plates

2004 ◽  
Author(s):  
A. V. S. Ravi Shastry ◽  
Pramod Kumar
Keyword(s):  
Finite Element ◽  
Frequency Response ◽  
Plate Theory ◽  
Composite Plates ◽  
Mode Shapes ◽  
Mindlin Plate ◽  
Integration Scheme ◽  
Shear Locking ◽  
Triangular Finite Element ◽  
Strain Components

A shear-locking free isoparametric three-node triangular finite element is considered for the study of frequency response of moderately thick and thin composite plates. The strain displacement relationship is based on Reissner-Mindlin plate theory that accounts for transverse shear deformations into the plate formulation to circumvent the shear locking effect. The element is developed with full integration scheme; hence the element remains kinematically stable. The performance of the element for the case of static load response using the shear correction terms to shear strain components applied to a composite plate has been studied. The natural frequencies and mode shapes in accordance with varying mode numbers, are deduced and the results are compared with the available analytical and finite element solutions in literature.

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