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.

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.

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.

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